Classical Dynamics from Self-Consistency Equations in Quantum Mechanics -- Extended Version
aa r X i v : . [ m a t h - ph ] S e p Classical Dynamics from Self-Consistency Equations inQuantum Mechanics – Extended Version
J.-B. Bru W. de Siqueira PedraSeptember 11, 2020
Abstract
During the last three decades, P. B ´ona has developed a non-linear generalization of quantummechanics, which is based on symplectic structures for normal states. One important applicationof such a generalization is a general setting which is very convenient to study the emergence ofmacroscopic classical dynamics from microscopic quantum processes. We propose here a newmathematical approach to Bona’s non-linear quantum mechanics. It is based on C -semigrouptheory and has a domain of applicability which is much brother than Bona’s original one. It high-lights the central role of self-consistency. This leads to a mathematical framework in which theclassical and quantum worlds are naturally entangled. In this new mathematical approach, webuild a Poisson bracket for the polynomial functions on the hermitian weak ∗ continuous function-als on any C ∗ -algebra. This is reminiscent of a well-known construction for finite-dimensionalLie algebras. We then restrict this Poisson bracket to states of this C ∗ -algebra, by taking quo-tients with respect to Poisson ideals. This leads to densely defined symmetric derivations on thecommutative C ∗ -algebras of real-valued functions on the set of states. Up to a closure, theseare proven to generate C -groups of contractions. As a matter of fact, in general commutative C ∗ -algebras, even the closableness of unbounded symmetric derivations is a non-trivial issue.Some new mathematical concepts are introduced, which are possibly interesting by themselves:the convex weak ∗ Gˆateaux derivative, state-dependent C ∗ -dynamical systems and the weak ∗ -Hausdorff hypertopology, a new hypertopology used to prove, among other things, that convexweak ∗ -compact sets generically have weak ∗ -dense extreme boundary in infinite dimension. Ourrecent results on macroscopic dynamical properties of lattice-fermion and quantum-spin systemswith long-range, or mean-field, interactions corroborate the relevance of the general approach wepresent here. Keywords: C -semigroups, Poisson algebras, quantum mechanics, classical mechanics, self-consistency, hypertopology. AMS Subject Classification:
Contents C ∗ -Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Phase Space of C ∗ -Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Generic Weak ∗ -Compact Convex Sets in Infinite Dimension . . . . . . . . . . . . . 122.4 Classical C ∗ -Algebra of Continuous Functions on the State Space . . . . . . . . . . 162.5 Classical C ∗ -Algebra of Continuous Functions on the Phase Space . . . . . . . . . . 171 Poisson Structures in Quantum Mechanics 18 C ∗ -Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3 Poisson Ideals Associated with State and Phase Spaces . . . . . . . . . . . . . . . . 213.4 Convex Weak ∗ Gateaux Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.5 Explicit Construction of Poisson Brackets for Functions on the State Space . . . . . 253.6 Poissonian Symmetric Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 C ∗ -Algebra . . . . . . . . . . . . . . . . . . . . . . . . 294.3 Self-Consistency Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.4 Classical Dynamics as Feller Evolution . . . . . . . . . . . . . . . . . . . . . . . . 32 C ∗ -Dynamical Systems 35 C ∗ -Algebras of Continuous Functions on State Space . . . . . . . . . . . . 355.2 State-Dependent Quantum Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 365.3 State-Dependent Symmetries and Classical Dynamics . . . . . . . . . . . . . . . . . 395.4 Reduction of Classical Dynamics via Invariant Subspaces . . . . . . . . . . . . . . . 405.5 Other Constructions Involving Algebras of C ∗ -Valued Functions . . . . . . . . . . . 40 ∗ -Hausdorff Hypertopology 40 ∗ -Hausdorff Hyperconvergence . . . . . . . . . . . . . . . . . . . . . . . . . . 476.4 Metrizable Hyperspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506.5 Generic Hypersets in Infinite Dimensions . . . . . . . . . . . . . . . . . . . . . . . 51 C ∗ -Algebras 64 An indian, son of a dangerous witch, ... said to his wife: “It is my wish that you return with meto my mother’s lodge – my home.” His wife, knowing well who he was and who his mother was,readily consented to accompany him; by so doing she was faithfully carrying out the policy whichher blind brother had advised her to pursue toward him. On their way homeward, while the husbandwas leading the trail, they came to a point where the path divided into two divergent ways which,however, after forming an oblong loop, reunited, forming once more only a single path. Here thewoman was surprised to see her husband’s body divide into two forms, one following the one pathand the other the other trail. She was indeed greatly puzzled by this phenomenon, for she was at aloss to know which of the figures to follow as her husband. Fortunately, she finally resolved to followthe one leading to the right. After following this path for some distance, the wife saw that the twotrails reunited and also that the two figures of her husband coalesced into one. It is said that thiscircumstance gave rise to the name of this strange man, which was Degiyan¯e’gˇe˜n‘; that is to say,“They are two trails running parallel.” legend [1]Recently, it was proven [2] that the Gross-Pitaevskii and Hartree hierarchies, which are infinitesystems of coupled PDEs mathematically describing Bose gases with mean-field interactions, areequivalent to Liouville’s equations for functions on a suitable phase space. This result is reminiscentof Hepp and Lieb’s seminal paper [3] from year 1973, making explicit, for the first time, the exis-tence of Poisson brackets in some space of functions, related to the classical effective dynamics for apermutation-invariant quantum-spin system with mean-field interactions. This research line was fur-ther developed by many other authors, at least until the nineties. For more details, see [4, Section 1].We focus here on B´ona’s impressive series of papers on the subject, starting in 1975 with [5]. In themiddle of the eighties, his works [6, 7] lead him to consider a non-linear generalization of quantummechanics. Based on his decisive progresses [8–11] on permutation-invariant quantum-spin systemswith mean-field interactions, B´ona presents a full-fledged abstract theory in 1991 [12], which is im-proved later in a mature textbook published in 2000 (and revised in 2012) [13]. This theory does notseem to be incorporated by the physics and mathematics communities, yet.Following [13, Section 1.1-a], B´ona’s original motivation was to “ understand connections be-tween quantum and classical mechanics more satisfactorily than via the limit ~ → .” This last limitrefers to the semi-classical analysis, a well-developed research field in mathematics. In physics, itrefers to Weyl quantization or, more generally, the quantization of classical systems with ~ as a de-formation parameter. See, e.g., [14, Chapter 13]. This is the common understanding of the relationbetween quantum and classical mechanics, which is seen as a limiting case of quantum mechanics,even if there exist physical features (such as the spin of quantum particles) which do not have a clearclassical counterpart. Nonetheless, classical mechanics does not only appear in the limit ~ → , asexplained for instance in [15, 16]. B´ona’s major conceptual contribution is to highlight the possibleemergence of classical mechanics without the disappearance of the quantum world, offering a gen-eral mathematical framework which is appropriate to study macroscopic coherence in large quantumsystems.Note that B´ona’s view point is different from recent approaches of theoretical physics like [17–22](see also references therein) which propose a general formalism to get a consistent description ofinteractions between classical and quantum systems, having in mind chemical reactions, decoherenceor the quantum measurement theory. In these approaches [16–22], neither B´ona’s papers nor Heppand Lieb’s results are mentioned, even if theoretical physicists are of course aware of the emergenceof classical dynamics in presence of mean-field interactions. See, e.g., [16] where the mean-field(classical) theory corresponds to the leading term of a “large N ” expansion while the quantum part ofthe theory (quantum fluctuations) is related to the next-to-leading order term. The approaches [17–22](see also references therein) refer to quantum-classical hybrid theories for which the classical spaceexists by definition, in a ad hoc way, because of measuring instruments for instance. By contrast,the classical dynamics in B´ona’s view point emerges as an intrinsic property of macroscopic quantumsystems, like in [23]. This is also similar to [24], which is however a much more elementary example referring to the Ehrenfest dynamics.In the present paper we revisit B´ona’s conceptual lines, but propose a new method to mathemati-cally implement them, with a broader domain of applicability than B´ona’s original version [13] (seealso [15, 25] and references therein). In contrast with all previous approaches, including those of The Seneca was an important tribe of the Iroquois, the so-called Five Nations of New York. There is still a Senecanation nowadays in the United States. At least in many textbooks on quantum mechanics. See for instance [14, Section 12.4.2, end of the 4th paragraph ofpage 178]. It corresponds to a quantum systems with two species of particles in an extreme mass ratio limit: the particles of onespecies become infinitely more massive than the particles in the other one. In this limit, the species of massive particles,like nuclei, becomes classical while the other one, like electrons, stays quantum mechanical. entangled , with backreaction (that is, feedbacks), as expected. Differently from B´ona’s technical setting, ours has ad-vantage of highlighting inherent self-consistency aspects, which are absolutely not exploited in [13],as well as in quantum-classical hybrid theories of physics (e.g., [17–24]).The relevance of the abstract setting we propose here is corroborated by recent results [4,26] on the macroscopic (i.e., infinite-volume) dynamical properties of lattice-fermion and quantum-spin systemswith long-range, or mean-field, interactions. Note that a simple illustration of them is available in[27, 28]. In fact, the outcomes of [4, 26] refer to objects that are far more general than quasi-freestates or permutation invariant models and required the development of an appropriate mathematicalframework to accommodate the macroscopic long-range dynamics, which turns out to be generallyequivalent to an intricate combination of classical and short-range quantum dynamics. [4, 26] aretherefore a strong motivation for a change of perspective, which is thus presented in a systematic way in the present paper. Several key ingredients of [4, 26] refer to abstract constructions discussedthis paper, like the Poisson structures elaborated here. In other words, [4, 26] represent importantapplications, to the quantum many-body problem, of the general setting presented here.To set up our approach, we use the algebraic formalism for quantum and classical mechanics [14,Chapter 12]. The most basic element of our mathematical framework is a generic non-commutativeunital C ∗ -algebra X , which will be called here the “primordial” algebra. For instance, X is the so-called CAR C ∗ -algebra for fermion systems or the spin C ∗ -algebras in the case of quantum spins.Then, the classical objects associated with X are defined as follows: • State and phase spaces (Sections 2.1-2.3). The state space is the convex weak ∗ -compact set E of all states on X . We define the phase space as being the (weak ∗ ) closure of the subset E ( E ) ⊆ E of all extreme points of E . Interestingly, in the case that the C ∗ -algebra X is an-tiliminal and simple (e.g., the CAR algebra associated with any separable infinite-dimensionalone-particle Hilbert space, the spin algebra of any infinite countable lattice, etc.), the phase andstate spaces coincide. More generally, by using a new (weak ∗ ) hypertopology, we show thatthis surprising property of the state space is not accidental, but generic in infinite-dimensionalseparable Banach spaces. Note that our definitions of phase and state spaces differ from B´ona’sones: he does not really distinguish both spaces and considers instead the set of all density ma-trices associated with a fixed Hilbert space [13, Section 2.1, see also 2.1-c]. In particular, B´ona’sdefinition of the phase/state space is representation-dependent, in contrast with our approach.In fact, in [13, Sections 2.1c, footnote], B´ona proposes as a mathematically and physically in-teresting problem to “ formulate analogies of [his] constructions on the space of all positivenormalized functionals on B ( H ) . This leads to technical complications .” In Section 3.2 wepropose a solution to this problem for any C ∗ -algebra X . • Classical algebra (Section 2.4). The classical (i.e., commutative) unital C ∗ -algebra in our We do not mean here the so-called quantum backreaction , commonly used in physics, which refers to the backreactioneffect of quantum fluctuations on the classical degrees of freedom. Note further that the phase spaces we consider are,generally, much more complex than those related to the position and momentum of simple classical particles. More properly, the phase space should be taken as being the set E ( E ) of extreme states on X itself. Note, however,that what is relevant in the algebraic approach is the algebra of continuous functions on the given topological space,and not the space itself. The algebras of continuous functions on the closure of E ( E ) is, of course, ∗ -isomorphic to a C ∗ -subalgebra of continuous functions on E ( E ) and the closure of E ( E ) is taken to get a compact phase space, only. Analogously to the above distinction between phase and state spaces, more properly, the algebra related to the “classi-cal world” should rather be the one of continuous functions on the phase space, but we expose in Section 2.5 the conceptuallimitations of the use of this algebra in quantum physics. Moreover, in the case the C ∗ -algebra X is antiliminal and sim-ple, both classical algebras are ∗ -isomorphic to each other. In fact, the phase space turns out to be always conserved bythe classical flows (in the state space) and we show that the classical dynamics studied in the present paper can always bepushed forward, by restriction of functions, from C to the algebra of weak ∗ continuous functions on the phase space. C . = C ( E ; C ) of continuous and complex-valued functions on the statespace E . • Poisson structures (Sections 3.4-3.5). By generalizing the well-known construction of a Poissonbracket for the polynomial functions on the dual space of finite dimensional Lie algebras [29,Section 7.1], we define a Poisson bracket for the polynomial functions on the hermitian contin-uous functionals (like the states) on any C ∗ -algebra X . Then, the Poisson bracket is localizedon the state or phase space associated with this algebra by taking quotients with respect to con-veniently chosen Poisson ideals. This leads to a Poisson bracket for polynomial functions ofthe classical C ∗ -algebra C .In our setting, we introduce state-dependent C ∗ -dynamical systems associated with the primordialalgebra X , as follows: • Secondary quantum algebra (Section 5.1). Similar to quantum-classical hybrid theories oftheoretical physics like in [17–22] we introduce an extended quantum algebra as being the tensor product C ⊗ X of the commutative C ∗ -algebra C with the primordial one X . This tensorproduct is nothing else (up to some ∗ -isomorphism) than the unital C ∗ -algebra X . = C ( E ; X ) ,named here the secondary algebra associated with the primordial one, X . There are naturalinclusions X ⊆ X and C ⊆ X by identifying elements of X with constant functions andelements of C with functions whose values are scalar multiples of the unit of the primordialalgebra X . Note that in B´ona’s approach, self-adjoint elements of X refer to what he calls“ non-linear observables” [13, Section 1.2.3]. • State-dependent quantum dynamics (Section 5.1). As in X , a (possibly non-autonomous)quantum dynamics on X is, by definition, a strongly continuous two-parameter family T ≡ ( T t,s ) s,t ∈ R of ∗ -automorphisms of X satisfying the reverse cocycle property: ∀ s, r, t ∈ R : T t,s = T r,s ◦ T t,r . If T preserves the classical algebra C ⊆ X , then we name the pair ( X , T ) state-dependent, orsecondary, C ∗ -dynamical system associated with the primordial algebra X .In this setting, the classical (i.e., commutative) and quantum (i.e., non-commutative) objects arestrongly related to each other as follows: • Any state-dependent C ∗ -dynamical system ( X , T ) associated with X , in the above sense, yieldsa classical dynamics on C , as explained in Section 5.2. This classical dynamics then inducesa Feller evolution system [30], which in turn implies the existence of corresponding Markovtransition kernels on E (which can be canonically identified with the Gelfand spectrum of thecommutative unital C ∗ -algebra C ). The full dynamics for (quantum) states on the primordialalgebra X can then be recovered from the Markov transition kernels. A Feller evolution withsimilar properties also exists for the phase space (i.e., the closure of E ( E ) ). • More interestingly, we remark in Section 4.2 that any classical differentiable Hamiltonian from C is associated with a state-dependent quantum dynamics on the primordial C ∗ -algebra X ,in a natural way. This observation is then used to derive, mathematically, in Sections 4.3-4.4, classical dynamics associated with the Poisson structure of the (polynomial subalgebraof the) classical algebra C . These define again Feller evolution systems which turn out tobe related to a self-consistency problem (Theorem 4.1). By Lemma 5.4, this yields, in turn,state-dependent quantum dynamics on the secondary (quantum) C ∗ -algebra X of continuous( X -valued) functions on states, associated with the primordial (quantum) algebra X . Because commutative C ∗ -algebras are nuclear, the norm making the completion of the algebraic tensor product C ⊗X into a C ∗ -algebra is unique.
5n the one hand, the classical world is embedded in the quantum world, as mathematically ex-pressed by the above defined inclusion C ⊆ X . On the other hand, our approach entangles the quantumand classical worlds through self-consistency. As a result, non-autonomous and non-linear dynamicscan emerge. Seeing both entangled worlds, quantum and classical, as “two sides of the same coin”looks like an oxymoron, but there is no contradiction there since everything refers to a primordial quantum world mathematically encoded in the structure of the non-commutative (unital) C ∗ -algebra X . In fact, the quantum algebra X is the arche of the theory. For instance, the state space E isthe imprint left by X in the classical world, whose observables are the self-adjoint elements of the commutative C ∗ -algebra C . = C ( E ; C ) , i.e., the continuous complex-valued functions on E . If X were a commutative algebra, note that the corresponding Poisson bracket and, hence, the associatedclassical dynamics would be trivial.Note that the abstract setting proposed in this paper is not really useful to portray quantum dynam-ics of finite systems. In fact, in this case, the time evolution is not state-dependent. Nevertheless, asdiscussed above, such a mathematical framework turn out to be natural for the study of macroscopicdynamics of lattice-fermion or quantum-spin systems with long-range, or mean-field, interactions,because, in this case, the Heisenberg dynamics turns out to be effectively state dependent, in thethermodynamic limit. See again [4, 26], which uses self-consistency equations in a essential way,similar to Theorem 4.1. Moreover, since quantum many-body systems in the continuum are also ex-pected to have, in general, a state-dependent Heisenberg dynamics in the thermodynamic limit (see,e.g., [32, Section 6.3]), the approach presented here is very likely relevant for future studies in thiscontext. We thus consider important to have a systematic approach that can be used beyond specificapplications, like [4, 26].Our approach is not too far, in its spirit, to the one developed in [13], although it differs in itsmathematical formulation. In comparison with [13], our formulation is more general in the case ofan infinite-dimensional underlying C ∗ -algebra, which generally has several inequivalent irreduciblerepresentations, as a consequence of the Rosenberg theorem [33]: Whereas [13] has to use a repre-sentation of the underlying C ∗ -algebra to be able to define Poisson brackets in the associated classicalalgebra, we provide a definition for such brackets with no reference to representations. This is ex-plained in more detail in Section 3.1. Notice at this point that, in condensed matter physics, the non-uniqueness of irreducible representations is intimately related to the existence of various inequivalentthermodynamically stable phases of the same material.Last but not least, we observe that a large set of symmetric derivations can be defined on all poly-nomial elements of C by using the Poisson bracket. See Section 3.6. These (unbounded) derivationsare not a priori closed operators, but this property is necessary to generate (classical) dynamics, in itsHamiltonian formulation, via strongly continuous semigroups . In contrast with our approach, B´onaavoids this problem by using Hamiltonian flows in symplectic leaves of the corresponding Poissonmanifold and by “gluing” together the flows within the leaves by showing continuity properties [13,Section 2.1-d].The closabledness of a symmetric derivation is usually proven from its dissipativity [34, Definition1.4.6, Proposition 1.4.7], which results from [34, Theorem 1.4.9] and the assumption that the squareroot of each positive element of the domain of the derivation also belongs to the same domain. Wecannot expect this property to be satisfied for symmetric derivations acting on a dense domain of C .As a matter of fact, the closableness of unbounded symmetric derivations in commutative C ∗ -algebraslike C is, in general, a non-trivial issue. This property might not even be true since there exists norm-densely defined derivations of C ∗ -algebras that are not closable [35]. For instance, in [36, p. 306], it iseven claimed that “Herman has constructed an extension of the usual differentiation on C (0 , which Following Aristotle’s use of the presocratic philosophical term “arche” ( ἀρχὴ ), here it means “the element or principleof a thing which, although undemonstrable and intangible in itself, provides the conditions of the possibility of that thing”.See [31, p. 143].
6s a non-closable derivation of C (0 , .” A complete classification of all closed symmetric derivationsof functions on a compact subset of a one-dimensional space was obtained around 1990. However,quoting [34, Section 1.6.4, p. 27], “for more than 2 dimensions only sporadic results in this directionare known.” See, e.g., [34, Section 1.6.4], [37], [38,39], and later [36, p. 306]. Since then, no progresshas been made on this classification problem, at least to our knowledge.In Section 4 (Theorem 4.5), via the analysis of certain self-consistency problems together withthe one-parameter semigroups theory [40], we naturally obtain infinitely many closed symmetricderivations with dense domain in C . As it turns out, this method is very natural and efficient for thestate space E , that is, a weak ∗ -compact convex subset of the dual X ∗ of the unital (not necessarilyseparable) C ∗ -algebra X , which is, in general, infinite - dimensional. In particular, E is generally not a subset of a finite-dimensional space. This construction of closed derivations of a commutative C ∗ -algebra via self-consistency problems is non-conventional and may motivate further studies. For moreinformation, see Section 4. Main results and structure of the paper.
Recall that E is the state space of a non-commutativeunital C ∗ -algebra X . Our main results are the following: • The weak ∗ - Hausdorff hypertopology (Definitions 2.3 and 6.1) is a new notion, proposed herein order to characterize generic convex weak ∗ -compact sets, by extending [41, 42] to weak ∗ topological structures. We show in particular that convex weak ∗ -compact sets of the dual space X ∗ of a (real or complex) Banach space X have generically weak ∗ -dense set of extreme pointsin infinite dimension, in the sense of this new (hyper)topology. This refers to Theorems 2.4and 2.5. These results has been extended in [43] for the dual space X ∗ , endowed with itsweak ∗ -topology, of any infinite-dimensional, separable topological vector space X . • Corollary 3.6 defines, in a natural way, a Poisson bracket {· , ·} on polynomial functions of C ( E ; C ) , while Corollary 3.7 shows that the restriction of {· , ·} to the phase space E ( E ) alsolead to a Poisson bracket on polynomial functions of C ( E ( E ); C ) . These Poisson brackets werepreviously used, for instance, in [4, 26]. • The convex weak ∗ Gateaux derivative (Definition 3.8) is used to give an explicit expression forthe Poisson bracket for functions on the state space E . This refers to Proposition 3.11, which isis an important result because it allows us to perform more explicit computations, both in thispaper and in [4, 26]. • Theorem 4.1 shows the well-posedness of self-consistency equations, allowing us to define, foran appropriate continuous family h ≡ ( h ( t )) t ∈ R ⊆ C ( E ; R ) , a classical flow ρ ̟ h ( s, t ) ( ρ ) , s, t ∈ R , in the state space E and thus, a (generally non-autonomous classical) dynamics ( V ht,s ) s,t ∈ R on C . = C ( E ; C ) : V ht,s ( f ) . = f ◦ ̟ h ( s, t ) , f ∈ C , s, t ∈ R . Physically, the functions h ( t ) , t ∈ R , are time-dependent classical energies. In Corollary 4.3,we show that the classical flow conserves both the set E ( E ) of extreme states and its weak ∗ closure E ( E ) , which is the phase space. • Proposition 4.4 proves that ( V ht,s ) s,t ∈ R is a strongly continuous two-parameter family of ∗ -automorphisms of C satisfying the reverse cocycle property, i.e., the classical dynamics is aFeller evolution system [30]. 7 Theorem 4.5 shows that, given an appropriate function h ∈ C ( E ; R ) , the Poissonian symmet-ric derivations f
7→ { h, f } ∈ C defined for any polynomial functions f in C is closable and is directly related to the generatorof the C -group ( V ht, ) t ∈ R for the constant energy function h . • Theorem 4.6 shows the non-autonomous evolution equations ∂ t V ht,s ( f ) = V ht,s ( { h ( t ) , f } ) and ∂ s V ht,s ( f ) = − (cid:8) h ( s ) , V ht,s ( f ) (cid:9) for any appropriate h ≡ ( h ( t )) t ∈ R ⊆ C ( E ; R ) , times s, t ∈ R and polynomial function f of C . In the autonomous case, i.e., when h ∈ C ( E ; R ) , one gets Liouville’s equation (Corollary4.7), i.e., ∂ t V ht, ( f ) = V ht, ( { h, f } ) = (cid:8) h, V ht, ( f ) (cid:9) . • In Section 5.2, we show how the above classical dynamics defines a state-dependent quan-tum dynamics with fixed-point algebra including the classical algebra C . This lead us to de-fine a state-dependent C ∗ -dynamical system (Definition 5.3). See Lemma 5.4 and discussionsafterwards. Such a quantum dynamics is relevant in the study of macroscopic dynamics oflattice-fermion systems or quantum-spin systems with long-range, or mean-field, interactionsperformed in [4, 26].The paper is organized as follows: We first introduce, in Section 2, classical systems associated witharbitrary unital C ∗ -algebras. The Poisson structures for these systems are built in Section 3. Section3 also gathers all the necessary definitions to describe, in Section 4, classical dynamics generated bya Poisson bracket, as is usual classical mechanics. Section 5 then explains the final setting of thetheory. In Section 5.3 we discuss the role of symmetries as well as the notion of “reduction” of theclassical dynamics. This is important in applications to simplify the self-consistency equations. Sec-tion 6 gives all arguments to deduce Theorems 2.4-2.5 by defining and studying the weak ∗ -Hausdorffhypertopology. The proof of the most important result, that is, Theorem 4.1, is performed in Section7, which also collects additional results used in Section 4.4. Finally, Section 8 is an appendix onliminal, postliminal and antiliminal C ∗ -algebras. Though these are standard notions in C ∗ -algebratheory, they may not be known by non-experts, but have major consequences on the structure of theset of states, which can be highly non-trivial and are relevant in our discussions. Notation 1.1 (i)
A norm on a generic vector space X is denoted by k · k X and the identity map of X by X . Thespace of all bounded linear operators on ( X , k · k X ) is denoted by B ( X ) . The unit element of anyalgebra X is denoted by , provided it exists. The scalar product of any Hilbert space H is denotedby h· , ·i H . (ii) For all topological spaces X and Y , C ( X ; Y ) denotes, as usual, the space of continuous mapsfrom X to Y . If X is a locally compact topological space and Y is a Banach space, then C b ( X ; Y ) denotes the Banach space of bounded continuous maps from X to Y along with the topology ofuniform convergence. For any p, n ∈ N , in the special case X = R n and Y = R , C pb ( R n ; R ) denotesthe Banach space of bounded continuous, real-valued, functions on R n along with the topology ofuniform convergence for the functions and all its m -th derivatives, where m ∈ { , . . . , p } . (iii) We adopt the term “automorphism” in the sense of category theory and its precise meaningthus depends on the structure of the corresponding domain: An automorphism of a ∗ -algebra is abijective ∗ -homomorphism from this algebra to itself, whereas an automorphism of a topological This is a very important property, excluding the definition given by Equation (68). pace is a self-homeomorphism, that is, a homeomorphism of the space to itself. In fact, in thecategory of topological spaces the morphisms are precisely the continuous maps and the morphismsof the category of ∗ -algebras are the ∗ -homomorphisms. Recall that in category theory invertiblemorphisms are called isomorphisms and isomorphisms whose domain and codomain coincide arecalled automorphisms. (iv) In the sequel, a primordial C ∗ -algebra X is fixed and its state space is denoted by E . Then,various sets of functions on E are defined. The most important are C . = C ( E ; C ) , X . = C ( E ; X ) and Y ≡ Y ( Y ) . = C ( E ; Y ) , Y being a Banach space, like R or C . These spaces appear many timesand we always use a shorter notation than the usual ones, like C ( E ; C ) , the letter of the codomainwithin the Fraktur alphabet. More generally, any capital letter in Fraktur alphabet always refers to aspace of functions on E . To denote real subspaces, we add the superscript R like in C R . = C ( E ; R ) or in X R , which is the real Banach space of all self-adjoint elements of X . C ∗ -Algebras Perhaps the philosophically most relevant feature of modern science is the emergence of abstractsymbolic structures as the hard core of objectivity behind – as Eddington puts it – the colorful tale ofthe subjective storyteller mind.
