Extensions of Bundles of C*-algebras
aa r X i v : . [ m a t h . OA ] F e b Extensions of Bundles of C*-algebras
Jeremy Steeger [email protected]
Department of PhilosophySavery Hall Box 353350University of Washington
Benjamin H. Feintzeig [email protected]
Department of PhilosophySavery Hall Box 353350University of Washington
February 25, 2021
Abstract
Bundles of C*-algebras can be used to represent limits of physical theories whosealgebraic structure depends on the value of a parameter. The primary example is the ~ → ~ → In this paper, we are interested in limiting procedures in which the algebraic structureof a theory varies depending on the numerical value of some parameter. In many cases,the physical quantities of such a theory can be represented by a C*-algebra of quantities,and the corresponding limit can be represented by a bundle of C*-algebras (Dixmier, 1977;Kirchberg and Wasserman, 1995). The primary example of this is the formulation of theclassical ~ → ~ → c → ∞ limit of relativistic theories. Thus, bundles of C*-algebras have importantapplications to limits of physical theories.However, in many cases the definition of the relevant bundle, and so the representationof the limiting procedure, is presented by stipulating the structure of the C*-algebra at thelimiting value of the parameter. For example, strict deformation quantizations are typicallypresented by starting with a classical Poisson algebra densely embedded in a commutativeC*-algebra at ~ = 0. One would like to know whether this limiting C*-algebra is somehowdetermined by the C*-algebras that form the rest of the bundle for ~ >
0, and similarlyfor the other examples. We pose the associated question as concerning the existence anduniqueness of extensions of bundles to enlarged parameter spaces. For example, for the ~ → ,
1] uniquely extendsto a bundle over [0 , ~ = 0 that canbe continuously glued to the existing bundle. We answer this question in the affirmative.Furthermore, as Landsman (2003a,b) has conjectured that quantization is functorial, onecan pose an analogous question for the inverse procedure of the classical limit, or for exten-sions of bundles of C*-algebras more generally. We use our existence and uniqueness resultshere to establish a number of senses in which limiting procedures represented by extensionsof bundles of C*-algebras are functorial. This holds for a variety of physically significantstructures on the fiber C*-algebras, including the C*-product, dynamical automorphisms,and the Lie bracket (in the ~ → ~ → §
2, we motivate our investigation in moredetail by reviewing how a strict deformation quantization determines a continuous bundleof C*-algebras. This section also takes the opportunity to define relevant notions, and theassociated Appendix A shows that the determination of a bundle from a strict quantiza-tion is functorial. In §
3, we provide a definition of a novel kind of bundle of C*-algebras,called a uniformly continuous bundle , which we will use to establish our results concerningextensions. We prove in the associated Appendix B that a category of uniformly contin-uous bundles is equivalent to a category of continuous bundles as standardly defined byKirchberg and Wasserman (1995). In §
4, we prove that our uniformly continuous bundleshave unique extensions to base spaces with limiting values of parameters. Finally, § § § § Motivation from Quantization
We represent the physical quantities of a quantum system with elements of a C*-algebra.A C*-algebra A is an associative, involutive, complete normed algebra satisfying the C*-identity: k A ∗ A k = k A k for all A ∈ A . The canonical examples of C*-algebras are com-mutative algebras of bounded functions on a locally compact topological space and possiblynon-commutative algebras of bounded operators on a Hilbert space. We employ commuta-tive algebras of functions on a classical phase space in classical physics and non-commutativealgebras of operators satisfying a version of the canonical (anti-) commutation relations inquantum theories. Thus, using C*-algebras provides a unified framework for investigatingthe relationship between classical and quantum theories. For mathematical background,we refer the reader to Sakai (1971), Dixmier (1977), Kadison and Ringrose (1997). SeeBratteli and Robinson (1987, 1996) and Haag (1992) for the C*-algebraic approach to quan-tum physics. With this background in place, we can use C*-algebras to analyze the classicallimit.A strict quantization provides extra structure to “glue together” a family of C*-algebrasindexed by the parameter ~ , as follows. Definition 1 (Landsman, 1998a) . A strict quantization of a Poisson algebra ( P , {· , ·} ) con-sists in a locally compact topological space I ⊆ R containing 0, a family of C*-algebras( A ~ ) ~ ∈ I and a family of linear quantization maps ( Q ~ : P → A ~ ) ~ ∈ I . We require that P ⊆ A , Q is the inclusion map, and for each ~ ∈ I , Q ~ [ P ] is norm dense in A ~ . Further,we require that the following conditions hold for all A, B ∈ P :(i)
Von Neumann’s condition. lim ~ → kQ ~ ( A ) Q ~ ( B ) − Q ~ ( AB ) k ~ = 0;(ii) Dirac’s condition. lim ~ → k i ~ [ Q ~ ( A ) , Q ~ ( B )] − Q ~ ( { A, B } ) k ~ = 0;(iii) Rieffel’s condition. the map ~
7→ kQ ~ ( A ) k ~ is continuous.A strict deformation quantization is a strict quantization that satisfies the additional re-quirement that for each ~ ∈ I , Q ~ [ P ] is closed under multiplication and is nondegenerate,i.e., Q ~ ( A ) = 0 if and only if A = 0 . Rieffel (1989, 1993) uses this approach to define a “deformed” product on the classicalalgebra, which he uses in turn to specify the quantization conditions. We work in theopposite direction, defining the elements of the classical algebra using information away from ~ = 0. We will mostly ignore the deformation condition, but still refer to strict deformationquantizations to distinguish them from other approaches like geometric quantization.In a strict deformation quantization, one can understand the classical limit of a quantity Q ~ ( A ) in A ~ in the quantum theory to be the classical quantity A ∈ P . Similarly, we cantake classical limits of states by defining a continuous field of states as a family of states ω ~ ∈ S ( A ~ ) for each ~ ∈ I such that the map ~ ω ~ ( Q ~ ( A )) is continuous for each A ∈ P .In this case, the classical limit of such a continuous field of states is understood to be theclassical state ω ∈ S ( A ). Thus, a strict deformation quantization provides enough structureto represent the classical limits of states and quantities in quantum theories. In fact, there isa close connection between the satisfaction of Rieffel’s condition in a strict quantization and3he existence of a continuous field of states converging to each classical state; see the proof ofTheorem 4 in Landsman (1993a, p. 33) for more details. However, in the present paper, wewill mostly ignore classical limits of states and restrict our focus kinematical quantities andtheir associated dynamics. We leave a more thorough treatment of states for future work.In general, there may be different strict deformation quantizations of the same Poissonmanifold. If two strict quantizations Q ~ and Q ′ ~ of a given Poisson algebra P employ the samefamily of C*-algebras, but possibly differ in their quantization maps, then the quantizationsare called equivalent just in caselim ~ → kQ ~ ( A ) − Q ′ ~ ( A ) k ~ = 0 (1)for all A ∈ P . Equivalent quantizations share the same behavior in the limit as ~ →
0. Onecan encode this common behavior in a further object, variously called a continuous bundleor field of C*-algebras, which itself has enough structure to understand the classical limits ofquantities and states. We consider an existing definition in the literature before providing analternative—and in a sense, equivalent —definition that will be more useful for our purposes.
Definition 2. (Kirchberg and Wasserman, 1995) A vanishingly continuous bundle of C*-algebras over a topological space I is a family of C*-algebras ( A ~ ) ~ ∈ I , a C*-algebra A calledthe collection of vanishingly continuous sections , and a family of *-homomorphisms ( φ ~ : A → A ~ ) ~ ∈ I called evaluation maps , which we require to satisfy the following conditions:(i) Fullness.
Each evaluation map φ ~ is surjective and the norm of each a ∈ A is givenby k a k = sup ~ ∈ I k φ ~ ( a ) k ~ .(ii) Vanishing completeness.
For each continuous function vanishing at infinity f ∈ C ( I )and a ∈ A , there is an element f a ∈ A such that φ ~ ( f a ) = f ( ~ ) φ ~ ( a ).(iii) Vanishing continuity.