Weyl, 1949 [44, Appendix B, p. 237]Fix once and for all a C ∗ -algebra X ≡ ( X , + , · C , × , ∗ , k·k X ) , that is, a (complex) Banach algebra endowed with an antilinear involution A A ∗ such that ( AB ) ∗ = B ∗ A ∗ and k A ∗ A k X = k A k X , A, B ∈ X . Here, AB ≡ A × B . We always assume that X is unital, i.e., the product of X has a unit ∈ X . The(real) Banach subspace of all self-adjoint elements of X is denoted by X R . = { A ∈ X : A = A ∗ } ≡ (cid:0) X R , + , · R , k·k X (cid:1) . (1)The C ∗ -algebra X is named the primordial C ∗ -algebra. Note that it is not necessarily separable.By [45, Theorem 3.10], the dual space X ∗ of X endowed with its weak ∗ topology (i.e., the σ ( X ∗ , X ) -topology of X ∗ ) is a locally convex space (in the sense of [45, Section 1.6]) whose dual is X . Recall that X ∗ is a Banach space when it is endowed with the usual norm for linear functionals ona normed space. A subset of X ∗ which is pivotal in the algebraic formulation of quantum mechanicsis the state space of X , defined as follows: Definition 2.1 (State space)
Let X be a unital C ∗ -algebra. The state space is the convex and weak ∗ -closed set E . = \ A ∈X { ρ ∈ X ∗ : ρ ( A ∗ A ) ≥ , ρ ( ) = 1 } of all positive normalized linear functionals ρ ∈ X ∗ . ρ ∈ X ∗ is a state iff ρ ( ) = 1 and k ρ k X ∗ = 1 . Note that any state is hermitian: forall ρ ∈ E and A ∈ X , ρ ( A ∗ ) = ρ ( A ) . From the Banach-Alaoglu theorem [45, Theorem 3.15], E isa weak ∗ -compact subset of the unit ball of X ∗ . Therefore, the Krein-Milman theorem [45, Theorem3.23] tells us that E is the weak ∗ closure of the convex hull of the (nonempty) set E ( E ) of its extremepoints : E = co E ( E ) . (2)The set E ( E ) is also called the extreme boundary of E . If X is separable then the weak ∗ topology ismetrizable on any weak ∗ -compact subset of X ∗ , by [45, Theorem 3.16]. In particular, the state space E of Definition 2.1 is metrizable, in this case, and by the Choquet theorem [46, p. 14], for any ρ ∈ E ,there is a probability measure µ ρ with support in E ( E ) such that, for any affine weak ∗ -continuouscomplex-valued function g on E , g ( ρ ) = Z E ( E ) g (ˆ ρ ) d µ ρ (ˆ ρ ) . (3)The measure µ ρ is unique for all ρ ∈ E , i.e., E is a Choquet simplex [36, Theorem 4.1.15], iff the C ∗ -algebra X is commutative, by [36, Example 4.2.6].If E is not metrizable, meaning that X is not separable, note that such a probability measure µ ρ is only pseudo–supported by E ( E ) , i.e., µ ρ ( B ) = 1 for all Baire sets B ⊇ E ( E ) . This refers to theChoquet-Bishop-de Leeuw theorem [46, p. 17]. Recall that the Baire sets are the elements of the σ -algebra generated by the compact G δ sets. If E ( E ) is a Baire set then E must be metrizable [47].The weak ∗ closure E ( E ) may even not be a G δ set, or more generally a Baire set, when E is notmetrizable. In fact, in the non-metrizable case, E ( E ) can have very surprising properties like being a zero-measure Borel set for µ ρ (cf. [48]).We use the state space E in the next section to define a classical algebra, the space C ( E ; C ) ofcomplex-valued weak ∗ -continuous functions on E . Note that our (quantum) state space E is differentfrom the one considered in [13, Section 2.1, see also 2.1-c]. In B´ona’s paper, the state space is definedto be the set of density matrices associated with a fixed Hilbert space. In relation to our approach,it corresponds to take, instead of all states of X , only those which are π -normal, for some fixed represetation π of the C ∗ -algebra X . Recall that the state ρ ∈ E is called “ π -normal” if the state ρ ◦ π on π ( X ) has a (unique) normal extention to the von Neumann algebra π ( X ) ′′ ⊇ π ( X ) . By contrast,our definition of the (quantum) state space is not representation-dependent. C ∗ -Algebras Before the pioneer works of Jacobi and Boltzmann, then of Gibbs and Poincar´e, the motion of apoint-like particle was seen as a trajectory within the three-dimensional space. However, in classicalmechanics, fixing only the position at a fixed time does not completely determine the trajectory, whichonly becomes unique after fixing the momentum. This leads to the term phase : If we regard a phase as represented by a point in space of n dimensions, the changes which takeplace in the course of time in our ensemble of systems will be represented by a current in such space. Gibbs, 1902 [49, p. 11, footnote]This view point required the idea of “high dimensional” spaces, which widespread only in the firstdecade of the 20th century. This space refers to the illustrious concept of phase space , which seemsto first appear in print in 1911 [50].The historical origins of the notion of phase space can be found in [51], which makes explicitthe “ tangle of independent discovery and misattributions that persist today ”, even if this concept is I.e., the points which cannot be written as – non–trivial – convex combinations of other elements of E . one of the most powerful inventions of modern science ”. For instance, the terminology ofphase space is widely used in classical mechanics, and also in [13, Section 2.1], but its use is regularlyconfusing in many textbooks, which often view the state and phase spaces as the same thing.The precise definition of phase space is an important, albeit non-trivial, issue in the understandingof a physical system because it is usually supposed to describe all its observable properties togetherwith a deterministic motion, once the initial coordinates of the system is fixed in this phase space. Inparticular, it has to be sufficiently large to support a deterministic, or causal, motion.In classical physics, the phase space is a locally compact Hausdorff space K , like R . In thealgebraic formulation of classical mechanics [52, Chapter 3], one starts with a commutative C ∗ -algebra. By the Gelfand theorem (see, for instance, [36, Theorem 2.1.11A] or [52, Theorem 3.1]),such an algebra is ∗ -isomorphic to the algebra C ( K ; C ) of all continuous functions f : K → C vanishing at infinity, where K is a unique (up to a homeomorphism) locally compact Hausdorff space.In this case, K is, by definition, the phase space of the physical system. The phase space K is compactiff the commutative C ∗ -algebra is unital.For non- commutative unital C ∗ -algebras, the definition of the associated phase space is lessstraightforward. To motivate the definition adopted here (Definition 2.2) for this space, we exhibitthe relation between the phase space K and the state space E of Definition 2.1 for a commutativeunital C ∗ -algebra seen as an algebra C ( K ; C ) ≡ (cid:16) C ( K ; C ) , + , · C , × , ( · ) , k·k C ( K ; C ) (cid:17) of continuous complex-valued functions on the compact Hausdorff space K . Extreme points of E arethe so-called characters of this C ∗ -algebra: E ( E ) = { c ( x ) ∈ E : x ∈ K } , where c is the continuous and injective map from K to E defined by [ c ( x )] ( f ) . = f ( x ) , f ∈ C ( K ; C ) , x ∈ K . (4)Recall that the characters of a given C ∗ -algebra are, by definition, the unital ∗ -homomorphisms fromthis algebras to C (i.e., the multiplicative hermitian functionals on the algebra). See [36, Proposition2.3.27]. In this special case, E ( E ) is weak ∗ -compact, like K , and the map c is a homeomorphism. Inparticular, the map f ˆ f from C ( K ; C ) to C ( E ( E ); C ) defined by ˆ f ( c ( x )) = [ c ( x )] ( f ) , f ∈ C ( K ; C ) , x ∈ K , (5)is a ∗ -isomorphism of the commutative unital C ∗ -algebras C ( K ; C ) and C ( E ( E ); C ) . (See again [36,Theorem 2.1.11A] or [52, Theorem 3.1].) Therefore, as is usual, the phase space of any commutativeunital C ∗ -algebra X can be identified with the weak ∗ -compact set E ( E ) of extreme states of thisalgebra. The set of all characters of the commutative C ∗ -algebra X is called its (Gelfand) spectrum and its generalization to arbitrary C ∗ -algebras is not straightforward: Remark, for instance, that thealgebra of N × N complex matrices, N ≥ , has no characters, in the above sense, at all, by thecelebrated Bell-Kochen-Specker theorem [52, Theorem 6.5]. The problem of properly defining anotion of spectrum for a general C ∗ -algebra is adressed, for instance, in [53, Chapters 3 & 4] in thecontext of decompositions of general representations of such an algebra in terms of its irreducible representations.Now, with regard to the definition of the phase space as the set E ( E ) = E of extreme states,we want to emphasize that, for a non-commutative unital C ∗ -algebra X , this set does not have to be I.e., a topological space whose open sets separate points ( → Hausdorff) and whose points always have a compactneighborhood ( → locally compact). C ( K ; C ) is separable iff K is metrizable. See [54, Problem (d) p. 245]. ∗ -closed (in E ), and so weak ∗ -compact. See, e.g., Lemma 8.5. As explained above, a classicalphysical system refers to the algebra of (complex-valued) continuous functions decaying at infinity ona locally compact Hausdorff space. Such an algebra is canonically ∗ -isomorphic, via the restriction offunctions, to a C ∗ -algebra of functions defined on any dense set of this Hausdorff space. Therefore,a natural definition of the (classical) phase space associated with a general quantum system, ensuringits compactness, is the weak ∗ closure E ( E ) , instead of the set E ( E ) of extreme states itself: Definition 2.2 (Phase space)
Let X be a unital C ∗ -algebra. The associated phase space is the weak ∗ closure E ( E ) of the extremeboundary of the state space E of Definition 2.1. The phase space is, by definition, only a weak ∗ -closed subset of the state space. However, inmathematical physics, the unital C ∗ -algebra associated with an infinitely extended (quantum) systemis usually an approximately finite-dimensional (AF) C ∗ -algebra, i.e., it is generated by an increasingfamily of finite-dimensional C ∗ -subalgebras. They are all antiliminal (Definition 8.3) and simple (Definition 8.6). See Section 8 for more details. In this case, by Lemma 8.5, E ( E ) is weak ∗ -dense in E , i.e., E = E ( E ) . (6)In other words, in general, the phase space of Definition 2.2 is the same as the state space of Definition2.1 for infinitely extended quantum systems. The set E of states has therefore a fairly complicated ge-ometrical structure. Compare, indeed, Equation (6) with (2). Provided the C ∗ -algebra X is separable,note that, surprisingly, (2) and (6) do not prevent E from having a unique center [55]. ∗ -Compact Convex Sets in Infinite Dimension Accidens vero est quod adest et abest praeter subiecti corruptionem . An accident in the Middle AgesThe existence of convex sets with dense extreme boundary is well-known in infinite-dimensionalvector spaces. For instance, the unit ball of any infinite-dimensional Hilbert space has a dense extremeboundary in the weak topology. In fact, a convex compact set with dense extreme boundary is notan accident in infinite-dimensional spaces, like Hilbert spaces or in the dual space of an antiliminalunital C ∗ -algebras (cf. (6) and Lemma 8.5).In 1959, Klee shows [41] that, for convex norm-compact sets within a Banach space, the propertyof having a dense set of extreme points is generic in infinite dimension. More precisely, by [41,Proposition 2.1, Theorem 2.2], the set of all such convex compact subsets of an infinite-dimensionalseparable Banach space Y is generic in the complete metric space of compact convex subsets of Y , endowed with the well-known Hausdorff metric topology [58, Definition 3.2.1]. Klee’s result isrefined in 1998 by Fonf and Lindenstraus [42, Section 4] for bounded norm-closed (but not necessarily I.e., a sort of maximally mixed point. Fr.:
L’accident est ce qui arrive et s’en va sans provoquer la perte du sujet.
See [56, V. L’accident]. It meansthat an accident is what is present or absent in a subject without affecting its essence. This comes from the
Isagoge ( ΕΙΣΑΓΩΓΗ , originally in greek) [56] written in the IIIe century by the Syrian Porphyry (of Tyr) as an introductionto
Aristotle’s Categories . The
Isagoge was a pivotal textbook in medieval philosophy and more generaly on early logicduring more than a millennium. Its reception by medieval (scholastic) philosophers has, in particular, initiated and fueledthe celebrated problem of universals [57] from the XIIe to the XIVe century. [41, Proposition 2.1, Theorem 2.2] seem to lead to the asserted property for all (possibly non-separable) Banachspaces, as claimed in [41, 42, 59]. However, [41, Theorem 1.5], which assumes the separability of the Banach space,is clearly invoked to prove the corresponding density stated in [41, Theorem 2.2]. We do not know how to remove theseparability condition. That is, the complement of a meagre set, i.e., a nowhere dense set. Y having so-called empty quasi-interior (as a necessary condition).In this case, [42, Theorem 4.3] shows that such sets can be approximated in the Hausdorff metrictopology by closed convex sets with a norm-dense set of strongly exposed points . See, e.g., [59,Section 7] for a recent review on this subject.In this section we demonstrate the same genericity in the dual space X ∗ of an infinite-dimensional,separable unital C ∗ -algebra X , endowed with its weak ∗ -topology. Of course, if one uses the usualnorm topology on X ∗ for continuous linear functionals, then one can directly apply previous results[41, 42] to the separable Banach space X ∗ . This is not anymore possible if one considers the weak ∗ -topology. In particular, [42, Theorem 4.3] cannot be used because, in general, weak ∗ -compact sets donot have an empty interior, in the sense of the norm topology. However, generic properties of convexweak ∗ -compact sets, like the state space E of Definition 2.1, are relevent in the present paper. Wethus prove, in this situation, results similar to [41, 42] in order to better understand the disconcertingstructure of the state and phase spaces, respectively E and E ( E ) defined above.In order to talk about generic properties of convex weak ∗ -compact sets, we first need to define anappropriate topological space of subsets of X ∗ . It is naturally based on the set CK ( X ∗ ) . = { K ⊆ X ∗ : K = ∅ is convex and weak ∗ -compact } . (7)By Equation (83) and Lemma 6.5, note that CK ( X ∗ ) = (cid:26) K ⊆ X ∗ : K = ∅ is convex, weak ∗ -closed and sup σ ∈ K k σ k X ∗ < ∞ (cid:27) . (8)This is a set of weak ∗ -closed sets in a locally convex Hausdorff space X ∗ . See, e.g., [60, Theorem10.8].We now make CK ( X ∗ ) into a topological (hyper)space by defining a hypertopology on it. Recallthat topologies for sets of closed subsets of topological spaces have been studied since the beginningof the last century and when such topologies, restricted to singletons, coincide with the original topol-ogy of the underlying space, we talk about hypertopologies and hyperspaces of closed sets. Thereexist several standard hypertopologies on the set of nonempty closed convex subsets of a topolog-ical space like, for instance, the slice topology [58, Section 2.4], the scalar and the linear topolo-gies [58, Section 4.3]. Because of [58, Theorem 2.4.5], note that the slice topology is unappropriatehere since it is not related to the weak ∗ -topology of X ∗ , but rather to its norm topology. In fact, wedo not use any of those standard hypertopologies, but another natural topology on CK ( X ∗ ) given bya family of pseudometrics inspired by the Hausdorff metric topology for closed subsets of C : Definition 2.3 (Weak ∗ -Hausdorff hypertopology for convex sets) The weak ∗ -Hausdorff hypertopology on CK ( X ∗ ) is the topology induced by the family of Hausdorffpseudometrics d ( A ) H defined, for all A ∈ X , by d ( A ) H ( K, ˜ K ) . = max (cid:26) max σ ∈ K min ˜ σ ∈ ˜ K | ( σ − ˜ σ ) ( A ) | , max ˜ σ ∈ ˜ K min σ ∈ K | ( σ − ˜ σ ) ( A ) | (cid:27) , K, ˜ K ∈ CK ( X ∗ ) . (9)Compare (9) with the definition of the Hausdorff distance, given by (81). Definition 2.3 is a restrictionof the weak ∗ -Hausdorff hypertopology of Definition 6.1. In this topology, an arbitrary net ( K j ) j ∈ J converges to K ∞ iff, for all A ∈ X , lim J d ( A ) H ( K j , K ∞ ) = 0 . (10) x ∈ K is a strongly exposed point of a convex set K ⊆ Y when there is f ∈ Y ∗ satisfying f ( x ) = 1 and such thatthe diameter of { y ∈ K : f ( y ) ≥ − ε } tends to as ε → + . (Strongly) exposed points are extreme elements of K . Recall that a pseudometric d satisfies all properties of a metric but the identity of indiscernibles. In fact, d ( x, x ) = 0 but possibly d ( x, y ) = 0 for x = y . CK ( X ∗ ) , by [54, Chapter 2, Theorem 9]. In fact, becausethis topology is generated by a family of pseudometrics, it is a uniform topology, see, e.g., [54,Chapter 6].It is completely obvious from the definition that any net ( σ j ) j ∈ J in X ∗ converges to σ ∈ X ∗ in the weak ∗ topology iff the net ( { σ j } ) j ∈ J converges in CK ( X ∗ ) to { σ } in the weak ∗ -Hausdorff(hyper)topology. In other words, the embedding of X ∗ into CK ( X ∗ ) is a bicontinuous bijectionon its image. This justifies the use of the name weak ∗ -Hausdorff hyper topology. We are not awarewhether this particular hypertopology has already been considered in the past. We thus give in Section6 its complete study along with interesting connections to other fields of mathematics and results thatare more general than those stated in Section 2.3.Endowed with the weak ∗ -Hausdorff hypertopology, CK ( X ∗ ) is a Hausdorff hyperspace. SeeCorollary 6.10. Observe also that the limit of weak ∗ -Hausdorff convergent nets within CK ( X ∗ ) isdirectly related to lower and upper limits `a la Painlev´e [61, § X is a separable Banach space, Corollary 6.18 tells us thatany weak ∗ -Hausdorff convergent net ( K j ) j ∈ J ⊆ CK ( X ∗ ) converges to its Kuratowski-Painlev´e limit K ∞ , which is thus the set of all weak ∗ accumulation points of nets ( σ j ) j ∈ J with σ j ∈ K j .Recall that, by the Krein-Milman theorem [45, Theorem 3.23], any nonempty convex weak ∗ -compact set K ∈ CK ( X ∗ ) is the weak ∗ -closure of the convex hull of the (nonempty) set E ( K ) of itsextreme points: K = co E ( K ) . The property K = E ( K ) (with respect to the weak ∗ topology) looks very peculiar. Nonetheless, as amatter of fact, typical elements of CK ( X ∗ ) have this property: Theorem 2.4 (Generic convex weak ∗ -compact sets) Let X be an infinite-dimensional separable Banach space. Then, the set D of all nonempty convexweak ∗ -compact sets K with a weak ∗ -dense set E ( K ) of extreme points is a weak ∗ -Hausdorff-dense G δ subset of CK ( X ∗ ) . Proof.
Combine Proposition 6.19 with Theorem 6.20. Note that the proof of Theorem 6.20 is craftedby following original Poulsen’s intuitive construction [62], like in the proof of [42, Theorem 4.3].The Hahn-Banach separation theorem [45, Theorem 3.4 (b)] plays a crucial role in this context.As a consequence, D is generic in the hyperspace CK ( X ∗ ) , that is, the complement of a meagre set,i.e., a nowhere dense set. In other words, D is of second category in CK ( X ∗ ) .The weak ∗ -Hausdorff hypertopology on CK ( X ∗ ) is finner than the scalar topology [58, Section4.3] restricted to weak ∗ -closed sets. The linear topology on the set of nonempty closed convex subsetsis the supremum of the scalar and Wijsman topologies. Since the Wijsman topology [58, Definition2.1.1] requires a metric space, one has to use the norm on X ∗ and the linear topology is not comparablewith the weak ∗ -Hausdorff hypertopology. If one uses the metric (108) generated the weak ∗ topologyon balls of X ∗ for a separable Banach space X , then the Wijsman and linear topologies for norm-closed balls of X ∗ are coarser than the weak ∗ -Hausdorff hypertopology, by Theorem 6.17. As a matterof fact, the Hausdorff metric topology is very fine, as compared to various standard hypertopologies(apart from the Vietoris hypertopology). Consequently, the weak ∗ -Hausdorff hypertopology can beseen as a very fine, weak ∗ -type, topology on CK ( X ∗ ) . It shows that the density of the subset of allconvex weak ∗ compact sets with weak ∗ -dense set of extreme points stated in Theorem 2.4 is a very strong property. Moreover, the genericity of such sets even holds true inside the state space E of anyseparable unital C ∗ -algebra: Theorem 2.5 (Generic weak ∗ -compact convex subset of the state space) Let X be a infinite-dimensional, separable and unital C ∗ -algebra and E the state space (Definition Vietoris and Hausdorff metric topologies are not comparable. .1). Denote by CK ( E ) the set of all nonempty convex weak ∗ -compact subsets of E and by D ( E ) the set of all K ∈ CK ( E ) with a weak ∗ -dense set E ( K ) of extreme points. Then, endowed with theweak ∗ -Hausdorff hypertopology, CK ( E ) is a compact and completely metrizable hyperspace with D ( E ) being a dense G δ subset. Proof.
Since any state ρ ∈ E has norm equal to k ρ k X ∗ = 1 , we deduce from Theorem 6.17 that CK ( E ) belongs to the weak ∗ -Hausdorff-compact and completely metrizable hyperspace CK ( X ∗ ) ,defined by (106). By Corollary 6.18 and because E is a weak ∗ -closed set, CK ( E ) is weak ∗ -Hausdorff-closed, and thus a compact and completely metrizable hyperspace. It remains to prove that D ( E ) is adense G δ subset of CK ( E ) .The fact that D ( E ) is a G δ subset of CK ( E ) can directly be deduced from the proof of Proposition6.19 by repacing F D,m with F m ( E ) . = { K ∈ CK ( E ) : ∃ ω ∈ K, B ( ω, /m ) ∩ E ( K ) = ∅} ⊆ CK ( E ) . To prove the weak ∗ -Hausdorff-density of D ( E ) ⊆ CK ( E ) , it suffices to reproduce the proof ofTheorem 6.20, by adding one essential ingredient: the decomposition of any continuous linear func-tional into non-negative components proven in [63] for real Banach spaces. By noting that (i) X R (1) is a real Banach space, (ii) all states are hermitian functionals over X , (iii) ( X R ) ∗ is canonicallyidentify with the real space of hermitian elements of X ∗ , and (iv) any σ ∈ X ∗ is decomposed as σ = Re { σ } + i Im { σ } with Re { σ } , Im { σ } ∈ ( X R ) ∗ , we deduce from [63] that any σ ∈ X ∗ can bedecomposed as σ = c ρ − c ρ + i ( c ρ − c ρ ) , c , c , c , c ∈ R +0 , ρ , ρ , ρ , ρ ∈ E . (11)At
Step 1 of the proof of Theorem 6.20, because of (11), we observe that there is a non-zero positivefunctional σ ∈ ( X ∗ \ span { ω , . . . , ω n ε } ) . So, we proceed by using σ as a (non-zero) positive functional with norm k σ k X ∗ ≤ and the state ω n ε +1 . = (1 − λ k σ k X ∗ ) ̟ + λ σ ∈ E , instead of (121). One then iterates the arguments, as explained in the proof of Theorem 6.20, usingalways a (non-zero) positive functional σ n with norm k σ n k X ∗ ≤ and ω n ε + n . = (1 − λ n k σ n k X ∗ ) ̟ n + λ n σ n ∈ E , instead of (128), as already explained. In doing so, we ensure that the convex weak ∗ -compact set K ∞ of Equation (132) belongs to D ( E ) ⊆ CK ( E ) .Note that Theorem 2.5 does not directly follow from Theorem 2.4 because the complement of CK ( E ) is open and dense in CK ( X ∗ ) .Important examples of (antiliminal and simple) C ∗ -algebras with state space E ∈ D ( E ) ⊆ D ⊆ CK ( X ∗ ) , i.e., satisfying (6), are the (even subalgebra of the) CAR C ∗ -algebras for (non-relativistic)fermions on the lattice. Quantum-spin systems, i.e., infinite tensor products of copies of some elemen-tary finite dimensional matrix algebra, referring to a spin variable, are also important examples. Theyare, for instance, widely used in quantum information theory as well as in condensed matter physics.In all these physical situations, the corresponding (non-commutative) C ∗ -algebra X is separable and E is thus a metrizable weak ∗ -compact convex set. It is not a simplex [36, Example 4.2.6], but E = [ n ∈ N P n (12)15s the weak ∗ -closure of the union of a strictly increasing sequence ( P n ) n ∈ N ⊆ D ( E ) of Poulsensimplices [62]. Equation (12) is a consequence of well-known results (see, e.g., [60, 64]) and wegive its complete proof in [4]. In other words, by Proposition 6.14, E is the weak ∗ -Hausdorff limit ofthe increasing sequence ( P n ) n ∈ N within the set D ( E ) of all K ∈ CK ( E ) with weak ∗ -dense set ofextreme points.Note that the Poulsen simplex P is not only a metrizable simplex with dense extreme boundary E ( P ) . It has also the following remarkable properties: • It is unique , up to an affine homeomorphism. Indeed, any two compact metrizable simplexeswith dense extreme boundary are mapped into each other by an affine homeomorphism, by [65,Theorem 2.3]. • It is universal in the sense that every compact metrizable simplex is affinely homeomorphic to a(closed) face of P , by [65, Theorem 2.5]. As a consequence, by [36, Example 4.2.6], the statespace of any classical system with separable phase space can be seen as a face of P . Moreover,by [66], every Polish space is homeomorphic to the extreme boundary of a face of P . • It is homogeneous in the sense that any two proper closed isomorphic faces of P are mappedinto each other by an affine automorphism of P . See [65, Theorem 2.3].Together with Equation (12) this demonstrates, for infinite-dimensional quantum systems, the amaz-ing structural richness of the state space E , while making mathematically clear the possible identifi-cation of the phase space E ( E ) as the state space E .In fact, because of Theorems 2.4-2.5, if the “primordial” (non-commutative) algebra X has infinitedimension , then, as is done without much attention in many textbooks, one should expect that thestate and phase spaces, as we define them in the present paper, are identical, even if this featurehas to be mathematically proven in each case (like for antiliminal and simple X ). For instance, if X is an infinite-dimensional, commutative and unital C ∗ -algebra, then the state and phase spaces,respectively E and E ( E ) , are cleary different from each other, even if E can always be approximatedin the weak ∗ -Hausdorff hypertopology by a convex weak ∗ -compact set K ⊆ E with weak ∗ -denseextreme boundary, by Theorem 2.5. C ∗ -Algebra of Continuous Functions on the State Space The space C ( E ; C ) of complex-valued weak ∗ -continuous functions on the state space E of Definition2.1, endowed with the point-wise operations and complex conjugation, is a unital commutative C ∗ -algebra denoted by C . = (cid:16) C ( E ; C ) , + , · C , × , ( · ) , k·k C (cid:17) , (13)where k f k C . = max ρ ∈ E | f ( ρ ) | , f ∈ C . (14)The (real) Banach subspace of all real-valued functions from C is denoted by C R C . If X isseparable then C is also separable, E being in this case metrizable. See, e.g., [54, Problem (d) p.245]. It is the (unique up to a homeomorphism) metrizable simplex with dense extreme boundary. A face F of a convex set K is defined to be a subset of K with the property that, if ρ = λ ρ + · · · + λ n ρ n ∈ F with ρ , . . . , ρ n ∈ K , λ , . . . , λ n ∈ (0 , and λ + · · · + λ n = 1 , then ρ , . . . , ρ n ∈ F . I.e., a separable topological space that is homeomorphic to a complete metric space. I.e, there is an affine homeomorphism between both faces. C ∗ -algebras, elements of theunital C ∗ -algebra X canonically define continuous affine functions ˆ A ∈ C by ˆ A ( ρ ) . = ρ ( A ) , ρ ∈ E, A ∈ X . (15)This is the well-known Gelfand transform . Note that A = B yields ˆ A = ˆ B , as states separateselements of X . Since X is a (unital) C ∗ -algebra, k A k X = max ρ ∈ E | ρ ( A ) | , A ∈ X R , (16)and hence, the map A ˆ A defines a linear isometry from the Banach space X R of all self-adjointelements (cf. Equation (1)) to the space C R of all real-valued functions on E .For any self-adjoint subspace B ⊆ X , we define the ∗ -subalgebras C B ≡ C B ( E ) . = C [ { ˆ A : A ∈ B} ] ⊆ C and C R B ≡ C R B ( E ) . = R [ { ˆ A : A ∈ B ∩ X R } ] ⊆ C R , (17)where K [ Y ] ⊆ C denotes the K -algebra generated by Y , i.e., the subspace of polynomials in theelements of Y , with coefficients in the field K ( = R , C ). The unit ˆ ∈ C , being the constant map ˆ ( ρ ) = 1 for ρ ∈ E (cf. Definition 2.1), belongs, by definition, to C B and C R B ⊆ C B . If B is dense in X then C B separates states. Therefore, by the Stone-Weierstrass theorem [67, Chap. V, § B ⊆ X , C B is dense in C , i.e., C = C B . C ∗ -Algebra of Continuous Functions on the Phase Space If the weak ∗ -compact set E ( E ) is supposed to play the role of a phase space (cf. Definition 2.2),then a classical dynamics should be defined on the space C ( E ( E ); C ) of complex-valued weak ∗ -continuous functions on E ( E ) . Endowed with the usual point-wise operations and complex conju-gation, it is again a unital commutative C ∗ -algebra. Of course, there is a natural ∗ -homomorphism C → C ( E ( E ); C ) , by restriction on E ( E ) of functions from C . Recall that C ( E ( E ); C ) is canonically ∗ -isomorphic, via the restriction on E ( E ) of functions, to a C ∗ -subalgebra of C ( E ( E ); C ) . In Corol-lary 4.3 and Equation (73), we show that the classical dynamics constructed in the present paper canbe pushed forward , through the restriction map, from C to either C ( E ( E ); C ) or C ( E ( E ); C ) . Thegenerator of the dynamics on C ( E ( E ); C ) can be expressed on polymonials via the Poisson bracketof Corollary 3.7, by Proposition 3.11.In standard classical mechanics, in the case of compact phase spaces, even if the C ∗ -algebra C isalways well-defined, note that C is usually never used, but rather C ( E ( E ); C ) , and a classical systemis always supposed to be in some extreme state. In fact, the same physical object cannot be at the sametime on two distinct points of the phase space, according to the spatio-temporal identity of classicalmechanics [68]. This refers to Leibniz’s Principle of Identity of Indiscernibles . This is related tothe fact that any extreme classical state is dispersion-free, see [52, Eq. (6.3), V being the state]. Inthe classical situation, the space C is therefore not fundamental: In this case, by the Riesz–Markovtheorem, the state space is the same as the set of probability measures on the phase space E ( E ) anda mixed, or non-extreme, state ρ ∈ E \E ( E ) of a classical system is only used to reflect the lack ofknowledge on the physical object along with a probabilistic interpretation. Compare with (3).For quantum systems, this property is not as evident as it is for classical ones, as conceptuallydiscussed for instance in [68]. The spatio-temporal identity of classical mechanics is questionable in This means that A ∈ B implies A ∗ ∈ B . Leibniz’s Principle of Identity of Indiscernibles [68, p. 1]: “
Two objects which are indistinguishable, in the sense ofpossessing all properties in common, cannot, in fact, be two objects at all. In effect, the Principle provides a guaranteethat individual objects will always be distinguishable. ” If one asks what, irrespective of quantum mechanics, is characteristic of the world of ideas of physics,one is first of all struck by the following: the concepts of physics relate to a real outside world... itis further characteristic of these physical objects that they are thought of as a range in a space-timecontinuum. An essential aspect of this arrangement of things in physics is that they lay clamed, ata certain time, to an existence independent of one another, provided these objects “are situated indifferent parts of space”.