For each a ∈ A , the function N a : ~
7→ k φ ~ ( a ) k ~ is in C ( I ).Other authors typically call these structures simply “continuous bundles of C*-algebras”.We add the modifier “vanishingly” due to the use of C ( I ), the continuous functions vanish-ing at infinity. This will help to distinguish these structures from the ones we define next.We will often think of the C*-algebras A ~ as fibers above the values ~ ∈ I , hence forming abundle structure over the base space I . A vanishingly continuous bundle of C*-algebras de-termines the continuity structure of the bundle by specifying the collection A of vanishinglycontinuous sections through the bundle.By a theorem of Landsman (1998a, Theorem 1.2.4, p. 111), given a strict quantization,there is a unique vanishingly continuous bundle of C*-algebras containing among its sectionsthe curves traced out by the quantization maps as ~ varies. More formally: given a strictquantization (( A ~ , Q ~ ) ~ ∈ I , P ), under modest conditions there is a unique vanishingly contin-uous bundle of C*-algebras (( A ~ , φ ~ ) ~ ∈ I , A ) such that for each A ∈ P , there is a continuoussection a ∈ A with φ ~ ( a ) = Q ~ ( A ) for all ~ ∈ I . We can thus speak of the vanishinglycontinuous bundle generated by a strict quantization. Moreover, Landsman’s theorem showsequivalent quantizations generate the same bundle. In this sense, the bundles encode aninvariant structure among different quantization maps capturing the same behavior in the ~ → Uniformly Continuous Bundles
The association of a C*-algebra of vanishingly continuous sections in this definition allowsone to use many familiar tools to analyze such bundles. Still, there are some drawbacksto vanishingly continuous bundles for our purposes. In the next section, we will formulateour central question about the determination of the classical limit, or limits of algebraicphysical theories more generally, as follows. Suppose one knows the quantum kinematics for ~ > A ~ , φ ~ ) ~ ∈ (0 , , A ) over the base space consisting only of parametervalues ~ >
0. Under what conditions is there a unique algebra A that, when appropriatelyglued to the given bundle of algebras, provides an extended continuous bundle?One would like to construct the sections of an extended bundle over [0 ,
1] by continuouslyextending sections of the restricted bundle over (0 ,
1] to the point ~ = 0 in the base space.But if one begins with a vanishingly continuous bundle of C*-algebras over (0 , ~ →
0, toward the 0 element of any C*-algebra one tries to glue on.Thus, this strategy can only lead to trivial limits. In other words, one cannot, in this way,directly recover non-trivial information about the corresponding classical algebra at ~ = 0.Notice, however, that we can make a slight change to the definition of our bundles to dealwith this issue. Consider a simplified toy example. Although every function in C ((0 , ~ →
0, the collection UC b ((0 , f ∈ UC b ((0 , ~ →
0, the limit may be non-zero. More generally, for arbitrary metric spaces (
I, d ), each f ∈ UC b ( I ) can be uniquely extended to the completion of I (Aliprantis and Border, 1999,Lemma 3.11, p. 77). This suggests the possibility of constructing bundles of C*-algebraswhose sections are uniformly continuous and bounded. We use this approach in an attemptto preserve the virtue of having a C*-algebra of sections while also gaining some control overnon-trivial limits. We propose the following definition for the task. Definition 3. A uniformly continuous bundle of C*-algebras over a metric space ( I, d ) is afamily of C*-algebras ( A ~ ) ~ ∈ I , a C*-algebra A called the collection of uniformly continuoussections , and a family of *-homomorphisms ( φ ~ : A → A ~ ) ~ ∈ I called evaluation maps , whichwe require to satisfy the following conditions:(i) Fullness.
Each evaluation map φ ~ is surjective and the norm of each a ∈ A is given by k a k = sup ~ ∈ I k φ ~ ( a ) k ~ .(ii) Uniform completeness.
For each f ∈ UC b ( I ) and a ∈ A , there is an element f a ∈ A such that φ ~ ( f a ) = f ( ~ ) φ ~ ( a ).(iii) Uniform continuity.
For each a ∈ A , the function N a : ~
7→ k φ ~ ( a ) k ~ is in UC b ( I ).Notice that the only difference between uniformly and vanishingly continuous bundles ofC*-algebras is the use of C ( I ) or U C b ( I ) in conditions (ii) and (iii). In general, we willrestrict our attention to both uniformly and vanishingly continuous bundles of C*-algebraswhose base space is a locally compact metric space.5e now formulate two relevant categories of bundles of C*-algebras that we will use inwhat follows. To that end, we define a notion of morphism, or structure-preserving map,between bundles. Recall that a metric map α : I → J is one satisfying d J ( α ( x ) , α ( y )) ≤ d I ( x, y ) for all x, y ∈ I , where d I and d J are the metrics on I and J , respectively. Definition 4. A homomorphism σ : A I → B J between (vanishingly or uniformly) continuousbundles of C*-algebras A I = (( A ~ , φ I ~ ) ~ ∈ I ) , A ) and B J = (( B ~ , φ J ~ ) ~ ∈ J ) , B ) is a pair of maps σ = ( α, β ) , (2)where α : I → J is a metric map, β : A → B is a *-homomorphism. We further require thefollowing condition of compatibility between α and β : for all a , a ∈ A and ~ ∈ I if φ I ~ ( a ) = φ I ~ ( a ), then φ Jα ( ~ ) ( β ( a )) = φ Jα ( ~ ) ( β ( a )) . A homomorphism σ = ( α, β ) is an isomorphism if α is an isometry and β is a *-isomorphism.The condition of compatibility between α and β ensures that β preserves fibers in the sensethat it defines a *-homorphism from the fiber A ~ to the fiber B α ( ~ ) (See Lemma 3 in § I of any of the structures considered is a locallycompact metric space. Definition 5.
The category
VBunC ∗ Alg consists in: • objects : vanishingly continuous bundles of C*-algebras whose base space is a locallycompact metric space, A VI = (cid:16)(cid:0) A ~ , φ I ~ (cid:1) ~ ∈ I , A (cid:17) ; • morphisms : bundle homomorphisms. Definition 6.
The category
UBunC ∗ Alg consists in: • objects : uniformly continuous bundles of C*-algebras whose base space is a locallycompact metric space, A UI = (cid:16)(cid:0) A ~ , φ I ~ (cid:1) ~ ∈ I , A (cid:17) ; • morphisms : bundle homomorphisms.The category VBunC ∗ Alg encodes the structures picked out by the definition of van-ishingly continuous bundles of C*-algebras. On the other hand, the category
UBunC ∗ Alg encodes the structures picked out by the definition of uniformly continuous bundles of C*-algebras. There is a sense in which our proposed definition of uniformly continuous bundlesis equivalent to the previous definition of vanishingly continuous bundles, i.e., one can con-struct a unique uniformly continuous bundle from each vanishingly continuous bundle, andvice versa. In Appendix B, we make this precise by establishing a categorical equivalencebetween
VBunC ∗ Alg and
UBunC ∗ Alg . We take this equivalence to justify our use ofuniformly continuous bundles of C*-algebras in what follows.Next, we will use uniformly continuous bundles of C*-algebras to analyze the existenceand uniqueness of limits of families of C*-algebras.6
Existence and Uniqueness of Extensions
We want to consider the case where we begin with only information about a quantum theoryat ~ > ~ = 0, or the analogoussituation for limits of other parameters.In other words, suppose we are given only the restriction of our uniformly continuousbundle of C*-algebras to a bundle over the base space given by the half open interval (0 , canonical restriction of a bundle A J = (( A ~ , φ J ~ ) ~ ∈ J , A ) overa base space given by a metric space J . If I is a further locally compact metric space and α : I → J is a metric embedding, we will say that the canonical restriction of A J from J to I along α is given by A J | I := (cid:16)(cid:0) A ~ , φ I ~ (cid:1) ~ ∈ I , A | I (cid:17) , where A | I := ( γ ∈ Y ~ ∈ I A ~ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) for some a ∈ A , γ ( ~ ) = φ Jα ( ~ ) ( a ) for all ~ ∈ I ) , and φ I ~ ( γ ) := γ ( ~ ) for all γ ∈ A | I and ~ ∈ I. (3)This restriction removes the fibers outside of α [ I ] and truncates continuous sections from J to α [ I ] ⊆ J . One can check that this is indeed a uniformly continuous bundle in its ownright. In particular, for I = (0 ,
1] and J = [0 , α : (0 , → [0 ,
1] to define the restricted bundle A J | I resulting from a strict deformationquantization as above. Such a restricted bundle represents only the information in thequantum theory for ~ > ~ = 0.Given such a restriction, our questions are: can one reconstruct the C*-algebra of classicalquantities A from this restricted continuous bundle? And can one continuously glue A tothe restricted bundle in a way that recovers the original information about the ~ → ~ = 0, which we call “extension-and-restriction.” Startingwith a bundle containing only information for ~ >
0, we (uniquely) extend the bundle toone containing information at the accumulation point ~ = 0; then we (uniquely) restrict thisnew bundle to the fiber algebra ~ = 0 to exactly recover the classical theory.We will prove our results in full generality for the case where our base space I is anarbitrary locally compact metric space. Our general result then applies immediately to thecase where the base space is either I = (0 ,
1] or I = { /N | N ∈ N } , which are the mosttypical base spaces used in analyzing limits of quantum theories. As such, we define a generalnotion of extension. Definition 7.
Let A I = (( A ~ , φ I ~ ) ~ ∈ I , A ) and B J = (( B ~ , φ J ~ ) ~ ∈ J , B ) be uniformly continuousbundles of C*-algebras over locally compact metric spaces I and J , respectively. • B J is an extension of A I if there is a monomorphism of continuous bundles of C*-algebras σ = ( α, β ) : A I → B J , i.e., a homomorphism where α and β are both injective. • B J is a minimal extension of A I if, moreover, α and β are both dense embeddings.In either case, we say that the extension B J is associated with α via σ .7e use the term “dense embedding” above to mean something slightly different for α andfor β . For α to be a dense embedding, it must be an isometric isomorphism between itsdomain and its image (equipped with the subspace topology) and its image must be densein J (equipped with the metric topology). Sometimes we will call α a “dense, isometricembedding” for emphasis; although, note that we do not require α to be bijective. For β tobe a dense embedding, nothing more is required other than that its image must be dense in B (according to the algebra’s norm)—this condition, in conjunction with injectivity, makes β a *-isomorphism (Kadison and Ringrose 1997, p. 243).We are especially interested in minimal extensions, which capture limits of physical the-ories for accumulation points in the parameter space (such as ~ = 0). One can show that aminimal extension is guaranteed to exist for any accumulation point of interest. Theorem 1.
Let A I = (( A ~ , φ I ~ ) ~ ∈ I , A ) be a uniformly continuous bundle of C*-algebras overa locally compact metric space I . Suppose α : I → J is a dense, isometric embedding. Thenthere exists a minimal extension ˜ A J of A I associated with α . We proceed to prove Theorem 1 through a series of lemmas.
Lemma 1.