Einstein, 1948 [69]The non-locality of quantum mechanics was in fact Einstein’s main criticism on this theory [70], morethan its weakly deterministic features.The non-locality of quantum mechanics has been experimentally verified, for instance via Bell’sinequalities, and it is not the subject of the present paper to discuss further related topics, like theexistence of hidden variables in quantum physics. The point in this brief discussion is that there is noclear reason to restrict ourselves to the phase space E ( E ) and not also consider the whole state space E , as, in contrast to classical physics, extreme states are not anymore dispersion-free for quantumsystems. See, e.g., [52, Proposition 2.10]; cf. also the Bell-Kochen-Specker theorem [52, Theorem6.5]. As a matter of fact, important phenomena, like the breakdown of the U (1) -gauge symmetry inthe BCS theory of superconductivity, are related with non-extreme states. See, as an example, [71,Theorem 6.5]. What’s more, the phase space and the state space turn out to be identical for importantclasses of (infinitely extended) quantum systems in condensed matter physics, as already explained.See Equation (6). If g is a finite dimensional Lie algebra, there is a standard contruction of a Poisson bracket for thepolynomial functions on its dual space g ∗ . See, for instance, [29, Section 7.1]. Observe that the (real)space X R of all self-adjoint elements of an arbitrary C ∗ -algebra X forms a Lie algebra by endowingit with the Lie bracket i [ · , · ] , i.e., the skew-symmetric biderivation on X R defined by the commutator i [ A, B ] . = i ( AB − BA ) ∈ X R , A, B ∈ X R . (18)One of the aims of our paper is to extend such a construction of a Poisson bracket to polynomialfunctions on the dual space of X R , which is possibly infinite-dimensional. Before doing that, we firstbriefly present B´ona’s setting [13, Sections 2.1b, 2.1c], which motivated the present work. B´ona [13, Sections 2.1b, 2.1c] proposes a Poisson structure for polynomial functions on the predual (instead of the dual) of a C ∗ -algebra. Recall that, if X is the C ∗ -algebra B ( H ) of all bounded operatorson a Hilbert space H , then its predual X ∗ can be identified with the Banach space L ( H ) of trace-classoperators on H , with the (trace) norm k A k . = Tr H √ A ∗ A , A ∈ L ( H ) . More precisely, for all A ∈ B ( H ) ( = X ), the linear map ˆ A defined by σ Tr H ( σA ) L ( H ) to C is continuous and, conversely, any linear continuous functional ˆ A : L ( H ) → C is of this form for a unique A ∈ B ( H ) . From this, one concludes that the dual of the real Banachspace L R ( H ) of self-adjoint trace-class operators on H is the real Banach space B ( H ) R of self-adjointbounded operators on the Hilbert space H . Thus, B ( H ) R ≡ ( L R ( H )) ∗ ⊆ C ( L R ( H ); R ) . (19)Let C R B ( H ) R . = R [ B ( H ) R ] ⊆ C ( L R ( H ); R ) be the subalgebra of polynomials in the elements of B ( H ) R with real coefficients. The elements of thissubalgebra are called “polynomial” functions on L R ( H ) , the predual of the Lie algebra ( B ( H ) R , i [ · , · ]) .In [13, Sections 2.1c], B´ona proves the existence of a unique Poisson bracket {· , ·} on C R B ( H ) R , i.e., ofa skew-symmetric biderivation satisfying the Jacobi identity on polynomial functions, such that { ˆ A, ˆ B } ( σ ) = Tr H ( i [ A, B ] σ ) = \ i [ A, B ]( σ ) , A, B ∈ B ( H ) R , σ ∈ L R ( H ) . It turns out that the Poisson manifold ( L R ( H ) , {· , ·} ) has a non-trivial symplectic foliation: For any σ ∈ L R ( H ) , we define its unitary orbit by O( σ ) . = { U σ U ∗ : U a unitary operator on H} ⊆ L R ( H ) . (20)If σ ∈ L R ( H ) has finite-dimensional range (i.e., dim ran( σ ) < ∞ ), then O( σ ) is a symplectic leafof the Poisson manifold ( L R ( H ) , {· , ·} ) . In particular, the restriction on such a leaf of the Poissonbracket of two functions f, g only depends on the restriction of f, g on the same leaf. Meanwhile,B´ona observes in [13, Lemma 2.1.7] that the union [ (cid:8) O( σ ) : σ ∈ L R ( H ) , σ ≥ , Tr H ( σ ) = 1 , dim ran( σ ) < ∞ (cid:9) is dense in the set S ∗ of all normalized positive elements (i.e., density matrices) of L R ( H ) . Usingthis observation, B´ona defines the Poisson bracket for polynomial functions defined on S ∗ ⊆ L R ( H ) ,but he proposes [13, Sections 2.1c, footnote] as a mathematically and physically interesting problemto “ formulate analogies of [his] constructions on the space of all positive normalized functionals on B ( H ) . This leads to technical complications .” In Sections 3.2 and 3.3 we give such a construction forthe dual space of any C ∗ -algebra X (and not only for the special case X = B ( H ) ). Sections 3.4-3.5contribute an alternative, more explicit, construction of the same Poisson structure. Remark 3.1
The construction given in the recent paper [72] for a Hamiltonian flow associated with Schr¨odinger’sdynamics of one quantum particle corresponds to B´ona’s symplectic leaf O( σ ) of density matrices σ of dimension one, i.e., dim ran( σ ) = 1 . However, the author of [72] does not seem to be aware ofB´ona’s works. C ∗ -Algebra Recall that ( X R , i [ · , · ]) is a (possibly infinite-dimensional) Lie algebra. See (18). It is easy to checkthat the continuous (real) linear functionals X R → R are in one-to-one correspondance to the hermi-tian continuous (complex) linear functionals X → C , simply by restriction to X R ⊆ X . Recall that a(complex) linear functional σ : X → C is, by definition, hermitian when σ ( A ∗ ) = σ ( A ) , A ∈ X .
19e denote by X ∗ R the (real) space of all hermitian elements of the (topological) dual space X ∗ and usethe identification X ∗ R ≡ ( X R ) ∗ , as already done in the proof of Theorem 2.5. The space X ∗ R with X = B ( H ) plays in our setting ananalogous role as L R ( H ) in B´ona’s approach [13, Sections 2.1b, 2.1c]. See Section 3.1.Similar to (15), for any A ∈ X , we define the weak ∗ -continuous (complex) linear functional ˆ A : X ∗ → C by ˆ A ( σ ) . = σ ( A ) , σ ∈ X ∗ . (21)(Note that we use the same notation as in (15), for the canonical identification of A ∈ X with alinear functional on X ∗ .) Any element of X ∗∗ is of this form. Note also that any weak ∗ -continuous(real) linear functional on X ∗ R uniquely extends to a weak ∗ -continuous (complex) linear hermitianfunctional on X ∗ . In this case, by hermiticity, the corresponding A ∈ X belongs to X R . Conversely,any A ∈ X R defines a weak ∗ -continuous (real) linear functional ˆ A : X ∗ R → C , by restriction of (21)to X ∗ R . Therefore, we identify the real Banach space X R of self-adjoint elements of the C ∗ -algebra X with the space of all weak ∗ -continuous (real) linear functionals X ∗ R → R , i.e., X R ≡ ( X ∗ R ) ∗ . (22)In this view point, X R ⊆ C ( X ∗ R ; R ) . Let C R X R ≡ C R X R ( X ∗ R ) . = R [ X R ] ⊆ C ( X ∗ R ; R ) be the subalgebra of polynomials in the elements of X R , with real coefficients. (Compare with (17)for B = X R .) The elements of this subalgebra are again called “polynomial” functions on X ∗ R , the dual of the Lie algebra ( X R , i [ · , · ]) .Note that such polynomials are Gateaux differentiable and, for any f ∈ C R X R and any σ ∈ X ∗ R ,the Gateaux derivative d G f ( σ ) is linear and weak ∗ continuous, i.e., d G f ( σ ) ∈ X R (see (22)). Inparticular, for any A ∈ X , by (21), d G ˆ A ( σ ) = A , σ ∈ X ∗ R . (23)Thus, we can define a skew-symmetric biderivation {· , ·} on C R X R as follows: Definition 3.2 (Poisson bracket)
The skew-symmetric biderivation {· , ·} on C R X R is defined by { f, g } ( σ ) . = σ (cid:0) i (cid:2) d G f ( σ ) , d G g ( σ ) (cid:3)(cid:1) , f, g ∈ C R X R . This skew-symmetric biderivation satisfies the Jacobi identity:
Proposition 3.3 (Usual properties of Poisson brackets) {· , ·} is a Poisson bracket, i.e., it is a skew-symmetric biderivation satisfying the Jacobi identity { f, { g, h } } + { h, { f, g } } + { g, { h, f } } = 0 , f, g, h ∈ C R X R . = R [ X R ] . Proof. {· , ·} is clearly skew-symmetric, by (18) and Definition 3.2. Note additionally that, for any f, g ∈ C R X R , d G ( f + g ) = d G f + d G g and d G ( f g ) = f d G g + g d G f , (24)where the products in the last equality are meant point-wise. As a consequence, {· , ·} is bilinearand satisfies Leibniz’s rule with respect to both arguments, by (18). In other words, {· , ·} is a skew-symmetric biderivation. Finally, by bilinearity, it suffices to prove the Jacobi identity for f, g, h being20onomials in the elements of X R . If the sum of the degree of the three monomials is , , or , thenthe Jacobi identity follows trivially. If the sum is exactly then the Jacobi identity follows from thecorresponding one for the commutators (18). (If one of the three monomials has zero degree thenall terms in the Jacobi identity trivially vanish.) If the sum is bigger than then at least one of themonomial has degree bigger than . Assume, without loss of generality, that this monomial is f .Then f = f f where the monomials f and f have degree at least , and, explicit computationsusing Leibniz’s rule and the skew-symmetry yield { f, { g, h } } + { h, { f, g } } + { g, { h, f } } = f (cid:0) { f , { g, h } } + { h, { f , g } } + { g, { h, f } } (cid:1) + f (cid:0) { f , { g, h } } + { h, { f , g } } + { g, { h, f } } (cid:1) . Since f and f have in this case degree stricly smaller than the degree of f , the Jacobi identity followsby induction. Corollary 3.4 (Poisson algebra)
The subspace C R X R of polynomials in the elements of X R ⊆ C ( X ∗ R ; R ) with real coefficients, endowedwith {· , ·} and the pointwise multiplications of C R X R , is a Poisson algebra, in the sense of [29, Defi-nition 1.1]. Let F ⊆ X ∗ R be any nonempty subset of X ∗ R and define the algebra C R X R ( F ) . = (cid:8) f | F : f ∈ C R X R ( X ∗ R ) (cid:9) (25)of polynomials on F . If the restriction to F of the Poisson bracket { f, g } of two polynomials f, g ∈ C R X R ( X ∗ R ) (Definition 3.2) only depends on the corresponding restrictions of f, g , then { f | F , g | F } . = { f, g } | F , f, g ∈ C R X R ( X ∗ R ) , is a well-defined Poisson bracket on C R X R ( F ) . Equivalently, this means that the subalgebra I F . = (cid:8) f ∈ C R X R : f ( F ) = { } (cid:9) of polynomials that vanish on F ⊆ X ∗ R is a Poisson ideal of the Poisson algebra ( C R X R , {· , ·} ) . Recallthat a subalgebra I of a Poisson algebra ( P , {· , ·} ) is called a Poisson ideal whenever, for all f ∈ I and g ∈ P , f g ∈ I and { f, g } ∈ I . See, e.g., [29, Section 2.2.1]. As a consequence of this fact, thePoisson algebras ( C R X R ( F ) , {· , ·} ) and ( C R X R ( X ∗ R ) , {· , ·} ) / I F are isomorphic. See [29, Section 2.2.1] for the definition of the quotient of a Poisson algebra by oneof its Poisson ideals. See also [29, Proposition 2.8].For any state ρ ∈ E , we apply these observations to the follium E ρ of states, defined by E ρ . = n h ϕ, π ρ ( · ) ϕ i H ρ : ϕ ∈ H ρ , k ϕ k H ρ = 1 o ⊆ E ⊆ X ∗ R , where the triplet ( H ρ , π ρ , Ω ρ ) is the GNS representation [36, Section 2.3.3] of ρ . Proposition 3.5 (Folia of states and Poisson ideals)
For any ρ ∈ E and any f, g ∈ C R X R ( X ∗ R ) , the restriction { f, g } | E ρ only depends on the correspondingrestriction of f, g . In particular, { f | E ρ , g | E ρ } . = { f, g } | E ρ , f, g ∈ C R X R ( X ∗ R ) , is a well-defined Poisson bracket on C R X R ( E ρ ) . roof. For any state ρ ∈ E with GNS representation ( H ρ , π ρ , Ω ρ ) , we define the unit sphere S ρ . = n ϕ ∈ H ρ : k ϕ k H ρ = 1 o . For any f ∈ C R X R ( X ∗ R ) , we define the continuous function f ρ ∈ C (S ρ ; R ) by f ρ ( ϕ ) . = f ( h ϕ, π ρ ( · ) ϕ i H ρ ) , ϕ ∈ S ρ . Let C ( ρ ) . = (cid:8) f ρ : f ∈ C R X R ( X ∗ R ) (cid:9) ⊆ C (S ρ ; R ) . Then, we prove the existence of a skew-symmetric biderivation {· , ·} ( ρ ) on C ( ρ ) satisfying { ˆ A ρ , ˆ B ρ } ( ρ ) = ( { ˆ A, ˆ B } ) ρ , A, B ∈ X R . (26)This last equality yields { f ρ , g ρ } ( ρ ) = ( { f, g } ) ρ , f, g ∈ C R X R ( X ∗ R ) , by linearity and Leibniz’s rule. In particular, for any ϕ ∈ S ρ , { f, g } ( h ϕ, π ρ ( · ) ϕ i H ρ ) = { f ρ , g ρ } ( ρ ) . As f ρ , g ρ only depend on the restrictions f | E ρ and g | E ρ , respectively, the assertion follows.Now, in order to prove the existence of a skew-symmetric biderivation {· , ·} ( ρ ) satisfying (26), let L R ( H ρ ) be the real Banach space of all self-adjoint trace-class operators on H ρ . For any f ∈ C ( ρ ) and ϕ ∈ S ρ , we denote by d Gρ f ( ϕ ) the Gateaux derivative at A = 0 of the map A f (cid:0) e iA ϕ (cid:1) from L R ( H ρ ) to R . For any f ∈ C ( ρ ) , this Gateaux derivative is linear and continuous, i.e., d Gρ f ( ϕ ) ∈B ( H ρ ) R . See, e.g., (19). Therefore, we can define a skew-symmetric biderivation {· , ·} ( ρ ) on C ρ by { f, g } ( ρ ) ( ϕ ) = (cid:10) ϕ, i (cid:2) d Gρ f ( ϕ ) , d Gρ g ( ϕ ) (cid:3) ϕ (cid:11) H ρ , f, g ∈ C ( ρ ) . For any A ∈ X R and ϕ ∈ S ρ , observe that d Gρ ˆ A ρ ( ϕ ) ( B ) = i h ϕ, [ π ρ ( A ) , B ] ϕ i H ρ = i Tr H ρ ([ P ϕ , π ρ ( A )] B ) , where P ϕ is the orthogonal projection whose range is C ϕ . In other words, d Gρ ˆ A ρ ( ϕ ) = i [ P ϕ , π ρ ( A )] ∈ B ( H ρ ) , ϕ ∈ S ρ , A ∈ X R . Since π ρ : X → B ( H ρ ) is a ∗ -homomorphism, by Equation (23) and Definition 3.2, it follows that,for any A, B ∈ X R , { ˆ A ρ , ˆ B ρ } ( ρ ) ( ϕ ) = i h ϕ, π ρ ([ A, B ]) ϕ i H ρ = ( { ˆ A, ˆ B } ) ρ ( ϕ ) , ϕ ∈ S ρ , i.e., Equation (26) holds true.The folia E ρ , ρ ∈ E , play here an analogous role as the symplectic leaves O( σ ) (20) of the Poissonmanifold ( L R ( H ) , {· , ·} ) in B´ona’s approach [13, Sections 2.1b, 2.1c]. See Section 3.1.22 orollary 3.6 (State space and Poisson ideals) For any f, g ∈ C R X R ( X ∗ R ) , the restriction { f, g } | E only depends on the corresponding restriction of f, g . In particular, { f | E , g | E } . = { f, g } | E , f, g ∈ C R X R ( X ∗ R ) , is a well-defined Poisson bracket on C R X R ( E ) ⊆ C R . = C ( E ; R ) . Proof.
The assertion is a direct consequence of Proposition 3.5 together with the obvious equality E = [ { E ρ : ρ ∈ E } . Equivalently, use that I E = \ (cid:8) I E ρ : ρ ∈ E (cid:9) , is a Poisson ideal of the Poisson algebra ( C R X R , {· , ·} ) .In Sections 3.4-3.5, we also present an explicit construction of the Poisson bracket of Corollary 3.6,because it is technically more convenient for the subsequent sections.Finally, recall that the phase space is the weak ∗ closure E ( E ) of the set E ( E ) of extreme pointsof the state space E , see Definition 2.2. Similar to Corollary 3.6, we prove from Proposition 3.5 theexistence of a Poisson bracket for polynomials acting on the phase space: Corollary 3.7 (Phase space and Poisson ideals)
For any f, g ∈ C R X R ( X ∗ R ) , the restriction { f, g } | E ( E ) only depends on the corresponding restrictionof f, g . In particular, { f | E ( E ) , g | E ( E ) } . = { f, g } | E ( E ) , f, g ∈ C R X R ( X ∗ R ) , is a well-defined Poisson bracket on C R X R ( E ( E )) ⊆ C ( E ( E ); R ) . Proof.
For any extreme (or pure) state ρ ∈ E ( E ) , we infer from [73, Proposition 2.2.4] that the folium E ρ ⊆ E ( E ) is a subset of extreme states and, hence, E ( E ) = [ { E ρ : ρ ∈ E ( E ) } . By Proposition 3.5 and continuity of polynomials, it follows that I E ( E ) = I E ( E ) = \ (cid:8) I E ρ : ρ ∈ E ( E ) (cid:9) is again a Poisson ideal of the Poisson algebra ( C R X R , {· , ·} ) . ∗ Gateaux Derivative
In order to construct the Poisson bracket {· , ·} of Corollary 3.6 more explicitly, as well as to analyzeits properties as generator of (generally non-autonomous) classical dynamics, we introduce the notionof convex Gateaux derivative on the space C ( E ; Y ) of weak ∗ -continuous functions on the convex andweak ∗ -compact set E of states with values in an arbitrary Banach space Y ≡ (cid:0) Y , + , · K , k·k Y (cid:1) , K = R , C . As far as only the construction of the Poisson bracket {· , ·} of Corollary 3.6 is concerned, the relevantexample is Y = R = K . 23e first define the Banach space A ( E ; Y ) . = { f ∈ C ( E ; Y ) : ∀ λ ∈ (0 , , ρ, υ ∈ E, f ((1 − λ ) ρ + λυ ) = (1 − λ ) f ( ρ ) + λf ( υ ) } of all affine weak ∗ -continuous Y -valued functions on E , endowed with the supremum norm k f k A ( E ; Y ) . = max ρ ∈ E k f ( ρ ) k Y , f ∈ A ( E ; Y ) . (27)Again, the norm is not used in the contruction of the Poisson bracket {· , ·} of Corollary 3.6, but onlyin Section 7.The convex Gateaux derivative of a weak ∗ -continuous Y -valued function on E at a fixed state isan affine weak ∗ -continuous Y -valued function on E defined as follows: Definition 3.8 (Convex weak ∗ -continuous Gateaux derivative) For any continuous function f ∈ C ( E ; Y ) and any state ρ ∈ E , we say that d f ( ρ ) : E → Y is the(unique) convex weak ∗ -continuous Gateaux derivative of f at ρ ∈ E if d f ( ρ ) ∈ A ( E ; Y ) and lim λ → + λ − ( f ((1 − λ ) ρ + λυ ) − f ( ρ )) = [d f ( ρ )] ( υ ) , ρ, υ ∈ E .
To our knowledge, the concept of convex weak ∗ -continuous Gateaux derivative defined above is new.A function f ∈ C ( E ; Y ) such that d f ( ρ ) exists for all ρ ∈ E is called differentiable and we usethe notation d f ≡ (d f ( ρ )) ρ ∈ E : E → A ( E ; Y ) . Explicit examples of spaces of such differentiable functions are given, for any n ∈ N , by Y n ≡ Y ( Y ) n . = n f ∈ C ( E, Y ) : ∃ { B j } nj =1 ⊆ X R , g ∈ C ( R n , Y ) such that f ( ρ ) = g ( ρ ( B ) , . . . , ρ ( B n )) o . (28)Functions of this kind are said to be cylindrical. In fact, for any n ∈ N and f ∈ Y n , [d f ( ρ )] ( υ ) = n X j =1 ( υ ( B j ) − ρ ( B j )) ∂ x j g ( ρ ( B ) , . . . , ρ ( B n )) , ρ, υ ∈ E . (29)We define the subspace of continuously differentiable Y -valued functions on the convex andweak ∗ -compact set E by Y ≡ Y ( Y ) . = C ( E ; Y ) . = { f ∈ C ( E ; Y ) : d f ∈ C ( E ; A ( E ; Y )) } . (30)We endow this vector space with the norm k f k Y . = max ρ ∈ E k f ( ρ ) k Y + max ρ ∈ E k d f ( ρ ) k A ( E ; Y ) , f ∈ Y , (31)in order to obtain a Banach space, also denoted by Y . Note again that we use “ max ” instead “ sup ” inthe definition of the norm, because of the continuity of f and d f together with the weak ∗ compactnessof E . Observe that ∂ + λ f ρ,υ ( λ ) . = lim ε → + ε − ( f ρ,υ ( λ + ε ) − f ρ,υ ( λ )) = 1(1 − λ ) [d f ((1 − λ ) ρ + λυ )] ( υ ) with f ρ,υ being the Y -valued function on the interval [0 , defined, at fixed states ρ, υ ∈ E , by f ρ,υ ( λ ) . = f ((1 − λ ) ρ + λυ ) , λ ∈ [0 , . ∂ + λ is closed on C ([ a, b ); Y ) for any real parameters a < b , in the senseof the supremum norm. Thus, by well-known properties of the uniform convergence of continuousfunctions, the normed vector space Y is complete.Remark that the family { Y n } n ∈ N is increasing with respect to inclusion and Y ∞ . = [ n ∈ N Y n ⊆ Y is the space of all cylindrical functions of Y . Additionally, if f ∈ A ( E ; Y ) then d f ( ρ ) = f − f ( ρ ) , ρ ∈ E , (32)which means in particular that affine weak ∗ -continuous Y -valued functions on E are continuouslydifferentiable, i.e., A ( E ; Y ) ⊆ Y . We use the convex weak ∗ Gateaux derivative in order to give an explicit expression for the Poissonbracket {· , ·} of Corollary 3.6. To this end, we only need the special case Y = R in Definition 3.8.We also exploit the following result: Proposition 3.9 (Affine weak ∗ –continuous real-valued functions over E ) For any unital C ∗ -algebra X , A ( E ; R ) = { ˆ A : A ∈ X R } , where A ˆ A is the linear isometry from X R to C R defined by (15). In particular, by (25), C R X R ( E ) = R [ A ( E ; R )] ⊆ C R . = C ( E ; R ) . Proof.
This statement is asserted without proof or references in [36, p 339]. A proof is only shortlysketched in [74, p 161] and we thus give it here for completeness and reader’s convenience. It is basedon preliminary results of convex analysis together with general properties of C ∗ -algebras: Clearly, { ˆ A : A ∈ X R } ⊆ A ( E ; R ) . Conversely, fix f ∈ A ( E ; R ) . Since E is a weak ∗ -compact subsetof X ∗ R , we deduce from [74, Corollary 6.3] the existence of an increasing sequence { f n } n ∈ N of affineweak ∗ -continuous real-valued functions on X ∗ R that uniformly converges to f , as n → ∞ . Meanwhile,observe that any affine weak ∗ -continuous real-valued functions g on X ∗ R is of the form g ( σ ) = σ ( A ) + g (0) , σ ∈ X ∗ R , for some self-adjoint element A ∈ X R , because the weak ∗ -continuous real-valued function g − g (0) on X ∗ R is linear. We thus deduce the existence of a sequence { A n } n ∈ N ⊆ X R such that f n ( σ ) = σ ( A n ) + f n (0) , σ ∈ X ∗ R . Since ρ ( ) = 1 for ρ ∈ E , by (16), the uniform convergence of { f n } n ∈ N to f on E yields that { A n + f n (0) } n ∈ N ⊆ X R is a Cauchy sequence, which thus converges to some A ∈ X R , as n → ∞ .It follows that f = ˆ A .Recall that A = B yields ˆ A = ˆ B for any A, B ∈ X . Therefore, by (16), (27), (30) and Proposition3.9, for any continuously differentiable real-valued function f ∈ Y ( R ) C there is a unique D f ∈ C ( E ; X R ) such that d f ( ρ ) = \ D f ( ρ ) , ρ ∈ E . (33)For instance, one infers from (29) that, for any n ∈ N and f ∈ Y ( R ) n , D f ( ρ ) = n X j =1 ( A j − ρ ( A j ) ) ∂ x j g ( ρ ( A ) , . . . , ρ ( A n )) , ρ ∈ E . (34)25y (16) and (27), note that k D f ( ρ ) k X = k d f ( ρ ) k A ( E ; R ) , ρ ∈ E . (35)Therefore, we can define a skew-symmetric biderivation on Y ( R ) for continuously differentiablereal-valued functions depending on the state space: Definition 3.10 (Skew-symmetric biderivation on Y ( R ) ) We define the map {· , ·} : Y ( R ) × Y ( R ) → C ( E ; R ) by { f, g } ( ρ ) . = ρ ( i [D f ( ρ ) , D g ( ρ )]) , f, g ∈ Y ( R ) . This map {· , ·} is clearly skew-symmetric, by (18) and Definition 3.10. This skew-symmetric bideriva-tion is precisely the one already constructed in Corollary 3.6 on polynomials: Proposition 3.11 (Poisson bracket)
Restricted to C R X R ( E ) , the skew-symmetric biderivation of Definition 3.10 coincides with the Poissonbracket defined by { f | E , g | E } . = { f, g } | E , f, g ∈ C R X R ( X ∗ R ) . See Corollary 3.6.
Proof.