Suppose A I and α : I → J are as in Theorem 1. Given any j ∈ α [ I ] , the set K j := n a ∈ A (cid:12)(cid:12)(cid:12) lim δ (cid:13)(cid:13) φ Ii δ ( a ) (cid:13)(cid:13) i δ = 0 for any net { i δ } with α ( i δ ) → j o is a closed two-sided ideal in A .Proof. In what follows, we let { i δ } denote an arbitrary net in I with α ( i δ ) → j .First, note that K j is well-defined: for each a ∈ A , define ˜ N a : α [ I ] → C by˜ N a ( j ) = N a ( α − ( j )) , and note that the limit lim δ ˜ N a ( α ( i δ )) = lim δ (cid:13)(cid:13) φ Ii δ ( a ) (cid:13)(cid:13) i δ exists and is unique by the uniformly continuous extension theorem (e.g., Aliprantis and Border,1999, Lemma 3.11). Now we go on to show that K j is a closed, two-sided ideal.First, K j is a vector subspace of A ; if k, k ′ ∈ K j and x, y ∈ C , thenlim δ (cid:13)(cid:13) φ Ii δ ( xk + yk ′ ) (cid:13)(cid:13) i δ ≤ lim δ x (cid:13)(cid:13) φ Ii δ ( k ) (cid:13)(cid:13) i δ + lim δ y (cid:13)(cid:13) φ Ii δ ( k ′ ) (cid:13)(cid:13) i δ = 0 . Further, consider an arbitrary k ∈ K j and a ∈ A ; note that, because each norm is submulti-plicative,lim δ (cid:13)(cid:13) φ Ii δ ( k · a ) (cid:13)(cid:13) i δ = lim δ (cid:13)(cid:13) φ Ii δ ( k ) · φ Ii δ ( a ) (cid:13)(cid:13) i δ ≤ lim δ (cid:13)(cid:13) φ Ii δ ( k ) (cid:13)(cid:13) i δ · (cid:13)(cid:13) φ Ii δ ( a ) (cid:13)(cid:13) i δ = 0 , and similarly lim δ (cid:13)(cid:13) φ Ii δ ( a · k ) (cid:13)(cid:13) i δ = 0 , so k · a and a · k are both in K j . Hence, K j is a two-sided ideal. Finally, consider a net { k λ } ⊆ K j that converges to k in A (which, recall, is equipped with the supremum norm).8ick some ǫ >
0. There is some λ ′ such that for all λ > λ ′ , k k − k λ k < ǫ/
2. Further, thereis some δ ′ such that for all δ > δ ′ , k φ Ii δ ( k λ ) k i δ < ǫ/
2. Thus, for all δ > δ ′ and λ > λ ′ , (cid:13)(cid:13) φ Ii δ ( k ) (cid:13)(cid:13) i δ ≤ (cid:13)(cid:13) φ Ii δ ( k − k λ ) (cid:13)(cid:13) i δ + (cid:13)(cid:13) φ Ii δ ( k λ ) (cid:13)(cid:13) i δ < ǫ/ ǫ/ ǫ. Since ǫ was arbitrary, we have lim δ k φ Ii δ ( k ) k i δ = 0. Hence, k ∈ K j , so K j is closed. Lemma 2.
With the definitions above, the canonical quotient A / K j is a C*-algebra.Proof. This follows immediately from Proposition 1.8.2 of (Dixmier, 1977, pp. 20-21).
Proof of Theorem 1.
With Lemmas 1 and 2, we define for each ~ ∈ J the C*-algebra ˜ A ~ := A / K ~ . Define φ J ~ : A → ˜ A ~ for each ~ ∈ J as the canonical quotient map. Then let˜ A J := (( ˜ A ~ , φ J ~ ) ~ ∈ J , A ). We will show that ˜ A J is an extension of A I associated with α .First, ˜ A J is a uniformly continuous bundle of C*-algebras:(i) Clearly, each map φ J ~ is surjective by the definition of the quotient, which establishesfullness.(ii) Given f ∈ UC b ( J ) and a ∈ A , we know f ◦ α ∈ UC b ( I ). Hence, if we define f a :=( f ◦ α ) a , then uniform completeness follows from the uniform completeness of A I .(iii) Uniform continuity follows immediately from the construction with the definition ofthe quotient norm.Finally, we show that ˜ A J is an extension of A I associated with α via σ := ( α, id A ). Weknow α is an isometric map, and id A is a *-homomorphism. Moreover, for any a , a ∈ A and ~ ∈ I , if φ I ~ ( a ) = φ I ~ ( a ), then ( a − a ) ∈ K α ( ~ ) , and hence φ Jα ( ~ ) ( a ) = φ Jα ( ~ ) ( a + ( a − a )) = φ Jα ( ~ ) ( a ) , so σ is a homomorphism. Clearly, α and id A are both injective, so σ is monomorphism.Now, we have established that minimal extensions always exist. But in order to use thisminimal extension to talk of, e.g., the classical theory determined by a quantum theory,we further require a sense in which this extension is unique. It turns out that all minimalextensions associated with a dense embedding α are isomorphic—so it makes sense to talkof both the minimal extension of a bundle and, e.g., the classical theory that it defines. Theorem 2.
Let A I = (( A ~ , φ I ~ ) ~ ∈ I , A ) be a uniformly continuous bundle of C*-algebrasover a locally compact metric space I . Suppose that B J = (( B ~ , φ J ~ ) ~ ∈ J , B ) and C J =(( C ~ , ψ J ~ ) ~ ∈ J , C ) are two minimal extensions of A I associated with a given dense, isomet-ric embedding α : I → J . Then B J and C J are isomorphic as bundles of C*-algebras. We again require a preliminary lemma, which establishes general conditions under which ahomomorphism of bundles generates isomorphisms of fiber algebras.9 emma 3.
Let A I and B J be uniformly continuous bundles of C*-algebras and suppose σ = ( α, β ) is any homomorphism between them. For each ~ ∈ I , define the fiberwise map σ ~ : A ~ → B α ( ~ ) by σ ~ (cid:0) φ I ~ ( a ) (cid:1) = φ Jα ( ~ ) ( β ( a )) for each a ∈ A .(i) σ ~ is a *-homomorphism.(ii) If σ is a monomorphism, then σ ~ is injective.(iii) If β is surjective, then σ ~ is surjective.(iv) If σ is a monomorphism and β is surjective, then σ ~ is a *-isomorphism. In particular, (cid:13)(cid:13) φ I ~ ( a ) (cid:13)(cid:13) ~ = (cid:13)(cid:13) φ Jα ( ~ ) ( β ( a )) (cid:13)(cid:13) α ( ~ ) . Proof.
We prove each part in turn.(i) First, notice that the definition fully specifies σ ~ because φ I ~ is surjective. Moreover, σ ~ iswell-defined because if a , a ∈ A are such that φ I ~ ( a ) = φ I ~ ( a ), then by the definitionof a bundle homomorphism, we know σ ~ ( φ I ~ ( a )) = φ Jα ( ~ ) ( β ( a )) = φ Jα ( ~ ) ( β ( a )) = σ ~ ( φ I ~ ( a )).Next, σ ~ is linear: for any x ∈ C and a , a ∈ A , σ ~ ( φ I ~ ( a + x · a )) = φ Jα ( ~ ) ( β ( a + x · a ))= φ Jα ( ~ ) ( β ( a )) + x · φ Jα ( ~ ) ( β ( a )) = σ ~ ( φ I ~ ( a )) + x · σ ~ ( φ I ~ ( a )) . Similarly, σ ~ is multiplicative: for any a , a ∈ A , σ ~ ( φ I ~ ( a · a )) = φ Jα ( ~ ) ( β ( a · a )) = φ Jα ( ~ ) ( β ( a )) · φ Jα ( ~ ) ( β ( a ))= σ ~ ( φ I ~ ( a )) · σ ~ ( φ I ~ ( a )) . Finally, σ ~ is *-preserving: for any a ∈ A , σ ~ ( φ I ~ ( a ∗ )) = φ Jα ( ~ ) ( β ( a ∗ )) = φ Jα ( ~ ) ( β ( a )) ∗ = σ ~ ( φ I ~ ( a )) ∗ . (ii) We prove the contrapositive. If σ ~ is not injective, then there is some a ∈ A for which φ I ~ ( a ) = 0, but σ ~ ( φ I ~ ( a )) = φ Jα ( ~ ) ( β ( a )) = 0. Let C ∗ ( a ) be the smallest C*-subalgebraof A containing a and consider the trivial bundle with fiber C ∗ ( a ) over the one-elementbase space {∗} . Let ι : {∗} → I be defined by ι ( ∗ ) = ~ , let β : C ∗ ( a ) → A be thenatural inclusion, and let β : C ∗ ( a ) → A be the map that multiplies every element by0. Define σ = ( ι, β ) and σ = ( ι, β ). Notice that σ ◦ σ = σ ◦ σ because for each a ′ ∈ C ∗ ( a ), we know φ Jα ◦ ι ( ∗ ) ( β ◦ β ( a ′ )) = 0 = φ Jα ◦ ι ( ∗ ) ( β ◦ β ( a ′ )) . But clearly σ = σ because σ ( a ) = a = 0 = σ ( a ). Thus, σ is not a monomorphism.10iii) Suppose β is surjective. Then for any b α ( ~ ) ∈ B α ( ~ ) , since φ Jα ( ~ ) is surjective, there issome b ∈ B with φ Jα ( ~ ) ( b ) = b α ( ~ ) . But since β is surjective, there is some a ∈ A suchthat β ( a ) = b . Hence, σ ~ ( φ I ~ ( a )) = φ Jα ( ~ ) ( β ( a )) = φ Jα ( ~ ) ( b ) = b α ( ~ ) . (iv) This follows immediately from Theorem 4.1.8(iii) of Kadison and Ringrose (1997, p.242).From this lemma, Theorem 2 quickly follows. Proof of Theorem 2.