By Equation (34),
D ˆ A ( ρ ) = A − ρ ( A ) , A ∈ X R , (36)and therefore, { ˆ A, ˆ B } ( ρ ) = ρ ( i [ A, B ]) , ρ ∈ E, A, B ∈ X R . (37)Hence, by Definition 3.2 and Equation (23), { ˆ A | E , ˆ B | E } = { ˆ A, ˆ B } | E , A, B ∈ X R . (38)Linearity and Leibniz’s rule then lead to the assertion.The Poisson bracket can easily be extended to a complex Poisson bracket, i.e., a Poisson bracketfor complex-valued polynomials: Since the sum of two affine functions stays affine, by Proposition3.9, observe that A ( E ; C ) = { ˆ A : A ∈ X } . Moreover, by (16), (27) and (30), for any continuously differentiable complex-valued function f ∈ Y ( C ) C there is a unique D f ∈ C ( E ; X ) satisfying (33). Then, the Poisson bracket {· , ·} ofDefinition 3.10 can be extended to all f, g ∈ Y ( C ) , as a skew-symmetric biderivation. In fact, since D ( f + g ) = D f + D g and D ( f g ) = f D g + g D f , this skew-symmetric biderivation satisfies { f, g } = { Re { f } , Re { g }}−{ Im { f } , Im { g }} + i ( { Im { f } , Re { g }} + { Re { f } , Im { g }} ) (39)for all f, g ∈ Y ( C ) . Note here that Re { f } , Im { f } ∈ Y ( R ) for all f ∈ Y ( C ) . Restricted to C X ≡ C X ( E ) , it is again a (complex) Poisson bracket, since it satisfies the Jacobi identity, by Proposition3.11 together with tedious computations. Remark 3.12 (Commutative case) If X is already a commutative unital C ∗ -algebra then the Poisson bracket is of course trivial, being thezero biderivation, and any classical dynamics generated by this Poisson bracket corresponds to theidentity map. This is reminiscent of the KMS dynamics, which becomes trivial when the correspondingvon Neumann algebra is commutative. (In this case, the modular operator is the identity operator.) .6 Poissonian Symmetric Derivations A derivation d (on C ) is a linear map from a dense ∗ -subalgebra dom( d ) (i.e., its domain) of C to theunital commutative C ∗ -algebra C (13) of complex-valued weak ∗ -continuous functions on E such that d ( f g ) = d ( f ) g + f d ( g ) , f, g ∈ dom ( d ) . (40)It is symmetric , or a ∗ -derivation, when d ( ¯ f ) = d ( f ) , f ∈ dom ( d ) . (41)For an exhaustive description of the theory of derivations, see [34, 36, 37] and references therein.An important class of symmetric derivations can be defined by using the Poisson bracket {· , ·} ofDefinition 3.10: Definition 3.13 (Poissonian symmetric derivations)
The Poissonian symmetric derivation associated with any continuously differentiable real-valuedfunction h ∈ Y ( R ) is the linear operator defined on its dense domain dom( d h ) = C X ⊆ C by d h ( f ) . = { h, f } , f ∈ C X . Recall at this point that C X ≡ C X ( E ) ⊆ C is the dense ∗ -subalgebra of all polynomials in theelements of { ˆ A : A ∈ X } , with complex coefficients. See (17). Because of Definition 3.10 andEquations (30)-(31), (33), (35) and (39), d h is a symmetric derivation satisfying (cid:13)(cid:13) d h ( f ) (cid:13)(cid:13) C ≤ k h k Y ( C ) k f k Y ( C ) , f ∈ C X ⊆ Y ( C ) . In particular, d h could be extended as a bounded symmetric derivation ˜ d h from Y ( C ) to C , i.e., as anelement of B ( Y ( C ) , C ) .At first sight, the extension ˜ d h of d h to all continuously differentiable complex-valued functionsof Y ( C ) seems to be natural, like for the usual differentiation on functions of the compact set [0 , .On second thoughts, ˜ d h may not be closable, even if this property would be true for d h . Of course, if ˜ d h is closable then d h is also closable.For a large class of symmetric derivations, the closableness is proven from dissipativity [34, Def-inition 1.4.6, Proposition 1.4.7]. This property is in turn deduced from a theorem proven by Kishi-moto [34, Theorem 1.4.9], which uses the assumption that the square root of each positive element ofthe domain of the derivation also belongs to the same domain. We cannot expect this last property tobe satisfied for symmetric derivations like d h or ˜ d h .The closableness of unbounded symmetric derivations of C ∗ -algebras is, in general, a non-trivialissue, even in the commutative case like C . This property is not generally true: there exist norm-densely defined derivations of C ∗ -algebras that are not closable [35]. For instance, in [36, p. 306], it iseven claimed that “ Herman has constructed an extension of the usual differentiation on C (0 , whichis a nonclosable derivation of C (0 , .” The general characterization of closed symmetric derivationsdepends heavily on the (Hausdorff) dimension of the locally compact set, here the weak ∗ -compactset E . Around 1990, a characterization of all closed symmetric derivations were obtained by usingspaces of functions acting on a compact subset of a one-dimensional space. However, “ for morethan 2 dimensions only sporadic results are known ”, as quoted in [34, Section 1.6.4, p. 27]. See,e.g., [34, Section 1.6.4], [37], [38, 39], and later [36, p. 306].In our approach, the closableness of unbounded symmetric derivations like d h or ˜ d h is a necessaryproperty to make sense of a classical dynamics, in its Hamiltonian formulation, via C -groups. InSection 4, we show that the symmetric derivation d h is closable, at least for all functions h in a densesubset of C , including C X . This is performed via a self-consistency problem together with the C -semigroup theory [40]. Our results are non-trivial since E is not a subset of a finite-dimensional spacewhen X has infinite dimension. See proof of Theorem 2.5.27 Hamiltonian Flows for States from Self-Consistent QuantumDynamics
Our approach to the construction of Hamiltonian flows and, in particular, closed derivations of acommutative C ∗ -algebra via self-consistency problems is non-conventional . However, it shares somesimilarity with the following simple example in the finite-dimensional case: Take A as being thecommutative unital C ∗ -algebra of all continuous, bounded and complex-valued functions on R N , N ∈ N , and fix a smooth and compactly supported function h : R N → R . From the Picard-Lindel¨ofiteration argument, the (Hamiltonian) vector field J ∇ h (where ∇ h is the gradient of h and J is the N -dimensional symplectic matrix) generates a global smooth flow φ t : R N → R N , t ∈ R . Let theone-parameter group { V t } t ∈ R of ∗ -automorphisms of A be defined by [ V t ( f )]( x ) = f ◦ φ − t ( x ) , x ∈ R N , t ∈ R . Because of the compactness of the support of h , this one-parameter group is strongly continuous andthe corresponding generator is a closed derivation in A , denoted by δ h . Moreover, it is straightforwardto check that, in the dense set of smooth functions, this derivation acts as δ h = { h, ·} , where {· , ·} isthe canonical Poisson bracket { f, g } ( x ) . = N X k,l =1 J ij [ ∂ x i f ( x )][ ∂ x j g ( x )] , x ∈ R N , for smooth functions f, g on R N . The analogy of the results presented in this section with thisexample is as follows: in our setting, the space E of all states on X replaces R N and the analogueof the global Hamiltonian flow { φ t } t ∈ R is a one-parameter family of weak ∗ automorphisms of E (orself-homeomorphisms of E ). Note that B´ona uses such a construction only on symplectic leavesof the corresponding Poisson manifold and “glues” them together in order to construct the globalflow [13, Section 2.1-d]. However, in strong contrast to this simple example, in our case, it is notclear at all how to construct the corresponding family of automorphisms from Hamiltonian vectorfields. Instead, we construct it as the solution to a self-consistency problem. Similar to the aboveexample, the closed derivations we obtain for the classical algebra C are closed extensions of denselydefined derivations of the form f
7→ { h, f } , f, h ∈ C X R , where C X R ⊆ C is the dense subalgebra ofpolynomials in the elements of X R , defined by (17) for B = X R .All this construction is performed in this section, supplemented with technical assertions provenin Section 7. We start by somehow tedious, albeit necessary, definitions and notation in Sections4.1-4.3, the self-consistency equations being asserted in Theorem 4.1 and exploited afterwards. Let C b ( R ; Y ( R )) be the Banach space of bounded continuous maps from R to Y ( R ) with the norm k h k C b ( R ; Y ( R )) . = sup t ∈ R k h ( t ) k Y ( R ) , h ∈ C b ( R ; Y ( R )) . (42)We identify Y ( R ) with the subalgebra of constant functions of C b ( R ; Y ( R )) , i.e., Y ( R ) ⊆ C b ( R ; Y ( R )) . (43)Let C ( E ; E ) be the set of weak ∗ -continuous functions from the state space E to itself endowedwith the topology of uniform convergence. In other words, a net ( f j ) j ∈ J ⊆ C ( E ; E ) converges to f ∈ C ( E ; E ) whenever lim j ∈ J max ρ ∈ E | f j ( ρ )( A ) − f ( ρ )( A ) | = 0 , for all A ∈ X . (44)28e denote by Aut ( E ) C ( E ; E ) the subspace of all automorphisms of E , i.e., elements of C ( E ; E ) with weak ∗ -continuous inverse. Equivalently, Aut ( E ) is the set of all bijective mapsin C ( E ; E ) , because E is a compact Hausdorff space. Recall that, here, the concept of an au-tomorphism depends on the structure of the corresponding domain: elements of Aut ( E ) are self-homeomorphisms while a automorphism of a C ∗ -algebra is a ∗ -automorphism of this C ∗ -algebra.Any continuous function h ∈ C b ( R ; Y ( R )) defines a non-autonomous, state-dependent, quantum dynamics on the C ∗ -algebra X via the family { D h ( t ) } t ∈ R ⊆ C ( E ; X R ) , satisfying (33) for each t ∈ R . This quantum dynamics can in turn be used to define a (classical) dynamics on the commutative C ∗ -algebra C of all continous complex-valued functions on E . This latter dynamics turns out to be theflow generated, as is usual in classical mechanics, by the Poisson bracket { h ( t ) , ·} of Definition 3.10(see also Corollary 3.6 and Proposition 3.11). We start with the state-dependent quantum dynamicson the primordial C ∗ -algebra X , in the next subsection. C ∗ -Algebra Fix h ∈ C b ( R ; Y ( R )) , which plays the role of a time-dependent family of classical Hamiltonians.Then, for each state ρ ∈ E and time t ∈ R , we define the symmetric bounded derivation X ρt ∈ B ( X ) by X ρt ( A ) . = i [D h ( t ; ρ ) , A ] . = i (D h ( t ; ρ ) A − A D h ( t ; ρ )) , A ∈ X , (45)where [ · , · ] is the usual commutator defined by (18) and D h ( t ; ρ ) . = [D h ( t )] ( ρ ) ∈ X R , ρ ∈ E, t ∈ R . By Equations (31) and (35), note that sup ρ ∈ E k X ρt k B ( X ) ≤ k h k C b ( R ; Y ( R )) (46)and, for any state-valued continuous function ξ ∈ C ( R ; E ) and times s, t ∈ R , k X ξ ( t ) t − X ξ ( s ) s k B ( X ) ≤ k h ( t ) − h ( s ) k Y ( R ) + 2 k D h ( s ; ξ ( t )) − D h ( s ; ξ ( s )) k X , from (16) and (33).Since D f ∈ C ( E ; X R ) when f ∈ Y ( R ) , for any function ξ ∈ C ( R ; E ) , ( X ξ ( t ) t ) t ∈ R is a norm-continuous family of bounded operators. Therefore, for any continuous functions h ∈ C b ( R ; Y ( R )) and ξ ∈ C ( R ; E ) , a norm-continuous two-parameter family ( T ξt,s ) s,t ∈ R of ∗ -automorphisms of X isuniquely defined in B ( X ) by the non-autonomous evolution equation ∀ s, t ∈ R : ∂ t T ξt,s = T ξt,s ◦ X ξ ( t ) t , T ξs,s = X , (47)or, equivalently, by ∀ s, t ∈ R : ∂ s T ξt,s = − X ξ ( s ) s ◦ T ξt,s , T ξt,t = X . (48)Note that ( T ξt,s ) s,t ∈ R clearly satisfies the (reverse) cocycle property ∀ s, r, t ∈ R : T ξt,s = T ξr,s ◦ T ξt,r . (49)The existence and uniqueness of a solution to these evolution equations follow from the usual theoryof non-autonomous evolution equations for bounded norm-continuous generators, see, e.g., [75]. Inthis case, it is explicitly given by Dyson series. The fact that it defines a family of ∗ -automorphismsof X results from the identity ∂ t n T ξt,s T ξs,t o = 0 , s, t ∈ R , and the fact that the corresponding generators are symmetric derivations.29 .3 Self-Consistency Equations Let C ( R ; C ( E ; E )) be the set of continuous functions from R to C ( E ; E ) . Any ξ ∈ C ( R ; C ( E ; E )) defines a function ξ ( · ; ρ ) ∈ C ( R ; E ) by ξ ( t ; ρ ) . = [ ξ ( t )] ( ρ ) , ρ ∈ E, t ∈ R . (50)Then, for any continuous functions h ∈ C b ( R ; Y ( R )) , ξ ∈ C ( R ; C ( E ; E )) and state ρ ∈ E , thenorm-continuous two-parameter family ( T ξ ( · ; ρ ) t,s ) s,t ∈ R of ∗ -automorphisms of X defined above (Section4.2) is used to define a family ( φ ( h, ξ ) t,s ) s,t ∈ R of maps from the state space E to itself, as follows: φ ( h, ξ ) t,s ( ρ ) . = ρ ◦ T ξ ( · ; ρ ) t,s , ρ ∈ E, s, t ∈ R . (51)By the reverse cocycle property (49) for ( T ξt,s ) s,t ∈ R , ( φ ( h, ξ ) t,s ) s,t ∈ R has the (non-reverse) cocycle prop-erty, i.e., φ ( h, ξ ) t,s = φ ( h, ξ ) t,r ◦ φ ( h, ξ ) r,s , s, t, r ∈ R . (52)By Lemma 7.1, ( φ ( h, ξ ) t,s ( ρ )) s,t ∈ R ∈ C (cid:0) R ; E (cid:1) , ρ ∈ E .
As a consequence, the family ( φ ( h, ξ ) t,s ) s,t ∈ R is a continuous flow on the state space E . Since { D h ( t ) } t ∈ R ⊆ C ( E ; X R ) , by Lemma 7.1 and Lebesgue’s dominated convergence theorem, note additionally that { φ ( h, ξ ) t,s } s,t ∈ R is a family of automorphisms (self-homeomorphisms) of E , i.e. { φ ( h, ξ ) t,s } s,t ∈ R ⊆ Aut ( E ) . To understand the relevance of this flow with respect to classical dynamics, it is enlightening toconsider the autonomous case for which h is the constant function ˆ H for some H ∈ X R . See (15) forthe definition of the function ˆ H , the Gelfand transform of H . In this case, choose a state ρ ∈ E andobserve from (45), (47) and (51), together with Definition 3.10 and Equation (36), that ∂ t ˆ A t,s = { h, ˆ A t,s } with ˆ A t,s . = ˆ A ◦ φ ( ˆ H ) t,s ∈ C for any A ∈ X and s, t ∈ R , noting that the flow φ ( ˆ H ) t,s ≡ φ ( h, ξ ) t,s , s, t ∈ R , does not depend on ξ ∈ C ( R ; C ( E ; E )) . Since φ ( ˆ H ) t,s = φ ( ˆ H ) t − s, for any s, t ∈ R , the flow defined by ( φ ( ˆ H ) t,s ) s,t ∈ R is associatedwith an autonomous classical dynamics, in the usual sense, on elementary elements { ˆ A : A ∈ X } .In the general case of (non-autonomous) classical dynamics generated by time-dependent Poisso-nian symmetric derivations of the form { h ( t ) , ·} , t ∈ R , a convenient (and non-trivial) choice of thefunction ξ in Equation (51) has to be made. We determine it via a self-consistency equation . This isour first main result: Theorem 4.1 (Self-consistency equations) (a)
Let X be a unital C ∗ -algebra and B a finite-dimensional real subspace of X R . (b) Take h ∈ C b ( R ; Y ( R )) such that, for some constant D ∈ R + , sup t ∈ R k D h ( t ; ρ ) − D h ( t ; ˜ ρ ) k X ≤ D sup B ∈ B , k B k =1 | ( ρ − ˜ ρ ) ( B ) | , ρ, ˜ ρ ∈ E .
Under Conditions (a)-(b), there is a unique function ̟ h ∈ C ( R ; Aut ( E )) such that ̟ h ( s, t ) = φ ( h, ̟ h ( α, · )) t,s | α = s , s, t ∈ R , (53) where we recall that Aut ( E ) C ( E ; E ) is the subspace of all automorphisms (or self-homeomorphisms)of E . roof. The theorem is a consequence of Lemmata 7.3 and 7.9.
Remark 4.2 (i)
Stronger results than Theorem 4.1 are proven in Section 7. See, in particular, Lemma 7.5. (ii) If X is separable, recall that the state space E of Definition 2.1 is metrizable, which is a veryuseful property. In Theorem 4.1, however, the separability of X is not necessary at the cost of takinga finite dimensional space B in Condition (b). Condition (b) of Theorem 4.1 is, for instance, satisfied for any cylindrical function h within theset Z . = n ( f ( t )) t ∈ R ∈ C : f ( t ; ρ ) = g ( t ; ρ ( B ) , . . . , ρ ( B n )) for t ∈ R and ρ ∈ E with n ∈ N , { B j } nj =1 ⊆ X R and g ∈ C b (cid:0) R ; C b ( R n , R ) (cid:1) o . (54)By (28), note that, for any h ∈ Z , there is n ∈ N such that h ( t ) ∈ Y n for all t ∈ R . See also (34).Observe that Z C is a dense subset since C R X R ⊆ Z . In (54) we are quite generous by assuming thatthe function g ( t ) belongs to C b ( R n , R ) for some n ∈ N , but even C b ( R n , R ) would be sufficient toget Condition (b). We assume more regularity for g ( t ) , t ∈ R , to be able to prove Theorem 4.6. Here, C pb ( R n ; R ) , p ∈ N , denotes the Banach space of bounded real-valued C p -functions on R n , whosenorm is the C p -norm, i.e., the sum of the supremum norm of all derivatives of order from to p .As explained in Section 2.5, for quantum systems, we shall not restrict our study to the phasespace E ( E ) of Definition 2.2, but we generally consider the whole state space E of Definition 2.1.We show next that both the set E ( E ) of extreme points and its weak ∗ closure E ( E ) are conserved bythe flow of Theorem 4.1, which is defined on the whole state space E : Corollary 4.3 (Conservation of the phase space)
Under Conditions (a)-(b) of Theorem 4.1, for any s, t ∈ R , ̟ h ( s, t ) ( E ( E )) ⊆ E ( E ) and ̟ h ( s, t ) ( E ( E )) ⊆ E ( E ) . Proof.
The proof is done by contradiction: Assume Conditions (a)-(b) of Theorem 4.1. Take ρ ∈E ( E ) and assume the existence of s, t ∈ R , λ ∈ (0 , and two distinct ρ , ρ ∈ E such that ̟ h ( s, t ) ( ρ ) = φ ( h, ̟ h ( s, · )) t,s ( ρ ) = (1 − λ ) ρ + λρ . See Theorem 4.1. By (49) and (51), it follows that ρ = (1 − λ ) ρ ◦ T ̟ h ( s, · )( ρ ) s,t + λρ ◦ T ̟ h ( s, · )( ρ ) s,t . This is not possible whenever ρ ∈ E ( E ) because ρ ◦ T ̟ h ( s, · )( ρ ) s,t and ρ ◦ T ̟ h ( s, · )( ρ ) s,t are two distinct states. This proves that the image of an extreme state by ̟ h ( s, t ) is always anextreme state. ̟ h ( s, t ) ∈ Aut ( E ) and thus preserves the phase space E ( E ) .31 .4 Classical Dynamics as Feller Evolution The continuous family ̟ h of Theorem 4.1 yields a family ( V ht,s ) s,t ∈ R of ∗ -automorphisms of C . = C ( E ; C ) defined by V ht,s ( f ) . = f ◦ ̟ h ( s, t ) , f ∈ C , s, t ∈ R . (55)By Corollary 4.3, such a map can also be defined in the same way on C ( E ( E ); C ) or C ( E ( E ); C ) ,where we recall that E ( E ) is the phase space of Definition 2.2. In any case, it is a strongly continuoustwo-parameter family defining a classical dynamics: Proposition 4.4 (Classical dynamics as Feller evolution system)
Under Conditions (a)-(b) of Theorem 4.1, ( V ht,s ) s,t ∈ R is a strongly continuous two-parameter family of ∗ -automorphisms of C satisfying the reverse cocycle property: ∀ s, r, t ∈ R : V ht,s = V hr,s ◦ V ht,r . (56) If, additionally, h ∈ Y ( R ) (cf. (43)), then V ht,s = V ht − s, for any s, t ∈ R and ( V ht, ) t ∈ R is a C -groupof ∗ -automorphisms of C . Proof.
The strong continuity of this family with respect to s, t ∈ R is a consequence of ̟ h ∈ C ( R ; Aut ( E )) and the fact that any continuous family of continuous functions on compacta isuniformly continuous. Recall that the topology of Aut ( E ) is the topology of uniform convergence ofweak ∗ -continuous functions from E to itself. (To prove continuity in such a strong sense, one couldalso use V ht,s ∈ B ( C ) and the density of C X in C .) Equation (56) follows from Corollary 7.7. Finally,if h ∈ Y ( R ) , while assuming Conditions (a)-(b) of Theorem 4.1, then the family ( T ξt,s ) s,t ∈ R definedby (47)-(48) for any ξ ∈ C ( R ; E ) satisfies T ξt,s = T ξ ( · + s ) t − s, for any s, t ∈ R , where ξ ( · + s ) ∈ C ( R ; E ) is the function ξ translated by the real number s . As a consequence, at any fixed s ∈ R and ρ ∈ E ,the function ξ ∈ C ( R ; E ) defined by ξ s ( t ) . = ̟ h (0 , t − s ; ρ ) , t ∈ R , is a solution to Equation (137). By Lemma 7.3, it follows that ̟ h (0 , t − s ) = ̟ h ( s, t ) , s, t ∈ R , i.e., V ht,s = V ht − s, for any s, t ∈ R . By using (56) at r = t − α + s for any α ∈ R , one verifies that theone-parameter family ( V ht, ) t ∈ R satisfies the group property.Under Conditions (a)-(b) of Theorem 4.1, ( V ht,s ) s,t ∈ R restricted on C R is automatically a Fellerevolution system in the following sense: • As a ∗ -automorphism of a C ∗ -algebra, V ht,s is positivity preserving and k V ht,s k B ( C R ) = 1 ; • ( V ht,s ) s,t ∈ R is a strongly continuous two-parameter family satisfying (56).Therefore, the classical dynamics defined on the real space C R from ( V ht,s ) s,t ∈ R can be associated inthis case with Feller processes in probability theory: By the Riesz-Markov representation theoremand the monotone convergence theorem, there is a unique two-parameter group ( p ht,s ) s,t ∈ R of Markovtransition kernels p ht,s ( · , · ) on E such that V ht,s f ( ρ ) = Z E f (ˆ ρ ) p ht,s ( ρ, dˆ ρ ) , f ∈ C R . The positivity and norm-preserving property are reminiscent of Markov semigroups. E to R . In fact, one can naturally extend ( V ht,s ) s,t ∈ R to this more general class of functions on E . See (55).Note that the notion of Feller evolution system, which is an extension of Feller semigroups to non-autonomous two-parameter families, has been (probably) introduced (only) in 2014 [30]. In contrastwith [30], here the usual cocycle property is replaced by the reverse one and C ∞ ( R d ) by C R , similarto [76, Section 8.1.15] or [77, Definition 1.6], because we do not have any differentiable structureon E . In fact, the term “Feller semigroup” can have different definitions in the literature. See,e.g., [76, Section 8.1.15] and [77, Section 1.1].For any constant function h ∈ Y ( R ) satisfying Conditions (a)-(b) of Theorem 4.1, ( V ht, ) t ∈ R istherefore a C -group of ∗ -automorphisms of C and we denote by k h its (well-defined) generator.By [40, Chap. II, Sect. 3.11], it is a closed (linear) operator densely defined in C . Since V ht, , t ∈ R ,are ∗ -automorphisms, we infer from the Nelson theorem [34, Theorem 1.5.4], or the Lumer-Phillipstheorem [36, Theorem 3.1.16], that ± k h are dissipative operators, i.e., k h is conservative. The ∗ -morphism property of V ht, , t ∈ R , is reflected by the fact that k h has to be a symmetric derivation of C . This closed derivation is directly related with a Poissonian symmetric derivation: Theorem 4.5 (Generators as Poissonian symmetric derivations)
Assume Conditions (a)-(b) of Theorem 4.1. (i)
The Poissonian symmetric derivation d h of Definition 3.13 is closable. Its closure ¯ d h is conservativeand equals the generator k h ⊇ ¯ d h on its domain. (ii) If i ⊇ ¯ d h is a conservative closed operator generating a C -group, then i = k h . (iii) If h ∈ C X R then ¯ d h = k h is the generator of the C -group ( V ht, ) t ∈ R . Proof.
Fix all assumptions of the theorem. Note first that one can compute k h for any (elementary)functions of { ˆ A : A ∈ X } , see (15). In the light of the self-consistency equation given by Theorem4.1, which is combined with (50)-(51) and (55), note that, for any ρ ∈ E , s, t ∈ R and A ∈ X , V ht,s ( ˆ A ) ( ρ ) = ρ ◦ T ̟ h ( s, · ; ρ ) t,s ( A ) , which, by (47), in turn leads to the equality ∂ t V ht,s ( ˆ A ) ( ρ ) = ̟ h ( s, t ; ρ ) ◦ X ̟ h ( s,t ; ρ ) t ( A ) . (57)Using Definitions 3.10, 3.13, Equations (39), (45) and (55) as well as the fact that ( V ht, ) t ∈ R is gener-ated by k h , we deduce from the last equality that k h ( ˆ A ) = d h ( ˆ A ) , A ∈ X . Since both k h and d h are symmetric derivations, it follows that k h | C X = d h . (58)The operator d h is therefore (norm-) closable: For any sequence ( f n ) n ∈ N ⊆ dom( d h ) = C X converg-ing to , if ( d h ( f n )) n ∈ N is a Cauchy sequence then it converges to , by (58) and the closedness of k h ,as a generator of a C -group. Since k h is conservative, we also infer from (58) that both the operator d h and its closure of d h are conservative. (See, e.g., [36, Proposition 3.1.15].) The generator k h isa closed, not necessarily minimal, extension of d h . This concludes the proof of Assertion (i). Thesecond one (ii) thus follows from [36, Proposition 3.1.15]. Feller semigroups have usually the same properties, but they can be defined on different classes of spaces in thelitterature.
33o prove Assertion (iii) we use (ii) and the Nelson theorem [34, Theorem 1.5.4]: Pick h , h ∈ C X .Assume without loss of generality that h , h are both not constant functions. Then, for any ℓ ∈ { , } there are n ℓ ∈ N , { B ℓ,j } n ℓ j =1 ⊆ X R , and g ℓ : R n ℓ → R being a polynomial of degree m ℓ ∈ N such that h ℓ ( ρ ) = g ℓ ( ρ ( B ℓ, ) , . . . , ρ ( B ℓ,n ℓ )) , ρ ∈ E .
Then, from Equation (34) and Definition 3.10, note that d h ( h ) ∈ C X with d h ( h ) ( ρ ) = n X j =1 n X j =1 ρ ( i [ B ,j , B ,j ]) ∂ x j g ( ρ ( B , ) , . . . , ρ ( B ,n )) (59) × ∂ x j g ( ρ ( B , ) , . . . , ρ ( B ,n )) for any ρ ∈ E . Note that, for any k ∈ N , n k k − Y j =0 ( j ( n + 1) + n ) ≤ n k ( k ( n + 1) + n ) k ≤ k ! exp ( n ( k ( n + 1) + n )) , (60)because x n ≤ n !e x for all x ≥ and n ∈ N . Thus, using (59)-(60) together with Equations (14), (16)and straightforward estimates, one gets that (cid:13)(cid:13) ( d h ) k ( h ) (cid:13)(cid:13) C ≤ k !2 k (1 + D ) k (1 + D ) k (1 + D ) exp ( n ( k ( n + 1) + n )) , k ∈ N . where D . = max ℓ ∈{ , } max j ∈{ ,...,n ℓ } k B ℓ,j k X , D ℓ . = max n ∈ N mℓ (cid:26) max ρ ∈ E | ∂ n g ℓ ( ρ ( B ℓ, ) , . . . , ρ ( B ℓ,n ℓ )) | (cid:27) for ℓ ∈ { , } . It follows that X k ∈ N t k k ! (cid:13)(cid:13) ( d h ) k ( h ) (cid:13)(cid:13) C < ∞ for some positive time t satisfying ≤ t < e − n ( n +1) D ) (1 + D ) (1 + D ) . Therefore, by density of C X in C , the conservative, densely defined, closed operator ¯ d h has a denseset of analytic elements. By the Nelson theorem [34, Theorem 1.5.4], ¯ d h is a conservative closedoperator generating a C -group of ∗ -automorphisms of C , whence Assertion (iii), following (ii).Note that Equation (57) holds true for any h ∈ C b ( R ; Y ( R )) satisfying Conditions (a)-(b) ofTheorem 4.1. It follows that, for any s, t ∈ R and polynomial function f ∈ C X , ∂ t V ht,s ( f ) = V ht,s ( { h ( t ) , f } ) . (61)A similar expression for ∂ s V ht,s like ∂ s V ht,s ( f ) = − (cid:8) h ( s ) , V ht,s ( f ) (cid:9) (62)is less obvious. First, we do not know, a priori, if V ht,s maps elements from C X to continuously dif-ferentiable complex-valued functions on E , i.e., if V ht,s ( C X ) ⊆ Y ( C ) . Secondly, even if V ht,s ( C X ) ⊆ Y ( R ) , one still has to prove that Equation (62) holds true. This is done in the next theorem:34 heorem 4.6 (Non-autonomous classical dynamics) Take h ∈ Z . Then, for any s, t ∈ R and f ∈ C X , (61)-(62) hold true. See (54) for the definition of Z . Proof.