Since β B and β C are both injective and dense, it follows from Theorem4.1.9 of Kadison and Ringrose (1997, p. 243) that they are *-isomorphisms. Define β : B → C as β := β C ◦ β − B . Then consider σ := (id J , β ). We need to show that σ is an isomorphismof bundles.Suppose that for ~ ∈ J and b , b ∈ B , we have φ J ~ ( b ) = φ J ~ ( b ). We know that there issome net ~ δ ∈ I such that α ( ~ δ ) → ~ in J . Hence, by part (iii) of the previous lemma, weknow that k φ J ~ ( β ( b )) − φ J ~ ( β ( b )) k ~ = k φ J ~ ( β C ◦ β − B ( b )) − φ J ~ ( β C ◦ β − B ( b )) k ~ = lim δ k φ Jα ( ~ δ ) ( β C ◦ β − B ( b )) − φ Jα ( ~ δ ) ( β C ◦ β − B ( b )) k α ( ~ δ ) = lim δ k φ I ~ δ ( β − B ( b )) − φ I ~ δ ( β − B ( b )) k ~ δ = lim δ k φ Jα ( ~ δ ) ( b ) − φ α ( ~ δ ) ( b ) k α ( ~ δ ) = k φ J ~ ( b ) − φ J ~ ( b ) k ~ = 0 . Therefore, φ J ~ ( β ( b )) = φ J ~ ( β ( b )), and hence, σ is a bundle homomorphism. Since id J and β are both invertible, σ is a bundle isomorphism.Theorems 1 and 2 together allow us to refer to ˜ A J as the minimal extension of a uniformlycontinuous bundle of C*-algebras A I associated with a given dense embedding α : I → J .Moreover, Theorem 2 allows us to refer to the algebra A j as the algebra at the accumulationpoint j ∈ ( J \ α [ I ]). We will do so for the remainder of the paper. In particular, to representthe classical limit of a quantum theory we can set I = (0 , J = [0 , α : (0 , → [0 , A j over an accumulation point j ∈ ( J \ α [ I ]) for the base space I .The limiting theory described by this algebra often carries additional structure, such as aprivileged family of dynamical automorphisms, or a Lie bracket. These structures can alsobe determined from corresponding dynamical automorphisms or Lie brackets on the fibersover the original base space I , and in the next section, we will proceed to demonstrate thefunctoriality of the determination of all of these structures. Before proceeding, we notethat there are corresponding existence and uniqueness results for the limiting dynamical11tructure and Lie bracket that one can extract from the discussion of §
5. Because they aresomewhat trivial, we refrain from proving the existence and uniqueness of limiting dynamicsand Lie brackets separately in this section; instead, we proceed directly to the discussion offunctoriality.
Now we prove the functoriality of the procedure for constructing limits via uniformly con-tinuous bundles of C*-algebras. We first define a functor F implementing the extension ofa uniformly continuous bundle of C*-algebras as in Theorems 1 and 2. Then, we define afunctor G representing the restriction from a bundle to the C*-algebra of the limit theory inthe fiber over a point in the base space (e.g., ~ = 0 in the classical limit). We understandthe limiting procedure to be represented by the functor L obtained from the composition G ◦ F . This is accomplished in § § § ~ > ~ = 0. We that show the commutatorfunctorially determines the classical Poisson bracket on the limiting algebra. We want to understand the extension from the base space (0 ,
1] to [0 ,
1] as an instance of ageneral procedure applicable to other base spaces. To that end, we understand [0 ,
1] as theone-point compactification of (0 , I of locally compact metric spaces asthe base spaces of possible bundles. Suppose further that one has an assignment to each I ∈ I of a further locally compact metric space C ( I ) and a dense isometric embedding α I : I → C ( I ). We will require that the assignment I ( C ( I ) , α I ) is functorial . Bythis, we mean to require that all suitable functions between spaces I and J in I can becontinuously extended to functions between C ( I ) and C ( J ), in a way that behaves well withthe embeddings α I and α J . It is well known that in general we cannot hope to extendarbitrary continuous functions between I and J to the larger spaces C ( I ) and C ( J ); instead,we must restrict to the proper metric maps between I and J , which are the functions betweenbase spaces that are suitable to be extended to continuous maps between these larger spaces(nLab authors, 2021). Recall that a map α : I → J between topological spaces I and J is called proper if for each compact set K ⊆ J , the inverse image α − [ K ] ⊆ I is compact.The restriction to proper maps is necessary in order to put the one-point compactificationof the base space in the purview of our extension results, as the one-point compactificationis functorial only when restricted to proper maps.12 efinition 8. An assignment I ( C ( I ) , α I ) is functorial if(i) for each proper metric map f : I → J , there is a map C ( f ) : C ( I ) → C ( J ) such that C ( f ) ◦ α I = α J ◦ f , or in other words, the following diagram commutes: I JC ( I ) C ( J ) fα I α J C ( f ) (ii) if f : I → J and g : J → K are proper metric maps, then C ( g ◦ f ) = C ( g ) ◦ C ( f ).With a functorial assignment I ( C ( I ) , α I ), we can simultaneously consider extendingeach bundle over I along α I to C ( I ). Such an assignment gives a standard for taking thesame kind of extension of all of the bundles considered. Moreover, the functoriality of theassignment establishes that the structure of C ( I ) is determined by the structure of I .For example, given any locally compact metric space I , let ˙ I = I ∪ { } denote its one-point compactifiation (we in general denote the added point by 0 because we have in mindthe extension from (0 ,
1] to [0 , ι : I → ˙ I (see, e.g., Engelking, 1989, p. 169,Theorem 3.5.11). Let I be the collection of locally compact, non-compact metric spaceswhose one-point compactification ˙ I is metrizable and whose embedding ι : I → ˙ I is anisometric map. This includes I = (0 , I = R . Then theassignment C ( I ) := ˙ I and α I = ι for each I ∈ I provides us with a starting point we canuse to take the extension of any bundle over a base space I ∈ I . It is well known thatthe one-point compactification is functorial in the sense outlined above for proper metricmaps. In this case, there is a strong sense in which the one-point compactification yields thesame kind of extension for each of the bundles considered, and the structure of the one-pointcompactification ˙ I is determined by the structure of the original space I .In what follows, we will have the one-point compactification in mind, but for now wewill only suppose that we have some functorial assignment or other of an enlarged space C ( I ) and a dense isometric embedding α I to each I ∈ I . Given such an assignment, we willrestrict attention to a subcategory UBunC ∗ Alg I of UBunC ∗ Alg consisting solely of theuniformly continuous bundles of C*-algebras over base spaces I ∈ I . Moreover, we restrictattention to morphisms that are proper maps between base spaces. Definition 9.
The category
UBunC ∗ Alg I consists in: • objects : uniformly continuous bundles of C*-algebras whose base space I belongs to I ; • morphisms : homomorphisms σ = ( α, β ) : A I → B J between uniformly continuousbundles of C*-algebras with I, J ∈ I , where α is proper.Similarly, letting C ( I ) := { C ( I ) | I ∈ I} , the category UBunC ∗ Alg C ( I ) is the subcategoryof UBunC ∗ Alg consisting of the uniformly continuous bundles of C*-algebras over basespaces C ( I ) ∈ C ( I ) with homomorphisms between them that act as proper maps betweenbase spaces. We now use the data contained in the functor C to define an extension functor F : UBunC ∗ Alg I → UBunC ∗ Alg C ( I ) . 13e define F on objects and morphisms by A I ˜ A C ( I ) ( α, β ) ( C ( α ) , β ) (4)where C ( I ) is the given enlarged base space assigned to I with dense metric embedding α I : I → C ( I ), and ˜ A C ( I ) is the unique minimal extension of A I associated with α I guaranteedby Theorems 1 and 2. Similarly, ( α, β ) is a morphism between uniformly continuous bundlesof C*-algebras A I = (( A ~ , φ I ~ ) ~ ∈ I , A ) and B J = (( B ~ , φ J ~ ) ~ ∈ J , B ), and C ( α ) is the functorialextension of α : I → J to C ( α ) : C ( I ) → C ( J ). In our construction of the extension ˜ A C ( I ) along α I , we defined the C*-algebra of continuous sections of ˜ A C ( I ) to be identical with theC*-algebra A of continuous sections in A I ; we only needed to extend their assignments byspecifying new evaluation maps φ C ( I ) ~ for ~ ∈ (cid:0) C ( I ) \ α I [ I ] (cid:1) . Hence, it makes sense to leave β : A → B unchanged in the extension. It is simple to check that this assignment respectsthe composition of morphisms, so F is indeed a functor. Proposition 1.
Given any functorial assignment C of dense metric embeddings α I : I → C ( I ) to locally compact metric spaces I ∈ I , the assignment F defined by (4) is a functor. This shows a sense in which, relative to the embeddings encoded in C , the constructionof bundle extensions is natural. It requires no further information or structure. Moreover,when I = (0 ,
1] and C is the one-point compactification, then the extension of a bundle over I to a bundle over C ( I ) = [0 ,
1] is an instance of this natural construction. This is preciselythe form of our extension of bundles for ~ > ~ = 0.We encode the remainder of the limit in the restriction from the base space [0 ,
1] to thestructure of the limiting algebra at ~ = 0. We now restrict ourselves to the case where I isthe collection of all locally compact, non-compact metric spaces whose one-point compacti-fication C ( I ) := ˙ I is such that the canonical embedding α I := ι : I → C ( I ) is an isometricmap. We will use this fixed C = C and I = I for the remainder of this section. Since thereis a unique point 0 I ∈ ( C ( I ) \ α I [ I ]), the restriction of a bundle over C ( I ) to the fiber over ~ = 0 I is a natural construction. We now define the restriction functor to the fiber over 0 I .The restriction functor will have the category UBunC ∗ Alg C ( I ) as its domain. Thecodomain category of the restriction functor is simply the category of C*-algebras. Definition 10.
The category C ∗ Alg consists in: • objects : C*-algebras; • morphisms : *-homomorphisms.Our restriction functor G : UBunC ∗ Alg C ( I ) → C ∗ Alg is now defined through thefollowing actions on objects and morphisms:˜ A C ( I ) = (cid:18)(cid:16) A ~ , φ C ( I ) ~ (cid:17) ~ ∈ C ( I ) , A (cid:19) A I σ = ( α, C ( α ) , β ) σ , (5)14here we specify σ (cid:16) φ C ( I )0 I ( a ) (cid:17) := φ C ( J )0 J ( β ( a )) (6)for any a ∈ A . Since C ( α ) must map 0 I to 0 J (which follows from the universal propertyof the one-point compactification), we know that σ is well-defined and a *-homomorphism(by Lemma 3). We can understand the morphism σ to be the classical limit in C ∗ Alg ofthe morphism σ in UBunC ∗ Alg C ( I ) . Furthermore, it is again simple to check that thisassignment respects composition of morphisms, so it follows that G is a functor. Proposition 2.
When C is the one-point compactification, the assignment G defined by(5) is a functor. This shows a sense in which the restriction of a bundle over C ( I ) to the fiber over 0 I isnatural given the structure of the category UBunC ∗ Alg C ( I ) . And finally, we can define afunctor L := G ◦ F , which provides a natural construction of the limiting C*-algebra. In this section, we show that the limiting construction is functorial even when we considerthe further structure of dynamics. We understand dynamics on a C*-algebra to be encodedin a one-parameter family of automorphisms. Our construction of a limiting C*-algebra fora bundle allows us to obtain limiting dynamics, under the condition that the dynamics scalescontinuously with the limiting parameter, which we make precise as follows.Suppose again that A I = (( A ~ , φ I ~ ) ~ ∈ I , A ) is a uniformly continuous bundle of C*-algebrasand allow ( τ t ; ~ ) t ∈ R to denote the dynamics on each fiber algebra A ~ . To enforce that thesedynamics scale continuously with ~ , we require that they lift to an automorphism group onthe algebra of sections. That is, we require the existence of a one-parameter automorphismgroup ( τ t ) t ∈ R on A satisfying τ t ; ~ ◦ φ I ~ = φ I ~ ◦ τ t for all ~ ∈ I, t ∈ R . (7)This is just the requirement that we are considering (in some sense) the “same” dynamicsat different scales. We now define objects that contain such dynamics. Definition 11. A dynamical bundle of C*-algebras is a uniformly continuous bundle of C*-algebras A I = (( A ~ , φ I ~ ) ~ ∈ I , A ) over a locally compact, non-compact metric space I ∈ I witha one-parameter group of automorphisms ( τ A t ) t ∈ R on A .Further, we define a category of dynamical bundles of C*-algebras that encodes the extrastructure of the dynamics. For the moment, we return to the case of some generality andsuppose only that I is a non-empty collection of locally compact metric spaces that serveas the base spaces of possible bundles, and that I ( C ( I ) , α I ) is a functorial assignmentof extended locally compact metric spaces C ( I ) with dense embeddings α I : I → C ( I ).As before, we now restrict attention to categories of dynamical bundles whose base spacesbelong to this collection. Definition 12.
The category
DBunC ∗ Alg I consists in:15 objects : dynamical bundles of C*-algebras ( A I , ( τ A t ) t ∈ R ) with I ∈ I ; • morphisms : homomorphisms σ = ( α, β ) : A I , → B J between uniformly continuousbundles of C*-algebras with I, J ∈ I , where α is proper and and for each t ∈ R τ B t ◦ β = β ◦ τ A t . Note that a morphism in
DBunC ∗ Alg is just a morphism in
UBunC ∗ Alg that furthermorepreserves the structure of the dynamics.We will first define an extension functor F D , and then define a restriction functor G D ;the classical limit will again be understood as the composition L D = G D ◦ F D .The extension functor F D : DBunC ∗ Alg I → DBunC ∗ Alg C ( I ) is defined to act onobjects and morphisms as follows: A I = (cid:16)(cid:0) A ~ , φ I ~ (cid:1) ~ ∈ I , A , ( τ A t ) t ∈ R ) (cid:17) (cid:0) F ( A I ) , ( τ A t ) t ∈ R (cid:1) σ F ( σ ) . (8) F D sends a dynamical bundle A I to its unique extension associated with α I with the samedynamics; and F D extends morphisms just as F does. It is immediate that the extension ofany dynamics-preserving morphism will also preserve dynamics, so that given a morphism σ in DBunC ∗ Al g I , F ( σ ) is indeed a morphism in DBunC ∗ Alg C ( I ) . This assignment is afunctor by Proposition 1. Proposition 3.
Given any functorial assignment C of dense metric embeddings α I : I → C ( I ) to locally compact metric spaces I ∈ I , the assignment F D is a functor. Next, we define a restriction functor G D . We specify now that C = C will be theone-point compactification, as above, and so I = I will again consist in those locallycompact, non-compact metric spaces whose embeddings in their one-point compactificationsare isometric maps. The codomain of our restriction functor will be a category of algebrasof quantities of classical theories, now with dynamics, defined as follows. Definition 13.
The category DC ∗ Alg consists in: • objects : pairs ( A , ( τ A t ) t ∈ R ), where A is a C*-algebra and ( τ A t ) t ∈ R is a one-parametergroup of automorphisms on A ; • morphisms : morphisms σ : ( A , ( τ A t ) t ∈ R ) → ( B , ( τ B t ) t ∈ R ), each of which is a *-homomorphism σ : A → B such that for each t ∈ R , τ B t ◦ σ = σ ◦ τ A t . With this category in hand, we define the functor G D : DBunC ∗ Alg C ( I ) → DC ∗ Alg as follows. Given any dynamical bundle of C*-algebras A I = (cid:16)(cid:0) A ~ , φ I ~ (cid:1) ~ ∈ I , A , ( τ A t ) t ∈ R (cid:17) , onecan restrict the dynamics to a fiber, defining for each t ∈ R an automorphism τ A t ; ~ of A ~ by τ A t ; ~ (cid:0) φ I ~ ( a ) (cid:1) := φ I ~ (cid:0) τ A t ( a ) (cid:1) , (9)16or all a ∈ A . We now use this definition in the case where the base space is C ( I ) and wefocus on the the fiber over 0 I ∈ C ( I ). Note that the way we construct limiting dynamics τ A t ;0 I at ~ = 0 I is the same as the way that G acts on morphisms because the dynamics arerepresented by *-homomorphisms (compare Equations (6) and (9)). In other words, we have τ A t ;0 I = G ( τ A t ). Hence, we define G D to act on objects by˜ A C ( I ) = (cid:16) ( A ~ , φ C ( I ) ~ ) ~ ∈ C ( I ) , A , ( τ A t ) t ∈ R (cid:17) (cid:16) A I , ( τ A t ;0 I ) t ∈ R (cid:17) σ G ( σ ) (10)It follows immediately from Proposition 2 that for any morphism σ in DBunC ∗ Alg C ( I ) , G D ( σ ) is a morphism in DBunC ∗ Alg , so G D is well-defined and a functor. Proposition 4.
When C is the one-point compactification, the assignment G D defined by(10) is a functor. Finally, we can define the classical limit functor for dynamics as the composition ofthe extension functor and restriction functor. Thus, we define L D := G D ◦ F D . It followsimmediately that L D is a functor, showing one sense in which the construction of limitingdynamics is natural.We close this section with two remarks about the limiting dynamics determined by L D .First, it follows trivially that the limiting dynamics exists and is unique (up to morphism in DC ∗ Alg ). Second, it follows from results in Landsman (1998a) that the dynamics definedby certain classes of Hamiltonians in quantum theories determine by L D a correspondingclassical Hamiltonian dynamics (For bounded Hamiltonians, this follows from Proposition2.7.1, p.138; for Hamiltonians at most quadratic in p and q , this follows from Corollary 2.5.2,p. 141). Thus, the functor L D defines dynamics that are not just natural, but also reproducethe physical dynamics that one expects upon convergence in the ~ → In this section, we restrict attention to the classical limit of quantum theory, and so we leavebehind other kinds of limits that may be captured by continuous bundles of C*-algebras.We will thus restrict attention to the kinds of bundles used for quantization and the classicallimit, but we here define a category that contains as objects slight generalizations of thebundles over (0 ,
1] generated by strict quantizations. Our generalization allows one to includebundles over base spaces that are not themselves identified with (0 , C as the one-point com-pactification and I as the collection of locally compact, non-compact topological spaceswhose embedding in their one-point compactification is an isometric map. For ease of nota-tion, given a base space I ∈ I , we let | ~ | I := d C ( I ) ( α I ( ~ ) , I ) (11)17enote the distance in the one-point compactification C ( I ) = ˙ I of ~ (as it is embedded by α I )from 0 I . For example, in the base space I = (0 ,
1] with ˙ I = [0 , | ~ | I = | ~ − | = ~ for each ~ ∈ I . Now we define a class of bundles generalizing those produced by strictquantizations, as follows. Definition 14.