Note that any function h ∈ Z satisfies Conditions (a)-(b) of Theorem 4.1. Equation (61) isalready discussed before the theorem: it results from (57) for h ∈ C b ( R ; Y ( R )) and properties ofderivatives and symmetric derivations (linearity and Leibniz’s rule, see, e.g., (40)). To prove (62), itsuffices to invoke Corollary 7.12, which says that ∂ s V ht,s ( ˆ A ) = −{ h ( s ) , V ht,s ( ˆ A ) } for any s, t ∈ R and A ∈ X . Since ( V ht,s ) s,t ∈ R is a family of ∗ -automorphisms of C , by using the(bi)linearity and Leibniz’s rule satisfied by the derivatives and the bracket {· , ·} , we deduce (62) forall polynomial functions f ∈ C X .This theorem applied to the autonomous situation leads to the dynamical equation of classicalmechanics (see, e.g., [78, Proposition 10.2.3]), i.e., (autonomous) Liouville’s equation , which readsin our case as follows:
Corollary 4.7 (Autonomous Liouville’s equation)
Take h ∈ Z constant in time. Then, for any t ∈ R and f ∈ C X , ∂ t V ht, ( f ) = V ht, ◦ k h ( f ) = V ht, ( { h, f } ) = (cid:8) h, V ht, ( f ) (cid:9) = k h ◦ V ht, ( f ) . (63) Proof.
Combine Theorem 4.6 with Theorem 4.5.In the non-autonomous case, ( V ht,s ) s,t ∈ R is a strongly continuous two-parameter family of ∗ -automorphisms of C solving the non-autonomous evolution equations ∀ s, t ∈ R : ∂ t V ht,s = V ht,s ◦ k h ( t ) , V hs,s = C , on C X , as explained before Theorem 4.6. See also Theorem 4.5. Theorem 4.6 suggests that, understronger conditions, ( V ht,s ) s,t ∈ R is the solution to the non-autonomous evolution equations ∀ s, t ∈ R : ∂ s V ht,s = − k h ( s ) ◦ V ht,s , V hs,s = C , (64)on some dense subspace. To prove this, one could look for assumptions on h such that the family ( k h ( t ) ) t ∈ R of closed dissipative operators satisfies sufficient conditions to generate an evolution familysolving (64), as explained in [75, 79–82]. Then, ( V ht,s ) s,t ∈ R would be the solution to the non-auto-nomous evolution equation (64). This looks doable, but at the cost of many technical arguments. Wethus refrain from doing such a study in this paper. C ∗ -Dynamical Systems C ∗ -Algebras of Continuous Functions on State Space The space C ( E ; X ) of X -valued weak ∗ -continuous functions on the weak ∗ -compact space E is aunital C ∗ -algebra with respect to the point-wise operations, denoted by X . = ( C ( E ; X ) , + , · C , × , ∗ , k·k X ) (65)where k f k X . = max ρ ∈ E k f ( ρ ) k X , f ∈ X . (66)35learly, X is commutative iff X is commutative. The (real) Banach subspace of all X R -valued func-tions from X is denoted by X R X . X is separable whenever X is separable, E being in this casemetrizable.We identify the primordial C ∗ -algebra X , on which the quantum dynamics is usually defined,with the subalgebra of constant functions of X . Meanwhile, the classical dynamics appears in thespace C . = C ( E ; C ) of complex-valued weak ∗ -continuous functions on E . See (13)-(14). This unital commutative C ∗ -algebra is thus identified with the subalgebra of functions of X whose values aremultiples of the unit ∈ X . Compare (65)-(66) with (13)-(14). Hence, we have the inclusions X ⊆ X and C ⊆ X . (67)Both classical and quantum dynamics can then be extended to X . This is explained in the next sub-section. Since C ⊆ X , there is a natural extension to X of the classical dynamics on C : The continuous family ̟ h of Theorem 4.1 yields a family ( V ht,s ) s,t ∈ R of ∗ -automorphisms of X defined by V ht,s ( f ) . = f ◦ ̟ h ( s, t ) , f ∈ X , s, t ∈ R . (68)In particular, by (55), V ht,s | C = V ht,s for any s, t ∈ R . However, it is not what we have in mind here:Emphasizing rather the inclusion X ⊆ X , the classical algebra C will become a subalgebra of the fixed-point algebra of the state-dependent dynamics we define below on X .In Section 4.2, we explain how a fixed function h ∈ C b ( R ; Y ( R )) is used to define (possibly non-autonomous) quantum dynamics ( T ξt,s ) s,t ∈ R on the primordial C ∗ -algebra X , for any ξ ∈ C ( R ; E ) .This primal dynamics induces classical dynamics on the (classical) C ∗ -algebra C . = C ( E ; C ) ofcontinuous functions on states, as discussed in Sections 4.3-4.4. By Theorem 4.1, it yields, in turn, a state-dependent quantum dynamics, referring in this case to a norm-continuous family (T ρt,s ) ( ρ,s,t ) ∈ E × R = ( T ̟ h ( s , · ; ρ ) t,s ) ( ρ,s,t ) ∈ E × R of ∗ -automorphisms of X for some fixed s ∈ R . This leads to a (state-dependent) dynamics on the(secondary) C ∗ -algebra X of continuous functions on states.As a matter of fact, any strongly continuous family (T ρ ) ρ ∈ E of linear contractions from X to itselfcan be viewed as a linear contraction T from X to itself defined by [ T ( f )] ( ρ ) . = T ρ ( f ( ρ )) , ρ ∈ E, f ∈ X . (69)Such contractions have the following properties: Lemma 5.1 (State-dependent quantum dynamics)
Let X be a unital C ∗ -algebra. For any s, t ∈ R , let (T ρt,s ) ρ ∈ E be any strongly continuous family oflinear contractions from X to itself, and T t,s be defined by (69) with T ρ = T ρt,s . (i) If T ρt,s is a ∗ -automorphism of X at s, t ∈ R for any ρ ∈ E , then T t,s is a ∗ -automorphism of X and the classical subalgebra C ⊆ X is contained in the fixed-point algebra of T t,s , i.e., T t,s ( f ) = f , f ∈ C . (ii) If (T ρt,s ) s,t ∈ R satisfies a reverse cocycle property for any ρ ∈ E , i.e., T ρt,s = T ρr,s ◦ T ρt,r , ρ ∈ E, s, t, r ∈ R , (70) then ( T t,s ) s,t ∈ R has also this property. (iii) If (T ρt,s ) ( ρ,s,t ) ∈ E × R is a strongly continuous family of contractions then so do ( T t,s ) s,t ∈ R . roof. Assertion (i)-(ii) directly follows from (69) and it remains to prove (iii). By contradiction,suppose that the family is not strongly continuous. Then, there is f ∈ X , times s, t ∈ R , two zero nets ( η j ) j ∈ J , ( κ j ) j ∈ J ⊆ R , a net ( ρ j ) j ∈ J ⊆ E of states and a positive constant D > such that inf j ∈ J (cid:13)(cid:13)(cid:13) T ρ j t + η j ,s + κ j (cid:0) f (cid:0) ρ j (cid:1)(cid:1) − T ρ j t,s (cid:0) f (cid:0) ρ j (cid:1)(cid:1)(cid:13)(cid:13)(cid:13) X ≥ D > . By weak ∗ compactness of E , we can assume without loss of generality that ( ρ j ) j ∈ J converges to some ρ ∈ E . Because (T ρt,s ) ( ρ,s,t ) ∈ E × R is a family of contractions, the above bound yields lim inf j ∈ J (cid:13)(cid:13)(cid:13) T ρ j t + η j ,s + κ j ( f ( ρ )) − T ρ j t,s ( f ( ρ )) (cid:13)(cid:13)(cid:13) X ≥ D > , which contradicts the strong continuity of this family.If (T ρt,s ) ( ρ,s,t ) ∈ E × R is a family of ∗ -automorphisms of X then ( T t,s ) s,t ∈ R is a family of ∗ -automorphismsof X and, by Lemma 5.1 (i), the classical subalgebra C ⊆ X is contained in the fixed-point algebra ofthe full quantum dynamics ( T t,s ) s,t ∈ R , i.e., for all f ∈ C and all s, t ∈ R , T t,s ( f ) = f . Any family ( T t,s ) s,t ∈ R of ∗ -automorphisms of X preserving each element of C is of this form, at least when X isseparable: Lemma 5.2 (State-dependent quantum dynamics and fixed-point algebra)
Let X be a separable, unital C ∗ -algebra. The classical subalgebra C ⊆ X is contained in the fixed-point algebra of a strongly continuous two-parameter family ( T t,s ) s,t ∈ R of ∗ -automorphisms of X iffthere is a strongly continuous family (T ρt,s ) ( ρ,s,t ) ∈ E × R of ∗ -automorphisms of X satisfying (69). Proof.
In order to obtain the equivalence stated in the lemma, it only remains to prove that anystrongly continuous family ( T t,s ) s,t ∈ R of ∗ -automorphisms of X whose fixed-point algebra contains C comes from the strongly continuous family (T ρt,s ) ( ρ,s,t ) ∈ E × R of ∗ -homomorphisms defined by T ρt,s ( A ) . = [ T t,s ( A )] ( ρ ) , ρ ∈ E, A ∈ X ⊆ X , s, t ∈ R . (71)To this end, recall that, if X is separable then E is metrizable. So, take a distance d ( · , · ) generatingthe weak ∗ topology on E . For any ρ ∈ E define the sequence { g n } n ∈ N ⊆ C of continuous functionsby g n (˜ ρ ) = 11 + nd (˜ ρ, ρ ) , ˜ ρ ∈ E, n ∈ N . Since, by assumption, T t,s is a ∗ -automorphism of X satisfying T t,s ( g n ) = g n for s, t ∈ R and n ∈ N ,we note that, for every fixed ρ ∈ E , s, t ∈ R , n ∈ N and all functions f ∈ X , [ T t,s ( f )] ( ρ ) = [ T t,s ( f g n − f ( ρ ) g n )] ( ρ ) + T ρt,s ( f ( ρ )) . Because T t,s is a contraction (for it is a ∗ -automorphism), by continuity of f ∈ X , it follows that lim n →∞ k T t,s ( f g n − f ( ρ ) g n ) k X = lim n →∞ k f g n − f ( ρ ) g n k X = 0 , and hence, [ T t,s ( f )] ( ρ ) = T ρt,s ( f ( ρ )) , ρ ∈ E, f ∈ X , s, t ∈ R . From the last equality we also conclude that T ρt,s is a ∗ -automorphism of X for all ( ρ, s, t ) ∈ E × R .The above situation motivates the following notion of state-dependent C ∗ -dynamical system:37 efinition 5.3 (State-dependent C ∗ -dynamical systems) If T ≡ ( T t,s ) s,t ∈ R is a strongly continuous two-parameter family of ∗ -automorphisms of X preservingeach element of C ⊆ X and satisfying the reverse cocycle property T t,s = T r,s ◦ T t,r , s, t, r ∈ R , then we name the pair ( X , T ) “state-dependent C ∗ -dynamical system”. An example of such a C ∗ -dynamical system is given from Theorem 4.1 via the family T h,s ≡ ( T h,s t,s ) s,t ∈ R of ∗ -automorphisms of X defined by h T h,s t,s ( f ) i ( ρ ) . = T ̟ h ( s , · ; ρ ) t,s ( f ( ρ )) , ρ ∈ E, f ∈ X , s, t ∈ R , for any fixed s ∈ R and every (time-depending classical Hamiltonian) h ∈ C b ( R ; Y ( R )) satisfyingall assumptions of Theorem 4.1. This is a state-dependent C ∗ -dynamical system: Lemma 5.4 (From self-consistency equations to state-dependent quantum dynamics)
Assume Conditions (a)-(b) of Theorem 4.1. Then, for any s ∈ R , ( X , T h,s ) is a state-dependent C ∗ -dynamical system. Proof.
Fix all parameters of the lemma. By Lemma 5.1 (i), T h,s t,s is ∗ -automorphism of X and theclassical subalgebra C ⊆ X is contained in the fixed-point algebra of T h,s t,s for any s, t ∈ R . FromLemma 5.1 (ii), T h,s clearly satisfies the reverse cocycle property. Moreover, by Lemma 7.1 andTheorem 4.1, we can infer from Lemma 5.1 (iii) that T h,s is strongly continuous.Exactly like the classical dynamics defined in Section 4.4, state-dependent C ∗ -dynamical systems ( X , T ) induce Feller dynamics within the (classical) commutative C ∗ -algebra C : • Recall that
Aut ( E ) is the space of all automorphisms (or self-homeomorphisms) of the statespace E , endowed with the topology of uniform convergence of weak ∗ -continuous functions. • From the family ( T t,s ) s,t ∈ R , we define a continuous family ( φ t,s ) s,t ∈ R ⊆ Aut ( E ) by φ t,s ( ρ ) . = ρ ◦ T ρt,s , ρ ∈ E, s, t ∈ R , (72)where (T ρt,s ) ( ρ,s,t ) ∈ E × R is a strongly continuous family of ∗ -automorphisms of X satisfying(69). See Lemma 5.2. Compare with Equation (51). Similar to Corollary 4.3, φ t,s ( E ( E )) ⊆ E ( E ) and φ t,s ( E ( E )) ⊆ E ( E ) . (73) • This family in turn yields a strongly continuous two-parameter family ( V t,s ) s,t ∈ R of ∗ -auto-morphisms of C defined by V t,s f . = f ◦ φ t,s , f ∈ C , s, t ∈ R . (74)Compare with Equation (55). Moreover, by (73), this map can also be defined in the same wayon C ( E ( E ); C ) , where we recall that E ( E ) is the phase space of Definition 2.2. • If (70) holds true, then ( V t,s ) s,t ∈ R satisfies a reverse cocycle property, i.e., for s, t, r ∈ R , V t,s = V t,r ◦ V r,s . This classical dynamics is a Feller evolution system , as defined in Section 4.4.Compare with Proposition 4.4. • If, for any ρ ∈ E , the strongly continuous family (T ρt,s ) s,t ∈ R of ∗ -automorphisms defined by (69)satisfies in B ( X ) some non-autonomous evolution equation, then the family ( V t,s ) s,t ∈ R wouldalso satisfy some non-autonomous evolution equation, as discussed at the end of Section 4.4.38 .3 State-Dependent Symmetries and Classical Dynamics Fix a state-dependent C ∗ -dynamical system ( X , T ) . See Definition 5.3. A state-dependent symmetry of ( X , T ) is defined as follows: Definition 5.5 (State-dependent symmetry)
A state-dependent symmetry G of ( X , T ) is a ∗ -automorphism of X satisfying G ◦ T t,s = T t,s ◦ G , s, t ∈ R , and with fixed-point algebra containing C ⊆ X . If G is a state-dependent symmetry of ( X , T ) , then, similar to Lemma 5.2, the equalities G ρ ( A ) . = [ G ( A )] ( ρ ) , ρ ∈ E, A ∈ X ⊆ X , define a strongly continuous family ( G ρ ) ρ ∈ E of ∗ -automorphisms of X . In this case, we define theweak ∗ -compact space E G . = { ρ ∈ E : ρ ◦ G ρ = ρ } (75)of G -invariant states. By Equation (72) and Definition 5.5, together with (69) for T = T t,s and T ρ = T ρt,s , it follows that φ t,s ( E G ) ⊆ E G and φ t,s ( E \ E G ) ⊆ E \ E G , s, t ∈ R . (76)In particular, by Equation (74), for any function f ∈ C and times s, t ∈ R , V t,s ( f | E G ) . = ( V t,s f ) | E G and V t,s (cid:0) f | E \ E G (cid:1) . = ( V t,s f ) | E \ E G (77)well define two two-parameter families of ∗ -automorphisms respectively acting on { f | E G : f ∈ C } and (cid:8) f | E \ E G : f ∈ C (cid:9) . (78)More generally, we can consider a faithful group homomorphism g G g from a group G to thegroup of ∗ -automorphisms of X . Then, a state-dependent symmetry group is defined as follows: Definition 5.6 (State-dependent symmetry group)
A state-dependent symmetry group of ( X , T ) is a group ( G g ) g ∈ G of state-dependent symmetries of ( X , T ) . If ( G g ) g ∈ G is a state-dependent symmetry group of ( X , T ) , then, defining the weak ∗ -compactspace E G . = (cid:8) ρ ∈ E : ρ ◦ G ρg = ρ for all g ∈ G (cid:9) (79)of G -invariant states, we observe that φ t,s ( E G ) ⊆ E G , φ t,s ( E \ E G ) ⊆ E \ E G , s, t ∈ R , (80)(cf. (76)) and, exactly like in Equations (77)-(78), we infer from (74) the existence of two-parameterfamilies of ∗ -automorphisms respectively defined on { f | E G : f ∈ C } and (cid:8) f | E \ E G : f ∈ C (cid:9) . .4 Reduction of Classical Dynamics via Invariant Subspaces Any family
B ⊆ X defines an equivalence relation ♥ B . = (cid:8) ( ρ , ρ ) ∈ E : ρ ( A ) = ρ ( A ) for all A ∈ B (cid:9) on the set E of states. We say that the subset E B ⊆ E represents E with respect to B whenever, for all ρ ∈ E , there is ρ ∈ E B such that ( ρ , ρ ) ∈ ♥ B . In particular, one can identify continuous functions f ∈ C B with their restrictions to E B .Fix now a state-dependent C ∗ -dynamical system ( X , T ) . See Definition 5.3. For any self-adjointsubspace B ⊆ X , consider the following conditions:
Condition 5.7 (Reduction of dynamics) (i) ( B ∩ X R , i [ · , · ]) is a real Lie algebra, [ · , · ] being the usual commutator in X . (ii) E B is a weak ∗ -compact space representing E with respect to B . (iii) T ρt,s ( B ) ⊆ B for all ρ ∈ E , s, t ∈ R , with T ρt,s defined by (69). (iv) φ t,s ( E B ) ⊆ E B for all s, t ∈ R , with ( φ t,s ) s,t ∈ R defined by (72). By (74), this condition yields that the polynomial algebra C B (17) is preserved by the family ( V t,s ) s,t ∈ R ,i.e., V t,s ( C B ) ⊆ C B , s, t ∈ R . In this case, the state space of the classical dynamics coming from ( T t,s ) s,t ∈ R can be restricted to theweak ∗ -compact subset E B ⊆ E with the corresponding Poisson algebra for observables being thesubalgebra C B ⊆ C ( E B ; C ) . C ∗ -Valued Functions We are not aware whether the C ∗ -algebra X of X -valued continuous functions on states has previouslybeen systematically studied. However, other constructions of C ∗ -algebras of X -valued, continuous ormeasurable, functions are well-known in the literature. For instance, in [83, Definition 1] C ∗ -algebrasof X -valued measurable functions on a locally compact group { G g } g ∈ G of ∗ -automorphisms of X areintroduced. This kind of construction goes under the name “covariance algebras”. In contrast, notethat the state space E has no natural group structure. Moreover, the product of covariances algebrasare convolutions and not point-wise products as in X .Covariance algebras are reminiscent of crossed products of C ∗ -algebras by groups acting on thesealgebras. Such products are relatively standard in the theory of operator algebras. For instance, theyare fundamental in Haagerup’s approach to noncommutative L p -spaces [84]. ∗ -Hausdorff Hypertopology Deep in the human unconscious is a pervasive need for a logical universe that makes sense. But thereal universe is always one step beyond logic. “The Sayings of Muad’Dib” by the Princess Irulan The aim of this section is to provide all arguments to deduce Theorems 2.4-2.5. We adopt a broadperspective on the weak ∗ -Hausdorff hypertopology because it does not seem to have been consideredin the past. This leads, hopefully, to a good understanding of this hypertopology along with interestingconnections to other fields of mathematics and more general results than those stated in Section 2.3. Dune by F. Herbert (1965). ∗ -compact subsetsin our arguments.Recall that, when the restriction to singletons of a topology for sets of closed subsets of topologicalspaces coincide with the original topology of the underlying space, we talk about hypertopologies andhyperspaces of closed sets. In all Section 6, X is not necessarily a C ∗ -algebra, but only a (real or complex) Banach space. Unlessit is explicitly mentioned, for convenience, we always consider the complex case, as in all othersections. We study subsets of its dual X ∗ , which, endowed with the weak ∗ -topology, is a locallyconvex Hausdorff space. See, e.g., [60, Theorem 10.8]. As is usual in the theory of hyperspaces [58],we start with the set F ( X ∗ ) . = { F ⊆ X ∗ : F = ∅ is weak ∗ -closed } of all nonempty weak ∗ -closed subsets of X ∗ . It is endowed below with some hypertopology.Recall that there are various standard hypertopologies on general sets of nonempty closed subsetsof a metric space ( Y , d ) : the Fell, Vietoris, Wijsman, proximal or locally finite hypertopologies, toname a few well-known examples. See, e.g., [58]. The most well-studied and well-known hypertopol-ogy, named the Hausdorff metric topology [58, Definition 3.2.1], comes from the Hausdorff distancebetween two sets F , F , associated with the metric d on Y : d H ( F , F ) . = max (cid:26) sup x ∈ F inf x ∈ F d ( x , x ) , sup x ∈ F inf x ∈ F d ( x , x ) (cid:27) ∈ R +0 ∪ {∞} . (81)In this case, the corresponding hyperspace of nonempty closed subsets of Y is complete iff the metricspace ( Y , d ) is complete. See, e.g., [58, Theorem 3.2.4]. The Hausdorff metric topology is thehypertopology used in [41, 42], the metric d being the one associated with the norm of a separableBanach space Y , in order to prove the density of the set of convex compact subsets of Y with denseextreme boundary.None of these well-known hypertopologies is used here for F ( X ∗ ) . Instead, we use a weak ∗ version of the Hausdorff metric topology. This corresponds to the weak ∗ -Hausdorff hypertopology ofDefinition 2.3, which is naturally extended to all weak ∗ -closed sets of F ( X ∗ ) : Definition 6.1 (Weak ∗ -Hausdorff hypertopology) The weak ∗ -Hausdorff hypertopology on F ( X ∗ ) is the topology induced (see (10)) by the family ofHausdorff pseudometrics d ( A ) H defined, for all A ∈ X , by d ( A ) H ( F, ˜ F ) . = max ( sup σ ∈ F inf ˜ σ ∈ ˜ F | ( σ − ˜ σ ) ( A ) | , sup ˜ σ ∈ ˜ F inf σ ∈ F | ( σ − ˜ σ ) ( A ) | ) ∈ R +0 ∪{∞} , F, ˜ F ∈ F ( X ∗ ) . (82)To our knowledge, this hypertopology has not been considered so far and we thus give here a de-tailed study of its main properties. Recall that it is an hyper topology because any net ( σ j ) j ∈ J in X ∗ converges to σ ∈ X ∗ in the weak ∗ topology iff the net ( { σ j } ) j ∈ J converges in F ( X ∗ ) to { σ } in theweak ∗ -Hausdorff (hyper)topology.Observe that (82) is always finite on the subspace K ( X ∗ ) . = (cid:26) K ∈ F ( X ∗ ) : sup σ ∈ K k σ k X ∗ < ∞ (cid:27) ⊆ F ( X ∗ ) (83)41f all nonempty weak ∗ -closed norm-bounded subsets of the dual space X ∗ . Its complement, i.e., theset of all nonempty weak ∗ -closed norm-unbounded subsets of X ∗ , is denoted by K c ( X ∗ ) . = F ( X ∗ ) \ K ( X ∗ ) . (84)Both sets are weak ∗ -Hausdorff closed since the weak ∗ -Hausdorff hypertopology immeasurably sepa-rates norm-unbounded sets from norm-bounded ones: Lemma 6.2 (Immeasurable separation of norm-unbounded sets from norm-bounded ones)
Let X be a Banach space. For any norm-unbounded weak ∗ -closed set F ∈ K c ( X ∗ ) , there is A ∈ X such that d ( A ) H ( F, K ) = ∞ , K ∈ K ( X ∗ ) . (85) Additionally, the union of any weak ∗ -Hausdorff convergent net ( K j ) j ∈ J ⊆ K ( X ∗ ) is norm-bounded. Proof.
Take any norm-unbounded F ∈ K c ( X ∗ ) . Then, there is a net ( σ j ) j ∈ J ⊆ F such that lim J k σ j k X ∗ = ∞ . By the uniform boundedness principle (see, e.g., [45, Theorems 2.4 and 2.5]), there is A ∈ X suchthat lim J | σ j ( A ) | = ∞ . (86)Now, pick any K ∈ K ( X ∗ ) . Then, by Definition 6.1 and the triangle inequality, for any j ∈ J , d ( A ) H ( F, K ) ≥ inf ˜ σ ∈ K | ( σ j − ˜ σ ) ( A ) | ≥ | σ j ( A ) | − sup ˜ σ ∈ K | ˜ σ ( A ) | . Since K is, by definition, norm-bounded, by (86), the limit over j of the last inequality obviouslyyields (85).Finally, any weak ∗ -Hausdorff convergent net ( K j ) j ∈ J ⊆ K ( X ∗ ) has to converge in K ( X ∗ ) , by thefirst part of the lemma. Therefore, using an argument by contradiction and the uniform boundednessprinciple (see, e.g., [45, Theorems 2.4 and 2.5]) as above, one also checks that the union of any net ( K j ) j ∈ J ⊆ K ( X ∗ ) that weak ∗ -Hausdorff converges must be norm-bounded.Because of Lemma 6.2, we say that the (nonempty) subhyperspaces K ( X ∗ ) and K c ( X ∗ ) areweak ∗ -Hausdorff- immeasurable with respect to each other. Corollary 6.3 (Weak ∗ -Hausdorff-clopen subhyperspaces) Let X be a Banach space. Then, K ( X ∗ ) is a weak ∗ -Hausdorff-closed subset of F ( X ∗ ) . Proof.
The assertion is a consequence of Lemma 6.2. Note that a subset of a topological space isclosed iff it contains the set of its accumulation points, by [54, Chapter 1, Theorem 5]. The accumu-lation points of a set are precisely the limits of nets whose elements are in this set, by [54, Chapter 2,Theorem 2].Note that K ( X ∗ ) is also a connected hyperspace: Lemma 6.4 ( K ( X ∗ ) as connected subhyperspace) Let X be a Banach space. Then, K ( X ∗ ) is convex and path-connected. Proof.
Take any K , K ∈ K ( X ∗ ) . Define the map f from [0 , to K ( X ∗ ) by f ( λ ) . = { (1 − λ ) σ + λσ : σ ∈ K , σ ∈ K } , λ ∈ [0 , . K ( X ∗ ) is convex.) By Definition 6.1, for any λ , λ ∈ [0 , , d ( A ) H ( f ( λ ) , f ( λ )) ≤ | λ − λ | max σ ∈ ( K − K ) | σ ( A ) | , A ∈ X . So, the map f is a continuous function from [0 , to K ( X ∗ ) with f (0) = K and f (1) = K . There-fore, K ( X ∗ ) is path-connected. The image under a continuous map of a connected set is connectedand, by [54, Chapter 1, Theorem 21], K ( X ∗ ) , being path-connected, is connected.Note that one can prove that K ( X ∗ ) is even a connected component of F ( X ∗ ) . There are pos-sibly many disconnected components, or even non-trivial weak ∗ -Hausdorff-clopen subsets of F ( X ∗ ) ,associated with different directions (characterized by some A ∈ X ) where the weak ∗ -closed sets F ∈ K c ( X ∗ ) are unbounded. This would lead to a whole collection of weak ∗ -Hausdorff-clopen sets,which could be used to form a Boolean algebra whose lattice operations are given by the union andintersection, as is usual in mathematical logics . This is far from the scope of the article and we thusrefrain from doing such a study here.Meanwhile, note that the weak ∗ -Hausdorff-closed set K ( X ∗ ) of all nonempty weak ∗ -closed norm-bounded subsets of X ∗ is nothing else than the set of all nonempty weak ∗ -compact subsets: Lemma 6.5 (Weak ∗ -compactness vs. norm-boundedness) Let X be a Banach space. Then, K ( X ∗ ) = { K ⊆ X ∗ : K = ∅ is weak ∗ -compact } . Proof.