A uniformly continuous bundle of C*-algebras A I = (( A ~ , φ I ~ )) ~ ∈ I , A ) overa locally compact, non-compact metric space I ∈ I is called a post-quantization bundle ifthere is a subset P A ⊆ A such that(i) P A is norm dense in A ;(ii) for each pair a, b ∈ P A , there is a c a,b ∈ P A such that φ I ~ ( c a,b ) = i | ~ | I [ φ I ~ ( a ) , φ I ~ ( b )] for all ~ ∈ I ; and(iii) for each pair a, b ∈ A , lim ~ → I k [ φ I ~ ( a ) , φ I ~ ( b )] k = 0.Condition (iii) guarantees that the classical limit algebra A is commutative. On the otherhand, conditions (i) and (ii) allow us to define a norm dense Poisson algebra of A .Given a post-quantization bundle, we can define a Poisson bracket {· , ·} A on the densesubset φ C ( I )0 I [ P A ] ⊆ A by n φ C ( I )0 I ( a ) , φ C ( I )0 I ( b ) o A := φ ( c a,b ) (12)for every a, b ∈ P A .We will make the structure of post-quantization bundles precise by defining a correspond-ing category of bundles. To do so, we consider morphisms between bundles that preservethe relevant additional structure. We will say that a morphism σ = ( α, β ) between post-quantization bundles A I = (( A ~ , φ I ~ ) ~ ∈ I , A , P A ) and B J = (( B ~ , φ J ~ ) ~ ∈ J , B , P B ) is smooth if β [ P A ] ⊆ P B . Since the subalgebra P A is the collection of quantities whose commutators scaleappropriately with ~ , the smoothness condition guarantees that the image of these quantitiesunder β belongs to P B and hence, have commutators that scale appropriately with ~ as well.This is a pre-condition for the morphism to respect the scaling of commutators with ~ , andfor the classical limit of a morphism to preserve the Poisson bracket.Further, we will say that a morphism σ = ( α, β ) between post-quantization bundles A I = (( A ~ , φ I ~ ) ~ ∈ I , A , P A ) and B J = (( B ~ , φ J ~ ) ~ ∈ J , B , P B ) is second-order if there is someconstant K ≥ (cid:12)(cid:12)(cid:12) | ~ | I − | α ( ~ ) | J (cid:12)(cid:12)(cid:12) → K as ~ → I . This is a generalization of thespecial case of most interest to us for the physics of the classical limit. The special case wehave in mind consists in bundles over the same base space I = (0 ,
1] (with C ( I ) = [0 ,
1] and0 I = 0) and with a re-scaling map α : [0 , → [0 , α has apower series expansion (in a neighborhood of ~ = 0) of the form α ( ~ ) = ∞ X n =0 a n ~ n . (13)In this case, the name “second-order” is meant to suggest that α agrees with the identitymap up to first order in ~ and differs only for powers n >
2. In other words, consider the18ase where a = 0 and a = 1. Thenlim ~ → (cid:12)(cid:12)(cid:12)(cid:12) ~ − α ( ~ ) (cid:12)(cid:12)(cid:12)(cid:12) = lim ~ → (cid:12)(cid:12)(cid:12)(cid:12) ~ + P ∞ n =2 a n ~ n − ~ P ∞ n =1 a n ~ n +1 (cid:12)(cid:12)(cid:12)(cid:12) = lim ~ → (cid:12)(cid:12)(cid:12)(cid:12) a ~ + P ∞ n =3 a n ~ n ~ + P ∞ n =2 a n ~ n +1 (cid:12)(cid:12)(cid:12)(cid:12) = | a | . (14)Hence, in this special case where α has the form α ( ~ ) = ~ + ∞ X n =2 a n ~ n , (15)we know α is second-order according to our definition with K = | a | . A rough physicalinterpretation of the second-order condition is that the scaling behavior of ~ in I agreesenough with α ( ~ ) in J in the limit as one approaches 0 I in I and 0 J in J . In other words,the condition requires that α preserve at least some metrical structure of the base spaces.These definitions allow us to define a category that slightly generalizes the continuousbundles obtained from strict deformation quantizations. Definition 15.
The category
PQBunC ∗ Alg I consists in: • objects : post-quantization bundles whose base space I belongs to I . • morphisms : smooth, second-order morphisms of uniformly continuous bundles of C*-algebras whose map between base spaces is proper.As noted in § PQBunC ∗ Alg I canbe thought of as a subcategory of uniformly continuous bundles of C*-algebras determinedby this construction.Similarly, we define a category of classical theories obtained through the classical limit.This category encodes additional Poisson structure of classical phase spaces. Definition 16.
The category
Class consists in: • objects : pairs ( A , ( P A , {· , ·} A )), where A is a commutative C*-algebra and P A is anorm-dense Poisson algebra with bracket {· , ·} A ; • morphisms : morphisms σ : ( A , ( P A , {· , ·} A )) → ( B , ( P B , {· , ·} B )), each of which is a*-homomorphism σ : A → B satisfying(i) σ [ P A ] ⊆ P B ; and(ii) the restriction of σ to P A is a Poisson morphism, i.e., for each A, B ∈ P A , { σ ( A ) , σ ( B ) } B = σ ( { A, B } A ) . The restrictions on morphisms in
Class again guarantee that smooth quantites get mappedto smooth quantities, and that the morphisms define the further classical structure of thePoisson bracket on the phase space. The canonical example of an object in
Class is a pair( C ( M ) , ( C ∞ c ( M ) , {· , ·} ) for a Poisson manifold ( M, {· , ·} ).19e define a classical limit functor L P analogous to L . Define the functor L P from PQBunC ∗ Alg I to Class on objects and arrows by A I = (cid:16)(cid:0) A ~ , φ I ~ (cid:1) ~ ∈ I , A , P A (cid:17) (cid:16) L ( A I ) , (cid:16) φ C ( I )0 I [ P A ] , {· , ·} A (cid:17)(cid:17) σ L ( σ ) = σ , (16)where {· , ·} A is defined by (12). As mentioned, it follows trivially that the Poisson bracket {· , ·} A both exists and is the unique Lie bracket (up to morphisms in Class ) on the fiber A that continuously extends the commutator for ~ >
0. The following guarantees L P iswell-defined and a functor. Proposition 5.
Suppose A I and B J are objects in PQBunC ∗ Alg I and σ : A I → B J isa morphism (i.e., smooth and second-order). Then L P ( σ ) is a morphism in Class (i.e.,restricts to a Poisson morphism). Hence, L P is a functor.Proof. Suppose σ : A I → B J is a morphism in PQBunC ∗ Alg , where σ = ( α, β ). We needto show that for each a, b ∈ P A , (cid:8) L P ( σ ) (cid:0) φ I I ( a ) (cid:1) , L P ( σ ) (cid:0) φ I I ( b ) (cid:1)(cid:9) = L P ( σ ) (cid:0)(cid:8) φ I I ( a ) , φ I I ( b ) (cid:9)(cid:1) . But, by the definition, we have that the right hand side is L P ( σ ) (cid:0)(cid:8) φ I I ( a ) , φ I I ( b ) (cid:9)(cid:1) = L P ( σ ) (cid:0) φ I I ( c a,b ) (cid:1) = φ J J ( β ( c a,b )) , and the left hand side is (cid:8) L P ( σ ) (cid:0) φ I I ( a ) (cid:1) , L P ( σ ) (cid:0) φ I I ( b ) (cid:1)(cid:9) = (cid:8) φ J J ( β ( a )) , φ J J ( β ( b )) (cid:9) = φ J J (cid:0) c β ( a ) ,β ( b ) (cid:1) . Moreover, comparing these two values, we find (cid:13)(cid:13) φ J J ( β ( c a,b )) − φ J J (cid:0) c β ( a ) ,β ( b ) (cid:1)(cid:13)(cid:13) J = lim ~ → I (cid:13)(cid:13) φ Jα ( ~ ) ( β ( c a,b )) − φ Jα ( ~ ) (cid:0) c β ( a ) ,β ( b ) (cid:1)(cid:13)(cid:13) α ( ~ ) = lim ~ → I (cid:13)(cid:13)(cid:13)(cid:13) i | ~ | I φ Jα ( ~ ) ( β ([ a, b ])) − i | α ( ~ ) | J (cid:2) φ Jα ( ~ ) ( β ( a )) , φ Jα ( ~ ) ( β ( b )) (cid:3)(cid:13)(cid:13)(cid:13)(cid:13) α ( ~ ) = lim ~ → I (cid:13)(cid:13)(cid:13)(cid:13) i | ~ | I φ Jα ( ~ ) ( β ([ a, b ])) − i | α ( ~ ) | J φ Jα ( ~ ) ( β ([ a, b ])) (cid:13)(cid:13)(cid:13)(cid:13) α ( ~ ) = lim ~ → I (cid:12)(cid:12)(cid:12)(cid:12) | ~ | I − | α ( ~ ) | J (cid:12)(cid:12)(cid:12)(cid:12) (cid:13)(cid:13) φ Jα ( ~ ) ( β ([ a, b ])) (cid:13)(cid:13) α ( ~ ) = 0 . This shows that the construction of the classical Poisson bracket from the Lie bracket com-mutator is also, in a sense, natural. 20
Conclusion
In this paper, we have shown that the bundles of C*-algebras used to represent the limits ofphysical theories dependent on a parameter can be uniquely extended to limiting values ofthat parameter. Moreover, by defining morphisms between bundles of C*-algebras, we haveshown the functoriality of the determination of limiting C*-algebraic structure, dynamicalstructure, and Lie brackets in the case of the classical limit.We remark on two avenues for further investigation. First, while our results on limitingdynamics apply to a class of Hamiltonians, it is not clear whether they apply to all physicallyinteresting dynamics. It might be of particular interest to analyze whether or when chaoticdynamics fall under the purview of our results in § Acknowledgements
The authors thank the participants of the 2020 Skylonda Salon for helpful comments anddiscussion, especially Gordon Belot, Clara Bradley, and Laura Ruetsche for their detailedwritten comments on an earlier draft. We are further indebted to the members of the UWQuantization Lab, especially Kade Cicchella and Michael Clancy for helpful discussions.BHF also thanks the participants of the 2019 Munich Workshop on Categorical Approachesto Reduction and Limits. Both authors were supported during the completion of this workby the National Science Foundation under Grant No. 1846560.