The proof of the norm-boundedness of a weak ∗ -compact set is completely standard (see,e.g., [58, Proposition 1.2.9]) and is given here only for completeness: Take any weak ∗ -compact set K ⊆ X ∗ and use, for any A ∈ X , the weak ∗ -continuity of the map ˆ A : σ σ ( A ) from X ∗ to C (cf. (15) and (21)) to show that σ ( K ) is a bounded set, by weak ∗ compactness of K . Then, thenorm-boundedness of any weak ∗ -compact set is a consequence of the uniform boundedness principle,see, e.g., [45, Theorems 2.4 and 2.5]. Since X ∗ is a Hausdorff space (see, e.g., [60, Theorem 10.8]),by [54, Chapter 5, Theorem 7], it follows that weak ∗ -compact set are weak ∗ -closed and norm-boundedsubsets of X ∗ . On the other hand, by the Banach-Alaoglu theorem [45, Theorem 3.15], weak ∗ -closedand norm-bounded subsets of X ∗ are also weak ∗ -compact and the assertion follows.By Lemma 6.5, for any K, ˜ K ∈ K ( X ∗ ) , the suprema and infima in (82) become respectivelymaxima and minima. In this case, Definition 6.1 is the same as Definition 2.3, extended to all weak ∗ -compact sets. Of course, by Lemma 6.5, K ( X ∗ ) includes the hyperspace CK ( X ∗ ) . = { K ⊆ X ∗ : K = ∅ is convex and weak ∗ -compact } ⊆ K ( X ∗ ) ⊆ F ( X ∗ ) (87)of all nonempty convex weak ∗ -compact subsets of X ∗ , already defined by (7) and used in Section 2.3. One fundamental property one shall ask regarding the hyperspace F ( X ∗ ) (or K ( X ∗ ) ) is whether it isa Hausdorff space, with respect to the weak ∗ -Hausdorff hypertopology, or not. The answer is negative for real Banach spaces of dimension greater than 1, as demonstrated in the next lemma: Lemma 6.6 (Non-weak ∗ -Hausdorff-separable points) Let X be a real Banach space. Take any set K ∈ CK ( X ∗ ) with weak ∗ -path-connected weak ∗ -closedset E ( K ) ⊆ K of extreme points . Then, E ( K ) ∈ K ( X ∗ ) and d ( A ) H ( K, E ( K )) = 0 for any A ∈ X . That is, a maximal connected subset. See Stone’s representation theorem for Boolean algebras. Cf. the Krein-Milman theorem [45, Theorem 3.23]. roof. Let X be a real Banach space. Recall that any A ∈ X defines a weak ∗ -continuous linearfunctional ˆ A : X ∗ → R by ˆ A ( σ ) . = σ ( A ) , σ ∈ X ∗ . See (21). Observe next that d ( A ) H ( K, E ( K )) = max (cid:26) max x ∈ ˆ A ( K ) min x ∈ ˆ A ( E ( K )) | x − x | , max x ∈ ˆ A ( E ( K )) min x ∈ ˆ A ( K ) | x − x | (cid:27) . (88)The right hand side is nothing else than the Hausdorff distance (81) between the sets ˆ A ( K ) and ˆ A ( E ( K )) , where the metric used in Y = R is the absolute-value distance. Now, clearly, ˆ A ( E ( K )) ⊆ ˆ A ( K ) ⊆ h min ˆ A ( K ) , max ˆ A ( K ) i . (89)By the Bauer maximum principle [60, Lemma 10.31] together with the affinity and weak ∗ -continuityof ˆ A , min ˆ A ( K ) = min ˆ A ( E ( K )) and max ˆ A ( K ) = max ˆ A ( E ( K )) . In particular, we can rewrite (89) as ˆ A ( E ( K )) ⊆ ˆ A ( K ) ⊆ h min ˆ A ( E ( K )) , max ˆ A ( E ( K )) i . (90)Since E ( K ) is, by assumption, path-connected in the weak ∗ topology, there is a weak ∗ -continuouspath γ : [0 , → E ( K ) from a minimizer to a maximizer of ˆ A in E ( K ) . By weak ∗ -continuity of ˆ A , itfollows that h min ˆ A ( E ( K )) , max ˆ A ( E ( K )) i = ˆ A ◦ γ ([0 , ⊆ ˆ A ( E ( K )) and we infer from (90) that ˆ A ( E ( K )) = ˆ A ( K ) = h min ˆ A ( K ) , max ˆ A ( K ) i = h min ˆ A ( E ( K )) , max ˆ A ( E ( K )) i . Together with (88), this last equality obviously leads to the assertion. Note that E ( K ) ∈ K ( X ∗ ) sinceit is, by assumption, a weak ∗ -closed subset of the weak ∗ -compact set K (Lemma 6.5). Corollary 6.7 (Non-Hausdorff hyperspaces)
Let X be a real Banach space of dimension greater than 1. Then, F ( X ∗ ) and K ( X ∗ ) are non-Hausdorff topological spaces. Proof.
This corollary is a direct consequence of Lemma 6.6 by observing that the dual of a realBanach space of dimension greater than 1 contains a two-dimensional closed disc.As a consequence, the Hausdorff property of the hyperspace F ( X ∗ ) does not hold true, in gen-eral. A restriction to the sub-hyperspace K ( X ∗ ) is also not sufficient to get the separation property.This fact, described in Lemma 6.6, also appears for other well-established hypertopologies, whichcannot distinguish a set from its closed convex hull. The so-called scalar topology for closed sets isa good example of this phenomenon, as explained in [58, Section 4.3]. Actually, similar to the scalartopology, CK ( X ∗ ) is a Hausdorff hyperspace. To get a better intuition of this fact, the followingproposition is instructive: Proposition 6.8 (Separation of the weak ∗ -closed convex hull) Let X be a Banach space and K , K ∈ K ( X ∗ ) . If d ( A ) H ( K , K ) = 0 for all A ∈ X , then co K =co K , where co F denotes the weak ∗ -closure of the convex hull of any set F ∈ F ( X ∗ ) . roof. Pick any weak ∗ -compact sets K , K satisfying d ( A ) H ( K , K ) = 0 for all A ∈ X . Let σ ∈ K .By Definition 6.1, it follows that min σ ∈ K | ( σ − σ ) ( A ) | = 0 , A ∈ X . (91)The dual space X ∗ of the Banach space X is a locally convex (Hausdorff) space in the weak ∗ topologyand its dual is X . Note also that co K is convex and weak ∗ -compact, because it is a norm-boundedweak ∗ -closed subset of X ∗ , see Lemma 6.5. Since { σ } is a convex weak ∗ -closed set, if σ / ∈ co K then we infer from the Hahn-Banach separation theorem [45, Theorem 3.4 (b)] the existence of A ∈X and x , x ∈ R such that max σ ∈ co K Re { σ ( A ) } < x < x < Re { σ ( A ) } , (92)which contradicts (91) for A = A . As a consequence, σ ∈ co K and hence, K ⊆ co K . Thisin turn yields co K ⊆ co K . By switching the role of the weak ∗ -compact sets, we thus deduce theassertion.Proposition 6.8 motivates the introduction of the weak ∗ -closed convex hull operator : Definition 6.9 (The weak ∗ -closed convex hull operator) The weak ∗ -closed convex hull operator is the map co from F ( X ∗ ) to itself defined by co ( F ) . = co F , F ∈ F ( X ∗ ) , where co F denotes the weak ∗ -closure of the convex hull of F or, equivalently, the intersection of allweak ∗ -closed convex sets containing F . It is a closure (or hull) operator [85, Definition 5.1] since it satisfies the following properties: • For any F ∈ F ( X ∗ ) , F ⊆ co( F ) (extensive); • For any F ∈ F ( X ∗ ) , co (co( F )) = co( F ) (idempotent); • For any F , F ∈ F ( X ∗ ) such that F ⊆ F , co ( F ) ⊆ co( F ) (isotone).Such a closure operator has probably been used in the past. It is a composition of (i) an algebraic (or finitary) closure operator [85, Definition 5.4] defined by F co F with (ii) a topological (orKuratowski) closure operator [54, Chapter 1, p.43] defined by F F on F ( X ∗ ) .As is usual, weak ∗ -closed subsets F ∈ F ( X ∗ ) satisfying F = co( F ) are by definition co -closed sets. In the light of Proposition 6.8, it is natural to propose the set co ( K ( X ∗ )) as the Hausdorffhyperspace to consider. This set is nothing else than the set of all nonempty convex weak ∗ -compactsets defined by (7) or (87): co ( K ( X ∗ )) = CK ( X ∗ ) . (93)We thus deduce the following assertion: Corollary 6.10 ( CK ( X ∗ ) as an Hausdorff hyperspace) Let X be a Banach space. Then, CK ( X ∗ ) is a Hausdorff hyperspace. Proof.
This is a direct consequence of Proposition 6.8.Note additionally that the restriction of the weak ∗ -closed convex hull operator to K ( X ∗ ) is aweak ∗ -Hausdorff continuous map from the hyperspace K ( X ∗ ) to CK ( X ∗ ) :45 roposition 6.11 (Weak ∗ -Hausdorff continuity of the weak ∗ -closed convex hull operator) Let X be a Banach space. Then, co preserves the set (84) of all nonempty weak ∗ -closed norm-unbounded subsets of X ∗ , i.e., co ( K c ( X ∗ )) ⊆ K c ( X ∗ ) . = F ( X ∗ ) \ K ( X ∗ ) , (94) and co restricted to K ( X ∗ ) is a weak ∗ -Hausdorff continuous map onto CK ( X ∗ ) . Proof.
Let X be a Banach space. Equation (94) and surjectivity of co seen as a map from K ( X ∗ ) to CK ( X ∗ ) are both obvious, by Definition 6.9 and (93). Now, take any weak ∗ -Hausdorff convergentnet ( K j ) j ∈ J ⊆ K ( X ∗ ) with limit K ∞ ∈ K ( X ∗ ) . Note that max σ ∈ co( K ∞ ) min ˜ σ ∈ co( K j ) | ( σ − ˜ σ ) ( A ) | = sup σ ∈ co K ∞ min ˜ σ ∈ co( K j ) | ( σ − ˜ σ ) ( A ) | , A ∈ X , (95)because, for any A ∈ X , j ∈ J , σ , σ ∈ co ( K ∞ ) and ˜ σ ∈ co ( K j ) , || ( σ − ˜ σ ) ( A ) | − | ( σ − ˜ σ ) ( A ) || ≤ | ( σ − σ ) ( A ) | , which yields (cid:12)(cid:12)(cid:12)(cid:12) min ˜ σ ∈ K j | ( σ − ˜ σ ) ( A ) | − min ˜ σ ∈ K j | ( σ − ˜ σ ) ( A ) | (cid:12)(cid:12)(cid:12)(cid:12) ≤ | ( σ − σ ) ( A ) | for any A ∈ X , j ∈ J and σ , σ ∈ co ( K ∞ ) . Fix n ∈ N , σ , . . . , σ n ∈ K ∞ and parameters λ , . . . , λ n ∈ [0 , such that n X k =1 λ k = 1 . For any A ∈ X and k ∈ { , . . . , n } , we define ˜ σ k,j ∈ K j such that min ˜ σ ∈ K j | ( σ k − ˜ σ ) ( A ) | = | ( σ k − ˜ σ k,j ) ( A ) | . Then, for all j ∈ J , min ˜ σ ∈ co( K j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X k =1 λ k σ k − ˜ σ ! ( A ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ n X k =1 λ k | ( σ k − ˜ σ k,j ) ( A ) | ≤ max σ ∈ K ∞ min ˜ σ ∈ K j | ( σ − ˜ σ ) ( A ) | . By using (95), we then deduce that, for all j ∈ J , max σ ∈ co( K ∞ ) min ˜ σ ∈ co( K j ) | ( σ − ˜ σ ) ( A ) | ≤ max σ ∈ K ∞ min ˜ σ ∈ K j | ( σ − ˜ σ ) ( A ) | , A ∈ X . (96)By switching the role of K ∞ and K j for every j ∈ J , we also arrive at the inequality max ˜ σ ∈ co( K j ) min σ ∈ co( K ∞ ) | ( σ − ˜ σ ) ( A ) | ≤ max ˜ σ ∈ K j min σ ∈ K ∞ | ( σ − ˜ σ ) ( A ) | , A ∈ X . (97)Since ( K j ) j ∈ J converges in the weak ∗ -Hausdorff hypertopology to K ∞ , Inequalities (96)-(97) com-bined with Definition 6.1 yield the weak ∗ -Hausdorff convergence of (co ( K j )) j ∈ J to co ( K ∞ ) . By [54,Chapter 3, Theorem 1], co restricted to K ( X ∗ ) is a weak ∗ -Hausdorff continuous map onto CK ( X ∗ ) . Corollary 6.12 ( CK ( X ∗ ) as a connected, weak ∗ -Hausdorff-closed set) Let X be a Banach space. Then, CK ( X ∗ ) is a convex, path-connected, weak ∗ -Hausdorff-closedsubset of K ( X ∗ ) . roof. By Corollary 6.10, CK ( X ∗ ) endowed with the weak ∗ -Hausdorff hypertopology is a Hausdorffspace. Hence, by [54, Chapter 2, Theorem 3], each convergent net in this space converges in theweak ∗ -Hausdorff hypertopology to at most one point, which, by Proposition 6.11, must be a convexweak ∗ -compact set. Additionally, by Lemma 6.4, Proposition 6.11 and the fact that the image undera continuous map of a path-connected space is path-connected, CK ( X ∗ ) is also path-connected.Convexity of CK ( X ∗ ) is also obvious.As is usual, the weak ∗ -closed convex hull operator co yields a notion of compactness, defined asfollows: A set K ∈ F ( X ∗ ) is co -compact iff it is co -closed and each family of co -closed subsets of K which has the finite intersection property has a non-empty intersection. Compare this definitionwith [54, Chapter 5, Theorem 1]. The set CK ( X ∗ ) of all nonempty convex weak ∗ -compact setsdefined by (7) or (87) is precisely the set of co -compact sets: Proposition 6.13 ( CK ( X ∗ ) as the space of co -compact sets) Let X be a Banach space. Then, CK ( X ∗ ) = { K ∈ F ( X ∗ ) : K is co -compact } . Proof.
By [54, Chapter 5, Theorem 1], we clearly have CK ( X ∗ ) ⊆ { K ∈ F ( X ∗ ) : K is co -compact } . Conversely, take any co -compact element K ∈ F ( X ∗ ) . If K is not norm-bounded, then we deducefrom the uniform boundedness principle [45, Theorems 2.4 and 2.5] the existence of A ∈ X such that ˆ A ( K ) ⊆ C is not bounded, where we recall that ˆ A : X ∗ → C is the weak ∗ -continuous (complex)linear functional defined by (21). Without loss of generality, assume that Re { ˆ A ( K ) } is not boundedfrom above. Define for every n ∈ N the set K n . = n σ ∈ K : Re { ˆ A ( σ ) } ≥ n o . Clearly, by convexity of K , K n is a convex weak ∗ -closed subset of K and the family ( K n ) n ∈ N has thefinite intersection property, but, by construction, \ n ∈ N K n = ∅ . (The intersection of preimages is the preimage of the intersection.) This contradicts the fact that K is co -compact. Therefore, K is norm-bounded and, being co -compact, it is also weak ∗ -closed andconvex. Consequently, K ∈ CK ( X ∗ ) (see, e.g., Equation 8).The last proposition goes beyond the specific topic of the present article, and the proof of theweak ∗ -Hausdorff density of convex weak ∗ -compact sets with dense extreme boundary. This is how-ever discussed here because, like (93), it is an elegant abstract characterization of CK ( X ∗ ) , onlygiven in terms of a closure operator, namely the weak ∗ -closed convex hull operator. It demonstratesconnections with other mathematical fields, in particular with mathematical logics where fascinatingapplications of closure operators have been developed, already by Tarski himself during the 1930’s. ∗ -Hausdorff Hyperconvergence In this subsection, we study weak ∗ -Hausdorff convergent nets. Even if only the hyperspace CK ( X ∗ ) of all nonempty convex weak ∗ -compact sets is Hausdorff (Corollary 6.10), we study the convergencewithin the hyperspace K ( X ∗ ) of all nonempty, weak ∗ -closed and norm-bounded subsets of X ∗ . Recallthat K ( X ∗ ) is a (path-) connected weak ∗ -Hausdorff-closed subset of F ( X ∗ ) , by Corollary 6.3 andLemma 6.4. 47t is instructive to relate weak ∗ -Hausdorff limits of nets to lower and upper limits of sets `a laPainlev´e [61, § lower limit of any net ( K j ) j ∈ J of subsets of X ∗ is defined by Li ( K j ) j ∈ J . = { σ ∈ X ∗ : σ is a weak ∗ limit of a net ( σ j ) j ∈ J with σ j ∈ K j for all j ∈ J } , (98)while its upper limit equals Ls ( K j ) j ∈ J . = { σ ∈ X ∗ : σ is a weak ∗ accumulation point of ( σ j ) j ∈ J with σ j ∈ K j for all j ∈ J } . (99)Clearly, Li( K j ) j ∈ J ⊆ Ls( K j ) j ∈ J . If Li( K j ) j ∈ J = Ls( K j ) j ∈ J then ( K j ) j ∈ J is said to be convergent tothis set. See [61, §
29, I, III, VI], which however defines Li and Ls within metric spaces. This refersin the literature to the Kuratowski or Kuratowski-Painlev´e convergence, see e.g. [86, Appendix B]and [58, Section 5.2]. By [45, Theorem 1.22], if X is an infinite-dimensional space, then its dual X ∗ , endowed with the weak ∗ or norm topology, is not locally compact. In this case, the Kuratowski-Painlev´e convergence is not topological [86, Theorem B.3.2]. See also [58, Chapter 5], in particular[58, Theorem 5.2.6 and following discussions] which relates the Kuratowski-Painlev´e convergence tothe so-called Fell topology.We start by proving the weak ∗ -Hausdorff convergence of monotonically increasing nets which arebounded from above within K ( X ∗ ) : Proposition 6.14 (Weak ∗ -Hausdorff hyperconvergence of increasing nets) Let X be a Banach space. Any increasing net ( K j ) j ∈ J ⊆ K ( X ∗ ) such that K . = [ j ∈ J K j ∈ K ( X ∗ ) F ( X ∗ ) (100) (with respect to the weak ∗ closure) converges in the weak ∗ -Hausdorff hypertopology to the Kuratowski-Painlev´e limit K = Li ( K j ) j ∈ J = Ls ( K j ) j ∈ J . Proof.
Let ( K j ) j ∈ J ⊆ K ( X ∗ ) be any increasing net, i.e., K j ⊆ K j whenever j ≺ j , satisfying(100). Because K ∈ K ( X ∗ ) , it is norm-bounded. By the convergence of increasing bounded nets ofreal numbers, it follows that, for any A ∈ X , lim J max ˜ σ ∈ K j min σ ∈ K | (˜ σ − σ ) ( A ) | = sup j ∈ J max ˜ σ ∈ K j min σ ∈ K | (˜ σ − σ ) ( A ) | ≤ max ˜ σ ∈ K min σ ∈ K | (˜ σ − σ ) ( A ) | = 0 . Therefore, by Definition 6.1, if lim sup J max σ ∈ K min ˜ σ ∈ K j | (˜ σ − σ ) ( A ) | = 0 , A ∈ X , (101)then the increasing net ( K j ) j ∈ J converges in K ( X ∗ ) to K , which clearly equals the Kuratowski-Painlev´e limit of the net. To prove (101), assume by contradiction the existence of ε ∈ R + suchthat lim sup J max σ ∈ K min ˜ σ ∈ K j | (˜ σ − σ ) ( A ) | ≥ ε ∈ R + (102)for some fixed A ∈ X . For any j ∈ J , take σ j ∈ K such that max σ ∈ K min ˜ σ ∈ K j | (˜ σ − σ ) ( A ) | = min ˜ σ ∈ K j | (˜ σ − σ j ) ( A ) | . (103) The idea of upper and lower limits is due to Painlev´e, as acknowledged by Kuratowski himself in [61, §
29, Footnote1, p. 335]. We thus use the name Kuratowski-Painlev´e convergence.
48y weak ∗ -compactness of K (Lemma 6.5), there is a subnet ( σ j l ) l ∈ L converging in the weak ∗ topologyto σ ∞ ∈ K . Via Equation (103) and the triangle inequality, we then get that, for any l ∈ L , max σ ∈ K min ˜ σ ∈ K jl | (˜ σ − σ ) ( A ) | ≤ | ( σ j l − σ ∞ ) ( A ) | + min ˜ σ ∈ K jl | (˜ σ − σ ∞ ) ( A ) | . By (100) and the fact that ( K j ) j ∈ J ⊆ K ( X ∗ ) is an increasing net, it follows that lim L max σ ∈ K min ˜ σ ∈ K jl | (˜ σ − σ ) ( A ) | = 0 . (104)By the convergence of decreasing bounded nets of real numbers, note that lim sup J max σ ∈ K min ˜ σ ∈ K j | (˜ σ − σ ) ( A ) | = lim inf J max σ ∈ K min ˜ σ ∈ K j | (˜ σ − σ ) ( A ) | and hence, (104) contradicts (102). As a consequence, Equation (101) holds true.Nonmonotone weak ∗ -Hausdorff convergent nets in K ( X ∗ ) are not trivial to study, in general. Inthe next proposition we give preliminary, but completely general, results on limits of convergent nets. Proposition 6.15 (Weak ∗ -Hausdorff hypertopology vs. upper and lower limits) Let X be a Banach space and K ∞ ∈ K ( X ∗ ) any weak ∗ -Hausdorff limit of a convergent net ( K j ) j ∈ J ⊆ K ( X ∗ ) . Then, Li ( K j ) j ∈ J ⊆ co ( K ∞ ) and K ∞ ⊆ co (Ls( K j ) j ∈ J ) , where we recall that co is the weak ∗ -closed convex hull operator (Definition 6.9). Proof.
Let X be a Banach space and ( K j ) j ∈ J ⊆ K ( X ∗ ) any net converging to K ∞ . Assume withoutloss of generality that Li ( K j ) j ∈ J is nonempty. Let σ ∞ ∈ Li ( K j ) j ∈ J , which is, by definition, theweak ∗ limit of a net ( σ j ) j ∈ J with σ j ∈ K j for all j ∈ J . Then, for any A ∈ X , min σ ∈ K ∞ | ( σ − σ ∞ ) ( A ) | ≤ | ( σ j − σ ∞ ) ( A ) | + min σ ∈ K ∞ {| ( σ − σ j ) ( A ) |} . Taking this last inequality in the limit with respect to J and using Definition 6.1, we deduce that min σ ∈ K ∞ | ( σ − σ ∞ ) ( A ) | = 0 , A ∈ X . (105)If σ ∞ / ∈ co ( K ∞ ) then, as it is done to prove (92), we infer from the Hahn-Banach separation theorem[45, Theorem 3.4 (b)] the existence of A ∈ X and x , x ∈ R such that max σ ∈ co( K ∞ ) Re { σ ( A ) } < x < x < Re { σ ∞ ( A ) } , which contradicts (105) for A = A . As a consequence, σ ∞ ∈ co ( K ∞ ) and, hence, Li ( K j ) j ∈ J ⊆ co ( K ∞ ) .Conversely, let σ ∞ ∈ K ∞ . Since K ∞ is by definition the limit of ( K j ) j ∈ J (see Definition 6.1), wededuce that lim J min σ ∈ K j | ( σ − σ ∞ ) ( A ) | = 0 , A ∈ X . By combining this equality with Lemma 6.2 and the Banach-Alaoglu theorem [45, Theorem 3.15],for any A ∈ X , there is σ A ∈ Ls ( K j ) j ∈ J such that σ A ( A ) = σ ∞ ( A ) . Consequently, one infers from the Hahn-Banach separation theorem [45, Theorem 3.4 (b)] that σ ∞ belongs to the weak ∗ -closed convex hull of the upper limit Ls ( K j ) j ∈ J .Applied to nonempty convex weak ∗ -compact subsets of the dual space X ∗ , Proposition 6.15 readsas follows: 49 orollary 6.16 (Weak ∗ -Hausdorff hypertopology and convexity vs. upper and lower limits) Let X be a Banach space and K ∞ ∈ CK ( X ∗ ) any weak ∗ -Hausdorff limit of a convergent net ( K j ) j ∈ J ⊆ CK ( X ∗ ) . Then, Li( K j ) j ∈ J = co (Li( K j ) j ∈ J ) ⊆ K ∞ ⊆ co (Ls( K j ) j ∈ J ) . Proof.
The assertion is an obvious application of Proposition 6.15 to the subset CK ( X ∗ ) ⊆ K ( X ∗ ) together with the idempotency of the weak ∗ -closed convex hull operator co . Note that Li( K j ) j ∈ J is aconvex set. We are interested in investigating metrizable sub-hyperspaces of F ( X ∗ ) . Metrizable topologicalspaces are Hausdorff, so, in the light of Corollaries 6.7 and 6.10, we restrict our analysis on the Haus-dorff hyperspace CK ( X ∗ ) of all nonempty convex weak ∗ -compact subsets of X ∗ , already defined byEquation (7) or (87).For a separable Banach space X , we show how the well-known metrizability of the weak ∗ topol-ogy on balls of X ∗ leads to the metrizability of the weak ∗ -Hausdorff hypertopology on uniformlynorm-bounded subsets of CK ( X ∗ ) : Let CK D ( X ∗ ) . = { K ∈ CK ( X ∗ ) : K ⊆ B (0 , D ) } (106)where B (0 , D ) . = { σ ∈ X ∗ : k σ k X ∗ ≤ D } ⊆ X ∗ (107)is the norm-closed ball of radius D ∈ R + in X ∗ . If X is separable then the weak ∗ topology ismetrizable on any ball B (0 , D ) , D ∈ R + , by the Banach-Alaoglu theorem [45, Theorem 3.15] and[45, Theorem 3.16]. Take any countable dense set ( A n ) n ∈ N of the unit ball of X and define the metric d ( σ , σ ) . = X n ∈ N − n | ( σ − σ ) ( A n ) | , σ , σ ∈ X ∗ . (108)This metric is well-defined and induces the weak ∗ topology on B (0 , D ) . Denote by d H the Hausdorffdistance between two elements K , K ∈ CK D ( X ∗ ) , associated with the metric d , as defined by (81),that is , d H ( K , K ) . = max (cid:26) max σ ∈ K min σ ∈ K d ( σ , σ ) , max σ ∈ K min σ ∈ K d ( σ , σ ) (cid:27) . (109)This Hausdorff distance induces the weak ∗ -Hausdorff hypertopology on CK D ( X ∗ ) : Theorem 6.17 (Complete metrizability of the weak ∗ -Hausdorff hypertopology) Let X be a separable Banach space and D ∈ R + . The family (cid:8) { K ∈ CK D ( X ∗ ) : d H ( K , K ) < r } : r ∈ R + , K ∈ CK D ( X ∗ ) (cid:9) is a basis of the weak ∗ -Hausdorff hypertopology of CK D ( X ∗ ) . Additionally, CK D ( X ∗ ) is weak ∗ -Hausdorff-compact and completely metrizable. Minima in (109) directly come from the compactness of sets and the continuity of d . The following maxima in (109)result from the compactness of sets and the fact that the minimum over a continuous map defines an upper semicontinuousfunction. roof. Recall that a topology is finer than a second one iff any convergent net of the first topologyconverges also in the second topology to the same limit. See, e.g., [54, Chapter 2, Theorems 4, 9].We first show that the topology induced by the Hausdorff metric d H is finer than the weak ∗ -Hausdorffhypertopology of CK D ( X ∗ ) at fixed radius D ∈ R + : Take any net ( K j ) j ∈ J converging in CK D ( X ∗ ) to K in the topology induced by the Hausdorff metric (109). Let A ∈ X and assume without lossof generality that k A k X ≤ . By density of ( A n ) n ∈ N in the unit ball of X , for any ε ∈ R + , there is n ∈ N such that, for all j ∈ J , d ( A ) H ( K, K j ) ≤ ε + d ( A n ) H ( K, K j ) ≤ ε + 2 n d H ( K, K j ) . Thus, the net ( K j ) j ∈ J converges to K also in the weak ∗ -Hausdorff hypertopology.Endowed with the Hausdorff metric topology, the space of closed subsets of a compact metricspace is compact, by [58, Theorem 3.2.4]. In particular, by weak ∗ compactness of norm-closed balls, CK D ( X ∗ ) endowed with the Hausdorff metric d H is a compact hyperspace. By Corollary 6.12, CK D ( X ∗ ) is closed with respect to the weak ∗ -Hausdorff hypertopology, and thus closed with respectto the topology induced by d H , because this topology is coarser than the weak ∗ -Hausdorff hyper-topology, as proven above. Hence, CK D ( X ∗ ) is also compact with respect to the topology inducedby d H . Since the weak ∗ -Hausdorff hypertopology is a Hausdorff topology (Corollary 6.10), as it iswell-known [45, Section 3.8 (a)], both topologies must coincide: Take any subset K ⊆ CK D ( X ∗ ) which is closed with respect to the Hausdorff metric d H . By compactness of ( CK D ( X ∗ ) , d H ) , K iscompact with respect to the Hausdorff metric d H (see, e.g., [54, Chapter 5, p. 140]) and, hence, alsowith respect to the weak ∗ -Hausdorff hypertopology. Because any compact set in a Hausdorff space isclosed [54, Chapter 5, Theorem 7], by Corollary 6.10, K is closed with respect to the weak ∗ -Hausdorffhypertopology.Note that Theorem 6.17 is similar to the assertion [58, End of p. 91]. It leads to a strong improvementof Proposition 6.14 and Corollary 6.16: Corollary 6.18 (Weak ∗ -Hausdorff hypertopology and Kuratowski-Painlev´e convergence) Let X be a separable Banach space. Then any weak ∗ -Hausdorff convergent net ( K j ) j ∈ J ⊆ CK ( X ∗ ) converges to the Kuratowski-Painlev´e limit K ∞ = Li( K j ) j ∈ J = Ls( K j ) j ∈ J ∈ CK ( X ∗ ) . Proof.
Recall that CK ( X ∗ ) ⊆ K ( X ∗ ) , see (87). By Lemma 6.2, the union of any weak ∗ -Hausdorffconvergent net in CK ( X ∗ ) is norm-bounded and, as a consequence, we can restrict, without loss ofgenerality, the study of weak ∗ -Hausdorff hyperconvergent nets to the sub-hyperspace CK D ( X ∗ ) forsome D ∈ R + . By Theorem 6.17 the weak ∗ -Hausdorff hypertopology is induced by the Hausdorffdistance d H defined by (109). The assertion thus follows from [61, §
29, Section IX, Theorem 2].