AppendixA From Quantization to Bundles
We remarked in § same continuous bundles. Wenow analyze the sense in which this holds using category-theoretic tools.First, we outline the construction of a vanishingly continuous bundle of C*-algebrasfrom a strict deformation quantization (( A ~ , Q ~ ) ~ ∈ I , P ). Here, we stick with vanishinglycontinuous bundles of C*-algebras to align our discussion with the literature, but the resultsof Appendix B show that one could pick any of the equivalent definitions. The idea isto define a (unique) C*-algebra of vanishingly continuous sections, which we will denote C ∗ ( Q ) ⊆ Q ~ ∈ I A ~ , satisfying the condition that for each A ∈ P , there is an a ∈ C ∗ ( Q )such that a ( ~ ) = Q ~ ( A ). To that end, we define the smallest *-subalgebra Q ⊆ Q ~ ∈ I A ~ a = [ ~
7→ Q ~ ( A )] as follows Q = span C ( a ∈ Y ~ ∈ I A ~ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a = [ ~
7→ Q ~ ( A ) · ... · Q ~ ( A n )] for some A , ..., A n ∈ P )! . (Here, we explicitly reference the scalars C in the linear span, denoted span C , to distinguishfrom what follows.) It should be clear that for this construction to work, we need to restrictattention to strict quantizations satisfying the condition that ~
7→ kQ ~ ( F ) k ~ is continuous forevery polynomial F of classical quantities A , ..., A n ∈ P . Further, to ensure the collectionof vanishingly continuous sections C ∗ ( Q ) satisfies the vanishing completeness condition, wemust also include each section that differs in norm from all elements of Q by a continuousfunction vanishing at infinity, as follows: C ∗ ( Q ) = ( a ′ ∈ Y ~ ∈ I A ~ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ ~
7→ k a ′ ( ~ ) − a ( ~ ) k ~ ] ∈ C ( I ) for each a ∈ span C Q ) Landsman (1998a, Theorem 1.2.3-4, p. 111) shows that this construction produces a uniquevanishingly continuous bundle of C*-algebras (( A ~ , φ I ~ ) ~ ∈ I , A ) by defining A = C ∗ ( Q ) φ I ~ ( a ) = a ( ~ ) for each a ∈ C ∗ ( Q ) . We would like to understand the construction of a bundle of C*-algebras from a strict de-formation quantization as functorial. To that end, we define a category of strict deformationquantizations that encodes the structure needed for the construction.
Definition 17.
The category
SDQuant consists in: • objects : strict deformation quantizations A DI = (( A ~ , Q A ~ ) ~ ∈ I , P A ), where I ⊆ R islocally compact and contains 0, and ~
7→ kQ ~ ( F ) k ~ is continuous for every polynomial F of classical quantities A , ..., A n ∈ P . • morphisms : a morphism σ : A DI → B DJ between strict deformation quantizations A DI = (( A ~ , Q A ~ ) ~ ∈ I , P A ) and B DJ = (( B ~ , Q B ~ ) ~ ∈ I , P B ) consists in a triple σ = (cid:0) α, ( σ ~ ) ~ ∈ I , ρ (cid:1) , where α : I → J is a metric map such that α (0) = 0, the map ρ : C ( I ) → C ( J ) isa *-homomorphism, and for each ~ ∈ I , σ ~ : A ~ → B α ( ~ ) is a *-homomorphism suchthat σ [ P A ] ⊆ P B and σ |P A is a Poisson morphism. Moreover, we require that thesemaps satisfy Q Bα ( ~ ) ( σ ( A )) = σ ~ (cid:0) Q A ~ ( A ) (cid:1) ρ ( f )( α ( ~ )) = f ( ~ ) . for each A ∈ P A and f ∈ C ( I ). 22e should pause to make one remark concerning the additional map ρ : C ( I ) → C ( J )in our definition of a morphism. To motivate the inclusion of ρ as part of a structure-preserving map, we note the importance of the structure C ( I ) in this construction of avanishingly continuous bundle. As Landsman (2003b) remarks, the C*-algebra of sectionsof a vanishingly continuous bundle of C*-algebras can be thought of as a C ( I )-algebra in acanonical way, and this structure is sufficient to determine the bundle (see Landsman (2017,p. 737-8) and Nilsen (1996)). Along with this, one can characterize the sections determinedby a quantization map directly in terms of the action of C ( I ). Proposition 6. C ∗ ( Q ) = span C ( I ) ( Q ) , where the overline denotes the closure under thesupremum norm in Q ~ ∈ I A ~ .Proof. We prove equality by proving ( ⊆ ) and ( ⊇ ).( ⊆ ) Let a ∈ C ∗ ( Q ). The argument in Lemma 1.2.2 of Landsman (2017, p. 110) implies a ∈ span C ( I ) ( Q ).( ⊇ ) Let a ∈ span C ( I ) ( Q ). Then there is a sequence a n ∈ span C ( I ) ( Q ) such that a n → a in norm as n → ∞ . But each a n ∈ C ∗ ( Q ) since C ∗ ( Q ) forms the algebra of vanishinglycontinuous sections of a bundle. Hence, since C ∗ ( Q ) is norm closed, a ∈ C ∗ ( Q ).Hence, there is reason to think of the action of C ( I ) on continuous sections as an essentialpart of the structure of a strict deformation quantization, at least as it is used to constructa bundle of C*-algebras. We encode this action in the structure of our category by requiringstructure-preserving morphisms of quantizations to preserve the action of C ( I ).Now we define a functor H : SDQuant → VBunC ∗ Alg that acts on objects andmoprhism by (cid:16)(cid:0) A ~ , Q A ~ (cid:1) ~ ∈ I , P A (cid:17) (cid:0)(cid:0) A ~ , φ I ~ (cid:1) , C ∗ (cid:0) Q A (cid:1)(cid:1)(cid:0) α, ( σ ~ ) ~ ∈ I , ρ (cid:1) ( α, β ) , where β : C ∗ ( Q A ) → C ∗ ( Q B ) is the unique linear continuous extension of β (cid:0)(cid:2) ~ f ( ~ ) Q A ~ ( A ) · ... · Q A ~ ( A n ) (cid:3)(cid:1) = (cid:2) ~ ρ ( f )( ~ ) Q B ~ ( σ ( A )) · ... · Q B ~ ( σ ( A n )) (cid:3) for f ∈ C ( I ) and A , ..., A n ∈ P A . Indeed, it follows from the definitions that ( α, β ) isa morphism in VBunC ∗ Alg because for each ~ ∈ I and section a ∈ C ∗ ( Q A ), we know σ ~ ( φ I ~ ( a )) = φ Jα ( ~ ) ( β ( a )) and σ ~ is well-defined. Hence, with this definition, H is a functor. B Continuity and Uniform Continuity
Each of our two definitions of bundles of C*-algebras in § § VBunC ∗ Alg and
UBunC ∗ Alg to analyzethe relationship between these definitions. We will prove the following result:
Proposition 7.
There is a categorical equivalence F UV : UBunC ∗ Alg ⇆ VBunC ∗ Alg : G V U . F UV takes each object and morphism in the cat-egory UBunC ∗ Alg and restricts it to the appropriate subcollection of sections whose normvanishes at infinity. For the reverse construction, we note that a collection of vanishinglycontinuous sections is dense in a collection of uniformly continuous sections in the locallyuniform topology. G V U extends each object in
VBunC ∗ Alg by taking the completion ofthe collection of vanishingly continuous sections in this topology and then restricting to thesubcollection of sections whose norm are uniformly continuous and bounded. Similarly, G V U extends morphisms continuously to the completion of the collection of sections and thenrestricts them. Proposition 7 establishes that these constructions are natural and preservestructure. This justifies us in working exclusively with uniformly continuous bundles ofC*-algebras, as we do throughout both this paper and the sequel.We now proceed to the proof establishing a categorical equivalence between
VBunC ∗ Alg and
UBunC ∗ Alg . Recall that a functor F : C → D is a categorical equivalence iff there isa functor G : D → C that is “almost inverse” to F in the sense that for each object X in C , there is an isomorphism η X from G ◦ F ( X ) to X in C that makes the following diagramcommute for all morhpisms f : X → Y in C : G ◦ F ( X ) XG ◦ F ( Y ) Y η X G ◦ F ( f ) fη Y (And vice versa for F ◦ G and objects in D .) In this case, η is a natural isomorphism between G ◦ F and the identity functor 1 C . One can characterize or establish a categorical equivalenceby providing this “almost inverse” functor (Awodey, 2010, Proposition 7.26, p. 173).In the case of VBunC ∗ Alg and
UBunC ∗ Alg , we will provide a pair of almost inversefunctors through an intermediary category. This intermediary category corresponds to afurther definition of a continuous bundle-like structure of C*-algebras that is often used inthe mathematical literature, which we prove to be equivalent along the way.
Definition 18 (Dixmier, 1977) . A continuous field of C*-algebras over a (now arbitrary)topological space I is a family of C*-algebras ( A ~ ) ~ ∈ I and a subset Γ ⊆ Q ~ ∈ I A ~ . In otherwords, each element a ∈ Γ is a map a : I ` ~ ∈ I A ~ such that a ( ~ ) ∈ A ~ . We call Γ thecollection of continuous sections , we require it to be a *-algebra under pointwise operations,and we require it to satisfy the following conditions:(i) Density.