By Corollary 6.10, recall that CK ( X ∗ ) is a weak ∗ -Hausdorff-closed subset of K ( X ∗ ) . Let D . = n K ∈ CK ( X ∗ ) : K = E ( K ) o ⊆ CK ( X ∗ ) (110)be the subset of all K ∈ CK ( X ∗ ) with weak ∗ -dense set E ( K ) of extreme points (cf. the Krein-Milman theorem [45, Theorem 3.23]).Recall that the so-called exposed points are particular examples of extreme ones: a point σ ∈ K in a convex subset K ⊆ X ∗ is exposed if there is A ∈ X such that the real part of the weak ∗ -continuous functional ˆ A : σ σ ( A ) from X ∗ to C (cf. (21)) takes its unique maximum on K at σ ∈ K . Considering exposed points instead of general extreme points is technically convenientbecause of the weak ∗ -density of the set of exposed points in the set of extreme points [87, Theorem6.2] is an important ingredient to show that D is a G δ subset of CK ( X ∗ ) :51 roposition 6.19 ( D as a G δ set) Let X be a separable Banach space. Then D is a G δ subset of CK ( X ∗ ) . Proof.
Let X be a separable Banach space. For any D ∈ R + , we can use the metric d defined by(108) and generating the weak ∗ topology on the norm-closed ball B (0 , D ) of radius D , defined by(107). For any D ∈ R + , we denote by B ( ω, r ) . = { σ ∈ B (0 , D ) : d ( ω, σ ) < r } (111)the weak ∗ -open ball of radius r ∈ R + centered at ω ∈ B (0 , D ) . Then, for any D ∈ R + and m ∈ N , let F D,m be the set of all nonempty convex weak ∗ -compact subsets K ⊆ B (0 , D ) such that B ( ω, /m ) ∩ E ( K ) = ∅ for some ω ∈ K , i.e., F D,m . = { K ∈ CK D ( X ∗ ) : ∃ ω ∈ K, B ( ω, /m ) ∩ E ( K ) = ∅} ⊆ CK D ( X ∗ ) . (112)Recall again that E ( K ) is the nonempty set of extreme points of K (cf. the Krein-Milman theorem [45,Theorem 3.23]). Now, by Equation (87), observe that the complement of D (110) in CK ( X ∗ ) equals CK ( X ∗ ) \D = [ D,m ∈ N F D,m . (113)Therefore, D is a G δ subset of CK ( X ∗ ) if F D,m is a weak ∗ -Hausdorff-closed set for any D, m ∈ N .By Theorem 6.17, the weak ∗ -Hausdorff hypertopology of CK D ( X ∗ ) is metrizable and CK D ( X ∗ ) ,being weak ∗ -Hausdorff-compact, is a weak ∗ -Hausdorff-closed subset of the Hausdorff hyperspace CK ( X ∗ ) (see Corollary 6.10 and [54, Chapter 5, Theorem 7]). So, fix D, m ∈ N and take any se-quence ( K n ) n ∈ N ⊆ F D,m converging with respect to the weak ∗ -Hausdorff hypertopology to K ∞ ∈ CK D ( X ∗ ) . For any n ∈ N , there is ω n ∈ K n such that B ( ω n , /m ) ∩ E ( K n ) = ∅ . By metrizabil-ity and weak ∗ compactness of the ball B (0 , D ) and Corollary 6.18, there is a subsequence ( ω n k ) k ∈ N converging to some ω ∞ ∈ K ∞ . Assume that, for some ε ∈ (0 , /m ) , there is σ ∞ ∈ E ( K ∞ ) such that d ( ω ∞ , σ ∞ ) ≤ m − ε . By the Mazur theorem (see, e.g., [87, Theorem 1.20]), the Straszewicz theorem extended to all weakAsplund spaces [87, Theorem 6.2] and the Milman theorem [60, Theorem 10.13], the set of exposedpoints of K ∞ is weak ∗ -dense in E ( K ∞ ) . As a consequence, we can assume without loss of generalitythat σ ∞ is an exposed point. In particular, there is A ∈ X such that max σ ∈ K ∞ Re { ˆ A ( σ ) } = ˆ A ( σ ∞ ) , (114)with σ ∞ being the unique maximizer in K ∞ . Recall that ˆ A is the map σ σ ( A ) from X ∗ to C (cf.(21)). Consider now the sets M n . = (cid:26) ˜ σ ∈ K n : max σ ∈ K n Re { ˆ A ( σ ) } = ˆ A (˜ σ ) (cid:27) , n ∈ N . By affinity and weak ∗ -continuity of the function ˆ A , together with the weak ∗ -compactness of K n , theset M n is a convex weak ∗ -compact subset of K n for any n ∈ N . In fact, M n is a (weak ∗ -closed)face of K n and thus, any extreme point of M n belongs to E ( K n ) . So, pick any extreme point σ n ∈ E ( K n ) of M n for each n ∈ N . Since max σ ∈ K n Re { ˆ A ( σ ) } − max ˜ σ ∈ K ∞ Re { ˆ A (˜ σ ) } = max σ ∈ K n min ˜ σ ∈ K ∞ Re { ˆ A ( σ − ˜ σ ) } ≤ max σ ∈ K n min ˜ σ ∈ K ∞ | ( σ − ˜ σ ) ( A ) | , max ˜ σ ∈ K ∞ Re { ˆ A (˜ σ ) } − max σ ∈ K n Re { ˆ A ( σ ) } = max ˜ σ ∈ K ∞ min σ ∈ K n Re { ˆ A (˜ σ − σ ) } ≤ max ˜ σ ∈ K ∞ min σ ∈ K n | ( σ − ˜ σ ) ( A ) | , It means that, if σ ∈ M n is a finite convex combination of elements σ j ∈ K n then all σ j ∈ M n .
52e deduce from Definition 2.3 and the weak ∗ -Hausdorff convergence of ( K n ) n ∈ N to K ∞ that lim n →∞ Re { ˆ A ( σ n ) } = lim n →∞ max σ ∈ K n Re { ˆ A ( σ ) } = max σ ∈ K ∞ Re { ˆ A ( σ ) } = ˆ A ( σ ∞ ) . Therefore, keeping in mind the convergence of the subsequence ( ω n k ) k ∈ N towards ω ∞ ∈ K ∞ , thereis a subsequence ( σ n k ( l ) ) l ∈ N of ( σ n k ) k ∈ N (itself being a subsequence of ( σ n ) n ∈ N ) converging to σ ∞ , asit is the unique maximizer of (114) and ˆ A is weak ∗ -continuous. Since, for any l ∈ N , d ( σ n k ( l ) , ω n k ( l ) ) ≤ d ( σ ∞ , ω ∞ ) + d ( ω ∞ , ω n k ( l ) ) + d ( σ n k ( l ) , σ ∞ ) ≤ m − ε + d ( ω ∞ , ω n k ( l ) ) + d ( σ n k ( l ) , σ ∞ ) with ε ∈ (0 , /m ) and σ n ∈ E ( K n ) for n ∈ N , we thus arrive at a contradiction. Therefore, K ∞ ∈F D,m . This means that F D,m is a weak ∗ -Hausdorff-closed set for any D, m ∈ N and hence, thecountable union (113) is a F σ set with complement being D . The assertion follows, as the complementof an F σ set is a G δ set.To show that D is weak ∗ -Hausdorff dense in the hyperspace CK ( X ∗ ) , like in the proof of [42,Theorem 4.3] and in contrast with [41], we design elements of D that approximate K ∈ CK ( X ∗ ) byusing a procedure that is very similar to the construction of the Poulsen simplex [62]. Note howeverthat Poulsen used the existence of orthonormal bases in infinite-dimensional Hilbert spaces . Here,the Hahn-Banach separation theorem [45, Theorem 3.4 (b)] replaces the orthogonality property com-ing from the Hilbert space structure. In all previous results [41, 42] on the density of convex compactsets with dense extreme boundary, the norm topology is used, while the primordial topology is herethe weak ∗ topology. In this context, the metrizability of weak ∗ and weak ∗ -Hausdorff topologies onnorm-closed balls is pivotal. See Theorem 6.17. We give now the precise assertion along with itsproof: Theorem 6.20 (Weak ∗ -Hausdorf density of D ) Let X be an infinite-dimensional separable Banach space. Then, D is a weak ∗ -Hausdorff dense subsetof CK ( X ∗ ) . Proof.
Let X be an infinite-dimensional separable Banach space and fix once and for all a convexweak ∗ -compact subset K ∈ CK ( X ∗ ) . The construction of convex weak ∗ -compact sets in D approx-imating K is done in several steps:Step 0: By Lemma 6.5, K belongs to some norm-closed ball B (0 , D ) of radius D ∈ R + , in otherwords, K ∈ CK D ( X ∗ ) , see (106)-(107). Therefore, we can use the metric d defined by (108)and generating the weak ∗ topology on B (0 , D ) . Then, for any fixed ε ∈ R + , there is a finite set { ω j } n ε j =1 ⊆ K , n ε ∈ N , such that K ⊆ n ε [ j =1 B ( ω j , ε ) , (115)where B ( ω, r ) ⊆ B (0 , D ) denotes the weak ∗ -open ball (111) of radius r ∈ R + centered at ω ∈ X ∗ .We then define the convex weak ∗ -compact set K . = co { ω , . . . , ω n ε } ⊆ span { ω , . . . , ω n ε } . (116)By (109) and (115), note that d H ( K, K ) ≤ ε . (117) In [62], Poulsen uses the Hilbert space ℓ ( N ) to construct his example of a convex compact set (in fact a simplex)with dense extreme boundary. B (0 , D ) is weak ∗ -separable, by its weak ∗ compactness (the Banach-Alaoglu theorem [45, Theorem 3.15]) and metrizability (cf. separability of X and [45, Theorem3.16]). Take any weak ∗ -dense countable set { ̺ ,k } k ∈ N of K . By infinite dimensionality of X ∗ , thereis σ ∈ X ∗ \ span { ω , . . . , ω n ε } with k σ k X ∗ = D . (118)As in the proof of Proposition 6.8, recall that X ∗ , endowed with the weak ∗ topology, is a locally con-vex (Hausdorff) space with X as its dual. Since { σ } is a convex weak ∗ -compact set and span { ω , . . . , ω n ε } is convex and weak ∗ -closed [45, Theorem 1.42], we infer from the Hahn-Banach separation theo-rem [45, Theorem 3.4 (b)] the existence of A ∈ X such that sup { Re { σ ( A ) } : σ ∈ span { ω , . . . , ω n ε }} < Re { σ ( A ) } . Since span { ω , . . . , ω n ε } is a linear space, observe that Re { σ ( A ) } = 0 , σ ∈ span { ω , . . . , ω n ε } . (119)Thus, by rescaling A ∈ X , we can assume without loss of generality that Re { σ ( A ) } = 1 . (120)Let ω n ε +1 . = (1 − λ ) ̟ + λ σ , with λ . = min (cid:8) , − D − ε (cid:9) , ̟ . = ̺ , ∈ K . (121)In contrast with the proof of [42, Theorem 4.3], we use a convex combination to automatically ensurethat k ω n ε +1 k X ∗ ≤ D , by convexity of the (norm-closed) ball B (0 , D ) . The inequality λ ≤ − D − ε yields d ( ω n ε +1 , ̟ ) ≤ k ω n ε +1 − ̟ k X ∗ ≤ − ε . (122)Define the new convex weak ∗ -compact set K . = co { ω , . . . , ω n ε +1 } ⊆ span { ω , . . . , ω n ε +1 } . Observe that ω n ε +1 is an exposed point of K , by (119) and (120). By (109), (116) and (122), notethat d H ( K , K ) ≤ − ε , which, by the triangle inequality and (117), yields d H ( K, K ) ≤ (cid:0) − (cid:1) ε (123)for an arbitrary (but previously fixed) ε ∈ R + .Step 2: Take any weak ∗ dense countable set { ̺ ,k } k ∈ N of K . By infinite dimensionality of X ∗ , thereis σ ∈ X ∗ \ span { ω , . . . , ω n ε +1 } with k σ k X ∗ = min (cid:8) D, − k A k − X λ (cid:9) . (124)As before, we deduce from the Hahn-Banach separation theorem [45, Theorem 3.4 (b)] the existenceof A ∈ X such that Re { σ ( A ) } = 1 and Re { σ ( A ) } = 0 , σ ∈ span { ω , . . . , ω n ε +1 } . (125)Let ω n ε +2 . = (1 − λ ) ̟ + λ σ , with λ . = min (cid:8) , − D − ε (cid:9) , ̟ . = ̺ , ∈ K . (126)54n this case, similar to Inequality (122), d ( ω n ε +2 , ̟ ) ≤ k ω n ε +2 − ̟ k X ∗ ≤ − ε . (127)Define the new convex weak ∗ -compact set K . = co { ω , . . . , ω n ε +2 } ⊆ span { ω , . . . , ω n ε +2 } . By (125), ω n ε +2 is an exposed point of K , but it is not obvious that the exposed point ω n ε +1 of K isstill an exposed point of K , with respect to A ∈ X . This property is a consequence of Re { ω n ε +2 ( A ) } = (1 − λ ) Re { ̟ ( A ) } + λ Re { σ ( A ) } < Re { ω n ε +1 ( A ) } = λ , (see (119), (121) and (126)), which holds true because Re { σ ( A ) } ≤ − λ < λ , by Equation (124). By (109), (123) and (127) together with the triangle inequality, d H ( K, K ) ≤ (cid:0) − + 2 − (cid:1) ε for an arbitrary (but previously fixed) ε ∈ R + .Step n → ∞ : We now iterate the above procedure, ensuring, at each step n ≥ , that the addition ofthe element ω n ε + n . = (1 − λ n ) ̟ n + λ n σ n , with λ n . = min (cid:8) , − ( n +1) D − ε (cid:9) , (128)in order to define the convex weak ∗ -compact set K n . = co { ω , . . . , ω n ε + n } ⊆ span { ω , . . . , ω n ε + n } , (129)does not destroy the property of the elements ω n ε +1 , . . . , ω n ε + n − being exposed. To this end, for any n ≥ , we choose σ n ∈ X ∗ \ span { ω , . . . , ω n ε + n − } such that k σ n k X ∗ = min (cid:8) D, − k A k − X λ , . . . , − k A n − k − X λ n − (cid:9) . (130)Compare with (118) and (124). Here, for any j ∈ { , . . . , n − } , A j ∈ X satisfies Re { σ j ( A j ) } = 1 and Re { σ ( A j ) } = 0 , σ ∈ span { ω , . . . , ω n ε + j − } . (131)Compare with (119)-(120) and (125). We also have to conveniently choose ̟ n ∈ K n − in order to getthe asserted weak ∗ density. Like in the proof of [42, Theorem 4.3] the sequence ( ̟ n ) n ∈ N is chosensuch that { ̟ n } n ∈ N = (cid:8) ̺ n,k (cid:9) n ∈ N ,k ∈ N and all the functionals ̺ n,k appear infinitely many times in the sequence ( ̟ n ) n ∈ N . In this case, weobtain a weak ∗ -dense set { ω n } n ∈ N in the convex weak ∗ -compact set K ∞ . = co {{ ω n } n ∈ N } ∈ CK D ( X ∗ ) , (132)which, by construction, satisfies d H ( K, K ∞ ) ≤ ∞ X n =0 − n ε = 2 ε ε ∈ R + .Step n = ∞ : It remains to verify that ω n ε + j , j ∈ N , are exposed points of K ∞ , whence K ∞ ∈ D . By(128) with ̟ n ∈ K n − (see (129)), for each natural number n ≥ j + 1 , there are α ( j ) n,j − , . . . , α ( j ) n,n ∈ [0 , and ρ ( j ) n ∈ co { ω , . . . , ω n ε + j − } such that α ( j ) n,j − + α ( j ) n,j + n X k = j +1 α ( j ) n,k λ k = 1 and ω n ε + n = α ( j ) n,j − ρ ( j ) n + α ( j ) n,j ω n ε + j + n X k = j +1 α ( j ) n,k λ k σ k . (133)Additionally, define α ( j ) n,k . = 1 for all natural numbers k ≥ n while α ( j ) n,k . = 0 for k ∈ N such that k ≤ j − . Using (130), (131) and (133), at fixed j ∈ N , we thus obtain that Re { ω n ε + n ( A j ) } = α ( j ) n,j Re { ω n ε + j ( A j ) } + n X k = j +1 α ( j ) n,k λ k Re { σ k ( A j ) }≤ λ j − − n X k = j +1 α ( j ) n,k λ k ! (134)for any n ≥ j + 1 , while, for any natural number n ≤ j − , Re { ω n ε + n ( A j ) } = 0 , using (131). Fix j ∈ N and let ω ∞ ∈ K ∞ be a solution to the variational problem max σ ∈ K ∞ Re { σ ( A j ) } = Re { ω ∞ ( A j ) } ≥ Re { ω n ε + j ( A j ) } = λ j . (135)( K ∞ is weak ∗ -compact.) By weak ∗ -density of { ω n } n ∈ N in K ∞ , there is a sequence ( ω n ε + n l ) l ∈ N con-verging to ω ∞ in the weak ∗ topology. Since K j is weak ∗ -compact and α ( j ) n,k ∈ [0 , for all k ∈ N and n, j ∈ N , by using a standard argument with a so-called diagonal subsequence, we can choosethe sequence ( n l ) l ∈ N such that ( ρ ( j ) n l ) weak ∗ -converges to ρ ( j ) ∞ ∈ K j − , and ( α ( j ) n l ,k ) l ∈ N has a limit forany fixed k ∈ N and j ∈ N . Using (128), (134) and the inequality n l X k = j +1 α ( j ) n l ,k λ k ≤ D − ε ∞ X k = j +1 − ( k +1) = 2 − ( j +1) D − ε together with Lebesgue’s dominated convergence theorem, we thus obtain that Re { ω ∞ ( A j ) } = lim l →∞ Re { ω n ε + n l ( A j ) } ≤ λ j − − ∞ X k = j +1 λ k lim l →∞ α ( j ) n l ,k ! . Because of (135), it follows that lim l →∞ α ( j ) n l ,k = 0 , k ∈ { j + 1 , . . . , ∞} , leading to ω ∞ ∈ K j , by (128), (133) and Lebesgue’s dominated convergence theorem. (Recall that K j is defined by (129) for n = j ∈ N .) Since ω n ε + j is by construction the unique maximizer of max σ ∈ K j Re { σ ( A j ) } = Re { ω n ε + j ( A j ) } and (135) holds true with ω ∞ ∈ K j , we deduce that ω ∞ = ω n ε + j , which is thus an exposed point of K ∞ for any j ∈ N . 56ur proof differs in several important aspects from the one of [42, Theorem 4.3], even if it has thesame general structure, inspired by Poulsen’s construction [62], as already mentioned. To be moreprecise, as compared to the proof of [42, Theorem 4.3], Step 0 is new and is a direct consequence ofthe compactness and metrizability of K , a property not assumed in [42, Theorem 4.3]. Step 1 to Step n → ∞ are similar to what is done in [42], but with the essential difference that convex combinationsare used to produce new (strongly) exposed points and the required bounds on { λ n , σ n } n ∈ N are thusquite different. Compare Equations (128) and (130) with the bounds on υ , υ , υ given in [42, p. 27-29], at parameters r ( t ) , r ( t ) , r ( t ) = 1 . In particular, [42, Lemma 4.2], which is essential to provethat the Poulsen-type construction leads to a dense set of (strongly) exposed points in [42, Theorem4.3], is never used here. Instead, we use other direct estimates on convex combinations to deduce thisproperty. This corresponds to Step n = ∞ .Note finally that [42, Theorem 4.3] shows the density of convex compact sets with dense set of strongly exposed points. A strongly exposed point σ in some convex set K ⊆ X ∗ is an exposed pointfor some A ∈ X with the additional property that any minimizing net of the real part of ˆ A (cf. (21))has to converge to σ in the weak ∗ topology . Observe that the only weak ∗ accumulation point ofsuch a minimizing net is the exposed point σ , by weak ∗ continuity of ˆ A . If K is weak ∗ -compact, thisyields that any minimizing net converges to σ in the weak ∗ topology. In other words, any exposedpoint is automatically strongly exposed in all convex weak ∗ -compact sets K ∈ CK ( X ∗ ) . The aim of this section is to prove Theorems 4.1 and 4.6. In fact, we prove here stronger results thanthese theorems. The proof of Theorem 4.1 is done in six lemmata and two corollaries. The proof ofTheorem 4.6 is a direct consequence of Corollary 7.12.We start with a useful estimate on the norm-continuous two-parameter family ( T ξt,s ) s,t ∈ R of ∗ -automorphisms of X defined by the non-autonomous evolution equations (47)-(48). Lemma 7.1 (Continuity of quantum dynamics)
Let X be a unital C ∗ -algebra. For any h ∈ C b ( R ; Y ( R )) , ξ , ξ ∈ C ( R ; E ) and s , s , t , t ∈ R , (cid:13)(cid:13)(cid:13) T ξ t ,s − T ξ t ,s (cid:13)(cid:13)(cid:13) B ( X ) ≤ | t − t | + | s − s | ) k h k C b ( R ; Y ( R )) + 2 Z t s k D h ( α ; ξ ( α )) − D h ( α ; ξ ( α )) k X d α . Proof.
Fix h ∈ C b ( R ; Y ( R )) , ξ , ξ ∈ C ( R ; E ) and s , s , t , t ∈ R . Via (49), observe that T ξ t ,s − T ξ t ,s = T ξ t ,s ◦ ( T ξ t ,t − X ) + ( T ξ s ,s − X ) ◦ T ξ t ,s + T ξ t ,s − T ξ t ,s . Using (47)-(48) together with (49), we thus obtain the equality T ξ t ,s − T ξ t ,s = Z t t T ξ α,s ◦ X ξ ( α ) α d α + Z s s X ξ ( α ) α ◦ T ξ t ,α d α + Z t s T ξ α,s ◦ (cid:0) X ξ ( α ) α − X ξ ( α ) α (cid:1) ◦ T ξ t ,α d α . (136)For any ξ ∈ C ( R ; E ) , ( T ξt,s ) s,t ∈ R is a two-parameter family of ∗ -automorphisms of X and the gener-ator X ξ ( t ) t defined by (45) has its operator norm bounded by (46). Therefore, the sum of the first twoterms in the right hand side of (136) is bounded by | t − t | k h k C b ( R ; Y ( R )) + 2 | s − s | k h k C b ( R ; Y ( R )) , One should not mistake the notion of strongly exposed points discussed here for the notion of weak ∗ strongly exposedpoints of [87, Definition 5.8] where a weak ∗ strongly exposed point is a (weak ∗ ) exposed point with the additional propertythat any minimizing net of the real part of ˆ A has to converge to σ in the norm topology of X ∗ . Z t s k D h ( α ; ξ ( α )) − D h ( α ; ξ ( α )) k X d α . We start now more specifically with the proof of Theorem 4.1, by showing the existence anduniqueness of the solution to the self-consistency equation. To this end, we basically use the Banachfixed point theorem.In contrast with Section 6, note that, below, the dual X ∗ of the unital C ∗ -algebra X is alwaysequipped with the usual norm for linear functionals on a normed space. In particular, X ∗ is in thiscase a Banach space. The set E of states is a weak ∗ -compact subset of X ∗ in the weak ∗ topology, butnot in the norm topology, unless X is finite-dimensional. This issue leads us to introduce Conditions(a)-(b) of Theorem 4.1, that is: Condition 7.2 (a)
Let X be a unital C ∗ -algebra and B a finite-dimensional real subspace of X R . (b) Take h ∈ C b ( R ; Y ( R )) and a constant D ∈ R + such that, for all t ∈ R , k D h ( t ; ρ ) − D h ( t ; ˜ ρ ) k X ≤ D sup B ∈ B , k B k =1 | ( ρ − ˜ ρ ) ( B ) | , ρ, ˜ ρ ∈ E .
We are now in a position to show the existence and uniqueness of the solution to the self-consistencyequation:
Lemma 7.3 (Self-consistency equations)
Under Condition 7.2, for any s ∈ R and ρ ∈ E , there is a unique solution ̟ ρ,s ∈ C ( R ; E ) to thefollowing equation in ξ ∈ C ( R ; E ) : ∀ t ∈ R : ξ ( t ) = ρ ◦ T ξt,s . (137) Moreover, ̟ ρ,s ( t ) = ̟ ̟ ρ,s ( r ) ,r ( t ) for any r, s, t ∈ R . Proof.
We prove the existence and uniqueness of a solution to (137) by using the Banach fixed pointtheorem, similar to the Picard-Lindel¨of theory for ODEs, keeping in mind that E is endowed with theweak ∗ topology: Pick a function h ∈ C b ( R ; Y ( R )) , an initial time s ∈ R and a state ρ ∈ E . For ǫ ∈ R + , define the map F from C ǫ,s . = C ([ s − ǫ, s + ǫ ]; X ∗ ) ∩ C ([ s − ǫ, s + ǫ ]; E ) to itself by F ( ξ ) ( t ) . = ρ ◦ T ξt,s , t ∈ [ s − ǫ, s + ǫ ] . (138)The continuity of F ( ξ ) in the Banach space C ([ s − ǫ, s + ǫ ]; X ∗ ) can directly be read from Lemma7.1 and Condition 7.2 (b). The same also yields the contractivity of F for sufficiently small ǫ ∈ R + ,uniformly with respect to s ∈ R and ρ ∈ E . Using the Banach fixed point theorem, there is aunique solution ̟ ρ,s to F ( ξ ) = ξ in C ǫ,s . By exactly the same arguments, observe that, for each r ∈ [ s − ǫ, s + ǫ ] , the following self-consistency equation ∀ t ∈ [ r − ˜ ǫ, r + ˜ ǫ ] : ξ ( t ) = ̟ ρ,s ( r ) ◦ T ξt,r , (139)has also a unique solution ̟ ̟ ρ,s ( r ) ,r in C ˜ ǫ,r for any ˜ ǫ ∈ (0 , ǫ ] . By the reverse cocycle property (49),at fixed s ∈ R and ρ ∈ E , ̟ ρ,s solves (139) for any r ∈ ( s − ǫ, s + ǫ ) and t ∈ [ s − ˜ ǫ, s + ˜ ǫ ] with ˜ ǫ = ǫ − | s − r | ∈ R + , whence ̟ ρ,s ( t ) = ̟ ̟ ρ,s ( r ) ,r ( t ) , r ∈ ( s − ǫ, s + ǫ ) , t ∈ [ s − ˜ ǫ, s + ˜ ǫ ] . (140)58ow, assume the existence and uniqueness of a solution ̟ ρ,s to F ( ξ ) = ξ in C ǫ ,s for someparameter ǫ ∈ R + . Take r ∈ ( s − ǫ , s − ǫ + ǫ ) ∪ ( s + ǫ − ǫ, s + ǫ ) . By combining the existenceand uniqueness of a solution ̟ ̟ ρ,s ( r ) ,r to (139) in C ˜ ǫ,r together with the reverse cocycle property (49),we deduce that ̟ ρ,s ( t ) = ̟ ̟ ρ,s ( r ) ,r ( t ) , t ∈ ( s − ǫ , s + ǫ ) , as well as the existence of a unique solution ̟ ρ,s to F ( ξ ) = ξ in C ǫ + ǫ,s . As a consequence, one caninfer from a contradiction argument the existence and uniqueness of a solution in C ( R ; X ∗ ) ∩ C ( R ; E ) to (137). Moreover, this solution must satisfy the equality ̟ ρ,s ( t ) = ̟ ̟ ρ,s ( r ) ,r ( t ) for any r, s, t ∈ R .Finally, to prove uniqueness in C ( R ; E ) , we observe from Lemma 7.1 and Condition 7.2 that anysolution in C ( R ; E ) (i.e., continuous with respect to the weak ∗ topology in E ) to (137) is automati-cally in C ( R ; X ∗ ) (i.e., continuous with respect to the norm topology in X ∗ ). Corollary 7.4 (Bijectivity of the solution to the self-consistency equation)
Under Condition 7.2, for any s, t ∈ R , ̟ s ( t ) ≡ ( ̟ ρ,s ( t )) ρ ∈ E is a bijective map from E to itself. Proof.
This is a straightforward consequence of Lemma 7.3, in particular the equality ̟ ρ,s ( t ) = ̟ ̟ ρ,s ( r ) ,r ( t ) for any r, s, t ∈ R . Lemma 7.5 (Differentiability of the solution – I)
Under Condition 7.2, for s ∈ R and ρ ∈ E , ̟ ρ,s ∈ C ( R ; X ∗ ) with derivative given by ∂ t ̟ ρ,s ( t ) = ρ ◦ T ̟ ρ,s t,s ◦ X ̟ ρ,s ( t ) t , t ∈ R . Proof.
This is a direct consequence of Equation (47) together with Lemma 7.3.
Lemma 7.6 (Continuity with respect to the initial condition)
Under Condition 7.2, for any s, t ∈ R , ̟ s ( t ) ≡ ( ̟ ρ,s ( t )) ρ ∈ E ∈ C ( E ; E ) . Proof.
Take s ∈ R and two states ρ , ρ ∈ E . Then, define the quantity X ( ǫ ) . = max t ∈ [ s − ǫ,s + ǫ ] max B ∈ B , k B k =1 (cid:12)(cid:12)(cid:0) ̟ ρ ,s ( t ) − ̟ ρ ,s ( t ) (cid:1) ( B ) (cid:12)(cid:12) , ǫ ∈ R + . Because ̟ ρ,s ( t ) = ρ ◦ T ̟ ρ,s t,s (Lemma 7.3) with ( T ξt,s ) s,t ∈ R being a family of ∗ -automorphisms of X for any ξ ∈ C ( R ; E ) , this positive number is bounded by X ( ǫ ) ≤ max B ∈ B , k B k =1 (cid:12)(cid:12) ( ρ − ρ ) ◦ T ̟ ρ ,s t,s ( B ) (cid:12)(cid:12) + Y ( ǫ ) , (141)where Y ( ǫ ) . = max t ∈ [ s − ǫ,s + ǫ ] (cid:13)(cid:13) T ̟ ρ ,s t,s − T ̟ ρ ,s t,s (cid:13)(cid:13) B ( X ) . (142)By Lemma 7.1, the last quantity is bounded by Y ( ǫ ) ≤ t ∈ [ s − ǫ,s + ǫ ] (cid:26)Z ts (cid:13)(cid:13) D h (cid:0) α ; ̟ ρ ,s ( α ) (cid:1) − D h (cid:0) α ; ̟ ρ ,s ( α ) (cid:1)(cid:13)(cid:13) X d α (cid:27) , (143)which, together with Condition 7.2 (b), leads to Y ( ǫ ) ≤ D ǫ X ( ǫ ) , ǫ ∈ R + . (144)By Inequality (141), it follows that (1 − D ǫ ) X ( ǫ ) ≤ max B ∈ B , k B k =1 (cid:12)(cid:12) ( ρ − ρ ) ◦ T ̟ ρ ,s t,s ( B ) (cid:12)(cid:12) , ǫ ∈ R + . (145)59ow, we combine ̟ ρ,s ( t ) = ρ ◦ T ̟ ρ,s t,s with (142) and (144)-(145) to get the inequality (cid:12)(cid:12) ̟ ρ ,s ( t ) ( A ) − ̟ ρ ,s ( t ) ( A ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) ( ρ − ρ ) ◦ T ̟ ρ ,s t,s ( A ) (cid:12)(cid:12) (146) + 2 D ǫ k A k X (1 − D ǫ ) max B ∈ B , k B k =1 (cid:12)(cid:12) ( ρ − ρ ) ◦ T ̟ ρ ,s t,s ( B ) (cid:12)(cid:12) for any s ∈ R , ρ , ρ ∈ E , A ∈ X , ǫ ∈ (0 , D / and t ∈ [ s − ǫ, s + ǫ ] . By finite dimensionality of B (Condition 7.2 (a)), the norm and weak ∗ topologies of B ∗ are the same and the weak ∗ continuityproperty of ̟ s ( t ) follows from (146) for any times s ∈ R and t ∈ [ s − ǫ, s + ǫ ] , provided ǫ < D / .Using now the equality ̟ ρ,s ( t ) = ̟ ̟ ρ,s ( r ) ,r ( t ) for any r, s, t ∈ R (Lemma 7.3), we thus deduce theweak ∗ continuity of ̟ s ( t ) for all times s, t ∈ R . Corollary 7.7 (Solution to the self-consistency equation as homeomorphism family)
Under Condition 7.2, at any fixed times s, t ∈ R , ̟ s ( t ) ≡ ( ̟ ρ,s ( t )) ρ ∈ E ∈ Aut ( E ) , i.e., ̟ s ( t ) is aautomorphism of the state space E . Moreover, it satisfies a cocycle property: ∀ s, r, t ∈ R : ̟ s ( t ) = ̟ r ( t ) ◦ ̟ s ( r ) . (147) Proof.
By Corollary 7.4 and Lemma 7.6, for any s, t ∈ R , ̟ s ( t ) is a weak ∗ -continuous bijective mapfrom E to itself. Recall that E is the (Hausdorff) topological space of all states on X with the weak ∗ topology. It is weak ∗ -compact. Therefore, the inverse of ̟ s ( t ) is also weak ∗ -continuous. Equation(147) is only another way to write the equality ̟ ρ,s ( t ) = ̟ ̟ ρ,s ( r ) ,r ( t ) of Lemma 7.3.Recall that the set Aut ( E ) of all automorphisms (or self-homeomorphisms) of E is endowedwith the topology of uniform convergence of weak ∗ -continuous functions from E to itself. See (44).Having this in mind, we obtain now the following lemma: Lemma 7.8 (Well-posedness of the self-consistency equation)
Under Condition 7.2, for any s ∈ R , ̟ s ≡ ( ̟ s ( t )) t ∈ R ≡ (( ̟ ρ,s ( t )) ρ ∈ E ) t ∈ R ∈ C ( R ; Aut ( E )) . Proof.
Take any net ( t j ) j ∈ J ⊆ R converging to some arbitrary time t ∈ R . Assume that ̟ s ( t j ) doesnot converge to ̟ s ( t ) , in the topology of uniform convergence of weak ∗ -continuous functions. Inthis case, by (44), there is a net ( ρ j ) j ∈ J ⊆ E of states, A ∈ X and ε ∈ R + such that lim inf j ∈ J (cid:12)(cid:12)(cid:12)h ̟ ρ j ,s ( t j ) − ̟ ρ j ,s ( t ) i ( A ) (cid:12)(cid:12)(cid:12) ≥ ε > . (148)By weak ∗ compactness of E , there is a subnet ( ρ j i ) i ∈ I weak ∗ -converging to some ρ ∈ E . By Lemmata7.3, 7.6 and Inequality (148), it follows that lim inf i ∈ I (cid:12)(cid:12)(cid:12)h ρ j i ◦ T ̟ ρji ,s t ji ,s − ̟ ρ,s ( t ) i ( A ) (cid:12)(cid:12)(cid:12) ≥ ε > . Using (142) and (144)-(145) together with the reverse cocycle property (49) and the fact that ( T ξt,s ) s,t ∈ R is a family of ∗ -automorphisms of X for any ξ ∈ C ( R ; E ) , we thus deduce from the last inequalitythat lim inf i ∈ I (cid:12)(cid:12)(cid:12)h ρ j i ◦ T ̟ ρ,s t ji ,s − ̟ ρ,s ( t ) i ( A ) (cid:12)(cid:12)(cid:12) ≥ ε > . (149)This is a contradiction because ( T ̟ ρ,s t,s ) s,t ∈ R is a norm-continuous two-parameter family. Hence, forany A ∈ X , lim i ∈ I ρ j i ◦ T ̟ ρ,s t ji ,s ( A ) = ρ ◦ T ̟ ρ,s t,s ( A ) = [ ̟ ρ,s ( t )] ( A ) . emma 7.9 (Joint continuity with respect to initial and final times) Under Condition 7.2, the solution to the self-consistency equation is jointly continuous with respectto initial and final times: ̟ ≡ ( ̟ s ) s ∈ R ≡ ( ̟ s ( t )) s,t ∈ R ≡ (( ̟ ρ,s ( t )) ρ ∈ E ) s,t ∈ R ∈ C (cid:0) R ; Aut ( E ) (cid:1) . Proof.
We use again the Banach fixed point theorem: Fix s ∈ R and ǫ ∈ R + . Similar to (138), wedefine the map F from C (cid:0) [ s − ǫ, s + ǫ ] ; X ∗ (cid:1) ∩ C (cid:0) [ s − ǫ, s + ǫ ] ; E (cid:1) to itself by F ( ζ ) ( r, t ) . = ρ ◦ T ζ ( r, · ) t,r , r, t ∈ [ s − ǫ, s + ǫ ] , where ζ ( r, · ) ∈ C ([ s − ǫ, s + ǫ ]; X ∗ ) ∩ C ([ s − ǫ, s + ǫ ]; E ) is the function defined, at fixed r ∈ [ s − ǫ, s + ǫ ] , by ζ ( r, t ) for any t ∈ [ s − ǫ, s + ǫ ] . By Lemma7.1 and Condition 7.2 (b), F is a contraction for sufficiently small times ǫ ∈ R + and we use similararguments as in the proof of Lemma 7.3 to show the existence of a unique solution ג to the followingequation in ζ ∈ C ([ s − ǫ, s + ǫ ] ; E ) : ∀ r, t ∈ [ s − ǫ, s + ǫ ] : ζ ( r, t ) = ρ ◦ T ζ ( r, · ) t,r . By uniqueness of the solution to (137) in C ( R ; E ) at any fixed s ∈ R , ̟ ρ,r ( t ) = ג ( r, t ) for any r, t ∈ [ s − ǫ, s + ǫ ] . By Corollary 7.7 and Lemma 7.8, it follows that ( ̟ ρ,s ( t )) s,t ∈ R ∈ C (cid:0) R ; E (cid:1) , ρ ∈ E .
Finally, by similar compactness arguments as in the proof of Lemma 7.8, we deduce the assertion.
Lemma 7.10 (Differentiability of the solution – II)
Fix n ∈ N , g ∈ C b ( R ; C b ( R n , R )) , { B j } nj =1 ⊆ X R and h ( t ; ρ ) . = g ( t ; ρ ( B ) , . . . , ρ ( B n )) , t ∈ R , ρ ∈ E . (150)
Then, for any s, t ∈ R and A ∈ X , ( ̟ ρ,s ( t ) ( A )) ρ ∈ E ≡ ( ̟ ρ,s ( t, A )) ρ ∈ E ∈ C ( E ; C ) (151) and, for any υ ∈ E , [d ̟ ρ,s ( t, A )] ( υ ) = υ (D ̟ ρ,s ( t, A )) = ( υ − ρ ) ◦ T ̟ ρ,s t,s ( A ) + z A [d ̟ ρ,s ( · , · ) ( υ )] where, for any continuous function ξ : R × X → C , z A [ ξ ] . = n X j,k =1 Z ts d α ξ ( α, B k ) ρ ◦ T ̟ ρ,s α,s (cid:0) i (cid:2) B j , T ̟ ρ,s t,α ( A ) (cid:3)(cid:1) (152) × ∂ x k ∂ x j g ( α ; ̟ ρ,s ( α, B ) , . . . , ̟ ρ,s ( α, B n )) . Moreover, for all s ∈ R and ρ ∈ E , the map ( t, A ) D ̟ ρ,s ( t, A ) from R × X to X is continuous. roof. Fix all parameters of the lemma. Observe first that Taylor’s theorem applied to ∂ x j g ( t ) foreach t ∈ R and j ∈ { , . . . , n } yields that, for all x, y ∈ R n , ∂ x j g ( t ; y ) = ∂ x j g ( t ; x ) + n X k =1 ( y k − x k ) (cid:0) ∂ x k ∂ x j g ( t ; x , . . . , x n ) + r k ( t, x, y ) (cid:1) (153)where, for any k ∈ { , . . . , n } , r k ( · , · , · ) is a continuous real-valued function on R × R n × R n suchthat lim y → x r k ( t, x, y ) = 0 , (154)uniformly for t and x in a compact set. Note additionally that the function h , as defined by (150),satisfies Condition 7.2.For any s, t ∈ R , ρ, υ ∈ E , λ ∈ (0 , and A ∈ X , we infer from Lemma 7.3 that i ( λ, t, A ; υ ) . = λ − (cid:0) ̟ (1 − λ ) ρ + λυ,s ( t, A ) − ̟ ρ,s ( t, A ) (cid:1) = ( υ − ρ ) ◦ T ̟ ρ,s t,s ( A ) + λ − ((1 − λ ) ρ + λυ ) ◦ (cid:0) T ̟ (1 − λ ) ρ + λυ,s t,s − T ̟ ρ,s t,s (cid:1) ( A ) . Through Equations (34), (45), (136), (150) and (153), we deduce that i ( λ, t, A ; υ ) = ( υ − ρ ) ◦ T ̟ ρ,s t,s ( A ) + n X j,k =1 Z ts d α i ( λ, α, B k ; υ ) (155) × ((1 − λ ) ρ + λυ ) ◦ T ̟ (1 − λ ) ρ + λυ,s α,s (cid:0) i (cid:2) B j , T ̟ ρ,s t,α ( A ) (cid:3)(cid:1) × (cid:0) ∂ x k ∂ x j g ( α ; x (0 , α )) + r k ( α ; x (0 , α ) , x ( λ, α )) (cid:1) , where x ( λ, α ) . = (cid:0) ̟ (1 − λ ) ρ + λυ,s ( α, B ) , . . . , ̟ (1 − λ ) ρ + λυ,s ( α, B n ) (cid:1) ∈ R n . From Equation (155), one sees that i ( λ, t, A ; υ ) is given by a Dyson-type series which is absolutelysummable, uniformly with respect to λ ∈ (0 , , because ( T ξt,s ) s,t ∈ R is a family of ∗ -automorphismsof X for any ξ ∈ C ( R ; E ) . By Lemmata 7.1 and 7.8 together with Condition 7.2 (b), lim λ → + ((1 − λ ) ρ + λυ ) ◦ T ̟ (1 − λ ) ρ + λυ,s α,s (cid:0) i (cid:2) B j , T ̟ ρ,s t,α ( A ) (cid:3)(cid:1) = ρ ◦ T ̟ ρ,s α,s (cid:0) i (cid:2) B j , T ̟ ρ,s t,α ( A ) (cid:3)(cid:1) while lim λ → + r k ( α, x (0 , α ) , x ( λ, α )) = 0 , using (154). (Both limits are uniform for α in a compact set.) Hence, we deduce from Lebesgue’sdominated convergence theorem that i (0 , t, A ; υ ) . = lim λ → + i ( λ, t, A ; υ ) = lim λ → + λ − (cid:0) ̟ (1 − λ ) ρ + λυ,s ( t, A ) − ̟ ρ,s ( t, A ) (cid:1) (156)exists for all s, t ∈ R , ρ, υ ∈ E and A ∈ X , as given by a Dyson-type series. In particular, forany υ ∈ E , the complex-valued function ( t, A ) i (0 , t, A, υ ) on R × X is the unique solution in ξ ∈ C ( R × X ; C ) to the equation ξ ( t, A ) = ( υ − ρ ) ◦ T ̟ ρ,s t,s ( A ) + z A [ ξ ] (157)with z A defined by (152). Compare with (155) taken at λ = 0 . Note that the integral equation D ( t, A ) = T ̟ ρ,s t,s ( A ) − ρ ◦ T ̟ ρ,s t,s ( A ) + n X j,k =1 Z ts d α D ( α, B k ) (158) × ρ ◦ T ̟ ρ,s α,s (cid:0) i (cid:2) B j , T ̟ ρ,s t,α ( A ) (cid:3)(cid:1) ∂ x k ∂ x j g ( α ; x (0 , α )) X ) Dyson-type series, a continuous map ( t, A ) D ( t, A ) from R × X to X , which, by (157), satisfies υ ( D ( t, A )) = i (0 , t, A ; υ ) . = lim λ → + λ − (cid:0) ̟ (1 − λ ) ρ + λυ,s ( t, A ) − ̟ ρ,s ( t, A ) (cid:1) (159)for all s, t ∈ R , ρ, υ ∈ E and A ∈ X . By Definition 3.8, the assertion follows. Lemma 7.11 (Differentiability of the solution – III)
Under the assumptions of Lemma 7.10, for any t ∈ R , ρ ∈ E and A ∈ X , ( ̟ ρ,s ( t ) ( A )) s ∈ R ≡ ( ̟ ρ,s ( t, A )) s ∈ R ∈ C ( R ; C ) with derivative given, for any A ∈ X , by ∂ s ̟ ρ,s ( t, A ) = − ρ ◦ X ρs ◦ T ̟ ρ,s t,s ( A ) + z A [ ∂ s ̟ ρ,s ] . (160) Here, z A is defined by (152) and ( t, A ) ∂ s ̟ ρ,s ( t, A ) is a continuous function on R × X . Proof.
By Lemma 7.3, for any ρ ∈ E , s, t ∈ R , A ∈ X and ε ∈ R \{ } , ˜ i ( ε, t, A ) . = ε − ( ̟ ρ,s + ε ( t, A ) − ̟ ρ,s ( t, A ))= ε − ρ ◦ ( T ̟ ρ,s t,s + ε − T ̟ ρ,s t,s ) ( A ) + ε − ρ ◦ ( T ̟ ρ,s + ε t,s + ε − T ̟ ρ,s t,s + ε ) ( A ) . Similar to (155), via Equations (34), (45), (136), (150) and (153) we deduce that ˜ i ( ε, t, A ) = ε − ρ ◦ ( T ̟ ρ,s t,s + ε − T ̟ ρ,s t,s ) ( A ) + n X j,k =1 Z ts + ε d α ˜ i ( ε, α, B k ) (161) × ρ ◦ T ̟ ρ,s + ε α,s (cid:0) i (cid:2) B j , T ̟ ρ,s t,α ( A ) (cid:3)(cid:1) (cid:0) ∂ x k ∂ x j g ( α ; y (0 , α )) + r k ( α ; y (0 , α ) , y ( ε, α )) (cid:1) with y ( ε, α ) . = ( ̟ ρ,s + ε ( α, B ) , . . . , ̟ ρ,s + ε ( α, B n )) ∈ R n . Again, one sees from Equation (161) that ˜ i ( ε, t, A ) is given by a Dyson-type series which is ab-solutely summable, uniformly with respect to ε in a bounded set. Recall that ( T ξt,s ) s,t ∈ R is a norm-continuous two-parameter family of ∗ -automorphisms of X satisfying in B ( X ) the non-autonomousevolution equation (48) for any fixed ξ ∈ C ( R ; E ) . Therefore, similar to (156), by Equation (154),Lemmata 7.1 and 7.9 together with Condition 7.2 (b) and Lebesgue’s dominated convergence theo-rem, we deduce that ∂ s ̟ ρ,s ( t, A ) . = lim ε → ˜ i ( ε, t, A ) = lim ε → ε − ( ̟ ρ,s + ε ( t, A ) − ̟ ρ,s ( t, A )) exists for all s, t ∈ R , ρ ∈ E and A ∈ X , as given by a Dyson-type series. In particular, the complex-valued function ( t, A ) ∂ s ̟ ρ,s ( t, A ) on R × X is the unique solution in ξ ∈ C ( R × X ; C ) to theequation ξ ( t, A ) = − ρ ◦ X ρs ◦ T ̟ ρ,s t,s ( A ) + z A [ ξ ] (162)with z A defined by (152). Compare with (161) taken at ε = 0 .We conclude this section with the derivation of Liouville’s equation for the time-evolution of(elementary) continuous and affine functions defined by (15), from which Theorem 4.6 is deduced. Corollary 7.12 (Liouville’s equation for affine functions)
Under the assumptions of Lemma 7.10, ∂ s V ht,s ( ˆ A ) = −{ h ( s ) , V ht,s ( ˆ A ) } , s, t ∈ R , A ∈ X , with ˆ A ∈ C being the elementary continuous and affine function defined by (15). In particular, bothside of the equation are well-defined functions in C . roof. Fix s ∈ R and ρ ∈ E . By (15) and (55), note that V ht,s ( ˆ A ) = ̟ ρ,s ( t ) ( A ) ≡ ̟ ρ,s ( t, A ) , t ∈ R , A ∈ X . By Lemma 7.10, the map ( t, A ) D V ht,s ( ˆ A ) ( ρ ) = D ̟ ρ,s ( t ; A ) = D ( t, A ) from R × X to X is continuous. See also Definition 3.8 and Equation (33). Therefore, the map ( t, A )
7→ −{ h ( s ) , V ht,s ( ˆ A ) } ( ρ ) . = − ρ (cid:16) i h D h ( s ; ρ ) , D V ht,s ( ˆ A ) ( ρ ) i(cid:17) from R × X to C is a well-defined continuous function. See Definition 3.10 and (39). By (158), itsolves Equation (162), like the well-defined continuous map ( t, A ) ∂ s V ht,s ( ˆ A ) ( ρ ) = ∂ s ̟ ρ,s ( t ) ( A ) ≡ ∂ s ̟ ρ,s ( t, A ) from R × X to C (Lemma 7.11). By uniqueness of the solution to (162), the assertion follows. C ∗ -Algebras La mˆeme structure qui, si vous montez, comporte une distance, si vous descendez, n’en comportepas. A. de Libera, 2015As explained in [53, p. 99], the notion of liminal C ∗ -algebras was first introduced in 1951 by Ka-plansky under the name of CCR -algebras. Remark that, in this context,
CCR does not mean “Canoni-cal Commutation Relations” but “Completely Continuous Representations”, “completely continuous”standing for “compact”.
CCR usually means nowadays “Canonical Commutation Relations” and thus,like Dixmier in his textbook [53] on C ∗ -algebras, we rather prefer the terminology “liminal”. Thisconcept is strongly related to the C ∗ -algebra K ( H ) of compact operators acting on a Hilbert space H via the concept of C ∗ -algebra representations. See also [89] for a recent compendium on operatoralgebras.Recall that a representation on the Hilbert space H of a C ∗ -algebra X is, by definition [36, Defi-nition 2.3.2], a ∗ -homomorphism π from X to the unital C ∗ -algebra B ( H ) of all bounded linear oper-ators acting on H . Injective representations are called faithful . The representation of a C ∗ -algebra X is not unique: For any representation π : X → B ( H ) , we can construct another one by doubling theHilbert space H and the map π , via a direct sum H ⊕ H of two copies H , H of H . Thus, recallalso the notion of “minimal” representations of C ∗ -algebras: If π : X → B ( H ) is a representationof a C ∗ -algebra X on the Hilbert space H , we say that it is irreducible , whenever { } and H are theonly closed subspaces of H which are invariant with respect to any operator of π ( X ) ⊆ B ( H ) .Every C ∗ -algebra which is isomorphic to the C ∗ -algebra K ( H ) of all compact operators acting onsome Hilbert space H is said to be elementary . The concept of liminal C ∗ -algebras generalizes thisnotion (see [53, Definition 4.2.1] or [89, Section IV.1.3.1]): Engl.:
The same structure which, if you go up, contains a distance, if you go down, does not contain it.
See [88, p.38]. This citation refers to the highly political and theological issues of hierarchies in Christianity, as discussed in theLate Middle Ages. Indeed, for some theologians of the XIIIe-XIVe centuries like Giles of Rome, the increasing hierarchyrefers to the existence of an order, implying in particular a distance (cf. “potentia dei ordinaria” ). From the top down, therelation can be direct, immediate, without distance (cf. “ potentia dei absoluta ”). In the mathematical context, from thebottom up, we have in mind the ordering of measures having same barycenter ρ in a compact convex space K to arrive,by “removing” the mass farther away from ρ , at a (maximal) measure only supported by extreme points, as proved andstated in the Choquet(-Bishop-de Leeuw) Theorem. See, e.g., (3). From the top down, we have in mind the disconcertingproperty that, for some K , extreme points are meanwhile dense, i.e., any x ∈ K is arbitrarily close to an extreme point. efinition 8.1 (Liminal C ∗ -algebras) A C ∗ -algebra X is called liminal if, for every irreducible representation π of X and each A ∈ X , π ( A ) is compact. All finite-dimensional C ∗ -algebras are of course liminal. All commutative C ∗ -algebras are also limi-nal. See [53, 4.2.1-4.2.2] or [89, Examples IV.1.3.3]. Note that the set of elements of a C ∗ -algebra X whose images under any irreducible representation are compact operators is the largest liminal closedtwo-sided ideal of X , by [53, Proposition 4.2.6].Later, Kaplansky and Glimm also introduced the term GCR for a generalization of Definition8.1, much later replaced by postliminal (see [53, Section 4.3.1] or [89, Section IV.1.3.1]). On the onehand, observe that the C ∗ -algebra K ( H ) of compact operators acting on a Hilbert space H is a closedtwo-sided ideal of the C ∗ -algebra B ( H ) of bounded operators acting on H . On the other hand, froma closed, self-adjoint two-sided ideal I of a C ∗ -algebra X and the quotient X / I , we can construct a C ∗ -algebra. Keeping this information in mind, the notion of postliminal C ∗ -algebras are defined asfollows [53, Section 4.3.1]: Definition 8.2 (Postliminal C ∗ -algebras) A C ∗ -algebra X is postliminal if every non-zero quotient C ∗ -algebra of X possesses a non-zeroliminal closed two-sided ideal. All liminal C ∗ -algebras are postliminal, by [53, Proposition 4.2.4], but the converse is false.Kaplansky and Glimm named important C ∗ -algebras that are not GCR (postliminal),
NGCR C ∗ -algebras. Such algebras were later called antiliminal [53, Section 4.3.1] (see also [89, SectionIV.1.3.1]): Definition 8.3 (Antiliminal C ∗ -algebras) A C ∗ -algebra X is antiliminal if the zero ideal is its only liminal closed two-sided ideal. Remark that a quotient C ∗ -algebra of an antiliminal C ∗ -algebra is not antiliminal, in general.If the image of X by an irreducible representation π would intersect the set of compact operators,then the set of compact operators would automatically be included in π ( X ) , by [53, Corollary 4.1.10].In other words, the image of a C ∗ -algebra by an irreducible representation either contains the set ofcompact operators or does not intersect it. Antiliminal and separable C ∗ -algebras are related to thesecond situation [90, Theorem 1 (b)]: Theorem 8.4 (Glimm)
Let X be a separable C ∗ -algebra. Then, the following conditions are equivalent: (i) X is antiliminal. (ii) X has a faithful type II representation. (iii) X has a faithful type III representation. (iv) X has a faithful representation which is a direct sum of a family of representations of X whoserange does not contain the compact operators. Proof.
By [53, Proposition 1.8.5], an antiliminal C ∗ -algebra does not possess any postliminal closedtwo-sided ideal, apart from the zero ideal. Therefore, the theorem is a direct consequence of [90,Theorem 1 (b)], keeping in mind that “completely continuous” in [90] is a synonym of “compact”.In other words, no irreducible representation of an antiliminal separable C ∗ -algebra X contains thecompact operators on the representation space. Additionally, antiliminal C ∗ -algebras are directly The definition of
GCR given in [89, Section IV.1.3.1] is different from the original one. In the non-separable situation, any of the assertions (ii)-(iv) yields (i), by [90, Theorem 1 (c)]. C ∗ -algebras are directlyassociated with von Neumann algebras of type I, by [90, Theorem 1 (a), (c)].Antiliminal (unital) C ∗ -algebras X have a set E of states with fairly complicated geometricalstructure, similar to the Poulsen simplex [62]. Recall that E is a weak ∗ -compact convex subset of X ∗ with (nonempty) set of extreme points denoted by E ( E ) . See (2). Then, one has the followingresult [53, Lemma 11.2.4]: Lemma 8.5 (Weak ∗ density of the set of extremes states) Let X be a antiliminal unital C ∗ -algebra. Assume that any two (different) non-zero closed two-sidedideals of X always have a non-zero intersection. Then E = E ( E ) , in the weak ∗ topology. C ∗ -algebras X relevant for mathematical physics often have a faithful type III representation. See,e.g., [91, Section 4] for UHF (uniformly hyperfinite) algebras. Note that every ∗ -representation of aUHF algebra, like for instance a CAR algebra, is faithful. For key statements on representations ofCAR C ∗ -algebras, see, e.g., [92, Theorem 2.4] or [93, Theorem 12.3.8] and references therein. Hence,by Theorem 8.4, many C ∗ -algebras X with physical applications are antiliminal. Note also that theyare generally separable and simple : Definition 8.6 (Simple C ∗ -algebras) A C ∗ -algebra X is simple if the only closed two-sided ideals of X are the trivial sets { } and X . C ∗ -algebras B ( H ) of all (bounded) linear operators acting on some finite-dimensional Hilbert space H are of course simple. See, e.g., [53, Corollary 4.1.7]. However, finite-dimensional C ∗ -algebras arenot generally simple, but semisimple only, as direct sums of simple algebras.In mathematical physics, unital C ∗ -algebras of infinitely extended (quantum) systems are usuallybuilt from a family of local finite-dimensional C ∗ -subalgebras. It refers to approximately finite-dimensional (AF) C ∗ -algebras, originally introduced in 1972 by Bratteli [94]. See also the quasi-local algebras [36, Definition 2.6.3]. AF C ∗ -algebras used in physics are usually simple, by [36,Corollary 2.6.19], because they are generally constructed from simple local algebras (typically assome inductive limit, with respect to boxes Λ , of a family of C ∗ -algebras B ( H Λ ) , with dim H Λ < ∞ ,like, for instance, Cuntz, lattice CAR or quantum-spin C ∗ -algebras). Therefore, by Lemma 8.5,for infinitely extended quantum systems, like fermions on the lattice or quantum-spin systems, thecorresponding set E of states has a dense subset of extreme points. This fact is well-known andalready discussed in [94, p. 226]. See also [36, Example 4.1.31] for a direct proof in the context ofthe so-called UHF (uniformly hyperfinite) C ∗ -algebras [36, Examples 2.6.12]. Acknowledgments:
This work is supported by CNPq (308337/2017-4), FAPESP (2017/22340-9), aswell as by the Basque Government through the grant IT641-13 and the BERC 2018-2021 program,and by the Spanish Ministry of Science, Innovation and Universities: BCAM Severo Ochoa ac-creditation SEV-2017-0718, MTM2017-82160-C2-2-P. We thank S. Rodrigues for pointing out thereference [18]. Finally, special thanks to S. Breteaux for having pointed out many typos and whomeanwhile suggested various improvements during his very detailed reading of the first version ofthis paper.
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