For each ~ ∈ I , the set { a ( ~ ) | a ∈ Γ } is dense in A ~ .(ii) Locally uniform closure. If a ∈ Q ~ ∈ I A ~ and for each ~ ∈ I and ǫ >
0, there is some a ′ ∈ Γ such that sup ~ ′ ∈ U k a ( ~ ′ ) − a ′ ( ~ ′ ) k ~ ′ ≤ ǫ for some neighborhood U of ~ , then a ∈ Γ. 24iii)
Continuity.
For each a ∈ Γ, the map N a : ~
7→ k a ( ~ ) k ~ is continuous.In Definition 18, note that although the collection Γ of continuous sections is a *-algebra,it is not in general a C*-algebra in a natural way. In general, elements of Γ may be unboundedin the supremum norm. One can see this, for example, because the condition of locallyuniform closure implies(ii*) Completeness.
For any continuous f : I → C and a ∈ Γ, there is an element f a ∈ Γsuch that f a ( ~ ) = f ( ~ ) a ( ~ ).Since there may be unbounded continuous functions on I , this implies that there may beunbounded continuous sections. However, it is known that one can associate with eachcontinuous field of C*-algebras (( A ~ ) ~ ∈ I , Γ) a canonical C*-algebraΓ A := { a ∈ Γ | N a ∈ C ( I ) } , which consists in continuous sections whose norm vanishes at infinity, with pointwise oper-ations. Similarly, one can associate with each continuous field of C*-algebras (( A ~ ) ~ ∈ I , Γ)over a metric space I a canonical C*-algebraΓ U A := { a ∈ Γ | N a ∈ UC b ( I ) } , which consists in continuous sections whose norm is uniformly continuous and bounded,again with pointwise operations. We now aim to understand the construction of canonicalvanishingly and uniformly continuous bundles associated with Γ A and Γ U A as a functor fromthe category of continuous fields of C*-algebras, which we define as follows. Definition 19.
The category
FieldC ∗ Alg consists in • objects : continuous fields of C*-algebras whose base space is a locally compact metricspace, A FI = (cid:0) ( A ~ ) ~ ∈ I , Γ A (cid:1) ; • morphisms : homomorphisms σ : A FI → B FJ , where each σ is a pair of maps σ = ( α, β ) , where α : I → J is a metric map, β : Γ A → Γ B is a *-homomorphism, and for all a , a ∈ Γ A and ~ ∈ I , if a ( ~ ) = a ( ~ ), then β ( a )( α ( ~ )) = β ( a )( α ( ~ )).We define a pair of functors relating FieldC ∗ Alg and
VBunC ∗ Alg that captures theconstruction described by Kirchberg and Wasserman (1995, p. 677-8) and we show that thisconstruction indeed provides a categorical equivalence. We then use an exactly analogousconstruction to establish a categorical equivalence between
FieldC ∗ Alg and
UBunC ∗ Alg .First, note that there is a difference between continuous fields of C*-algebras and (vanishinglyor uniformly) continuous bundles of C*-algebras that is somewhat trivial: in a continuousfield, the sections are directly specified as functions from the base up to the fibers, whereasin a bundle, the sections are given by an “independent” C*-algebra that is connected to25he fibers via the evaluation maps. To aid in the translation, if we are given a vanishinglycontinuous bundle of C*-algebras (( A ~ , φ I ~ ) ~ ∈ I , A ), we defineˆ A := ( γ a ∈ Y ~ ∈ I A ~ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ a ( ~ ) = φ ~ ( a ) for some a ∈ A ) . Similarly, if we are given a uniformly continuous bundle of C*-algebras (( A ~ , φ I ~ ) ~ ∈ I , A ), wedefine ˆ A := ( γ a ∈ Y ~ ∈ I A ~ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γ a ( ~ ) = φ ~ ( a ) for some a ∈ A ) . Now we can define a functor F F V : FieldC ∗ Alg → VBunC ∗ Alg on objects and mor-phisms by A FI = (cid:0) ( A ~ ) ~ ∈ I , Γ A (cid:1)
7→ A VI := (cid:16)(cid:0) A ~ , φ I ~ (cid:1) ~ ∈ I , Γ A (cid:17) ( α, β ) (cid:16) α, β | Γ A (cid:17) , where φ I ~ ( a ) := a ( ~ ) for all a ∈ Γ A and ~ ∈ I . One can easily check this induces a mapfrom Hom FieldC ∗ Alg ( A FI , B FJ ) to Hom VBunC ∗ Alg ( A VI , B VJ ), i.e., β ( a ) has vanishingly continu-ous norm for any a with vanishingly continuous norm. This follows from Lemma 3.Further, define a functor G V F : VBunC ∗ Alg → FieldC ∗ Alg on objects and morphismsby A VI = (cid:16)(cid:0) A ~ , φ I ~ (cid:1) ~ ∈ I , A (cid:17)
7→ A FI := (cid:0) ( A ~ ) ~ ∈ I , A Γ (cid:1) ( α, β ) ( α, β Γ ) , where A Γ is defined as the completion of ˆ A in the topology τ lu of locally uniform convergence,i.e., A Γ := τ lu ˆ A . More explicitly, τ lu is the vector space topology that has a neighborhood base at 0 consistingin sets of the form O ( K, ǫ ) := (cid:26) γ a ∈ ˆ A (cid:12)(cid:12)(cid:12)(cid:12) sup ~ ∈ K k γ a ( ~ ) k ~ < ǫ (cid:27) for each ǫ > K ⊆ I . Elements of A Γ are equivalence classes of τ lu -Cauchynets with pointwise operations and limiting norm (cf. Ex. 4.1 of Bagarello et al., 2006).Similarly, β Γ := τ lu ˆ β , where ˆ β ( γ a ) = γ β ( a ) for each a ∈ A , and the overline denotes the unique linear, continuous(relative to τ lu ) extension of the map ˆ β to the completion A Γ . We have the following result. Lemma 4. F F V : FieldC ∗ Alg ⇆ VBunC ∗ Alg : G V F is a categorical equivalence. roof. We specify directly two natural isomorphisms χ : G V F ◦ F F V → F , where and 1 F is the identity functor on FieldC ∗ Alg , and η : F F V ◦ G V F → V , where 1 V is the identityfunctor on VBunC ∗ Alg .First, for each object A FI = (( A ~ ) ~ ∈ I , Γ A ) in FieldC ∗ Alg , we define χ A : G V F ◦ F F V (cid:0) A FI (cid:1) → A FI by χ A := (id I , β A ), where id I is the identity map on I so it only remains to specify the map β A : τ lu ˆΓ A → Γ A . We define β A for each [ a δ ] ∈ τ lu ˆΓ A by β A ([ a δ ]) = τ lu lim δ a δ , where [ a δ ] is the equivalence class of nets b δ such that ( a δ − b δ ) → τ lu . We need to showthat β A is a *-isomorphism. First, note that β A is well-defined because if [ a δ ] = [ b δ ], thenthe limits lim δ a δ = lim δ b δ exist and are identical (where these limits are taken relative tothe topology τ lu ). Similarly, β A is injective because if lim δ a δ = lim δ b δ , then [ a δ ] = [ b δ ].Further, β A is surjective: consider any a ∈ Γ A . We will construct a net in ˆΓ A convergingto a with respect to τ lu . Our net will be indexed by the directed set of pairs ( K, ǫ ), where K is a compact subset of I and ǫ > K, ǫ ) (cid:22) ( K ′ , ǫ ′ ) iff K ⊆ K ′ and ǫ ′ ≤ ǫ . Foreach compact subset K ⊆ I and ǫ >
0, we consider the function N a | K : ~ ∈ K
7→ k a ( ~ ) k ~ .The Tietze extension theorem (Munkres, 2000, Theorem 35.1, p. 219) implies there is a f K ∈ C ( I ) such that f K | K = N a | K . Now define g K,ǫ : I → R by g K,ǫ ( ~ ) := f K ( ~ ) N a ( ~ ) + ǫ for all ~ ∈ I . Since g K,ǫ is continuous, a K,ǫ := g K,ǫ a ∈ Γ A . Then for each ~ ∈ I , k a K,ǫ ( ~ ) k ~ = | g K,ǫ ( ~ ) | · k a ( ~ ) k ~ = | f K ( ~ ) | · N a ( ~ ) N a ( ~ ) + ǫ ≤ | f K ( ~ ) | , and hence, N a K,ǫ ∈ C ( I ). Further,sup ~ ∈ K k ( a − a K,ǫ )( ~ ) k ~ = sup ~ ∈ K k (1 − g K,ǫ )( a ( ~ )) k ~ = sup ~ ∈ K | − g K,ǫ ( ~ ) | · N a ( ~ ) ≤ ǫ · N a ( ~ ) N a ( ~ ) + ǫ < ǫ. Hence, for each compact K ⊆ I and each ǫ >
0, for all pairs ( K ′ , ǫ ′ ) (cid:23) ( K, ǫ ), we havesup ~ ∈ K k ( a − a K ′ ,ǫ ′ )( ~ ) k ~ ≤ sup ~ ∈ K ′ k ( a − a K ′ ,ǫ ′ )( ~ ) k ~ < ǫ ′ ≤ ǫ, which implies β A ([ a K,ǫ ]) = a . Hence, β A is surjective.27inally, β A is linear, multiplicative, and *-preserving because operations are definedpointwise on the completion of ˆΓ A with respect to τ lu . Hence, we have established that χ A = (id I , β A ) is an isomorphism in FieldC ∗ Alg .Moreover, clearly the following diagram commutes for each arrow σ : A FI → B FJ in FieldC ∗ Alg : G V F ◦ F F V (cid:0) A FI (cid:1) F (cid:0) A FI (cid:1) G V F ◦ F F V (cid:0) B FJ (cid:1) F (cid:0) B FJ (cid:1) χ A G V F ◦ F F V ( σ ) 1 F ( σ ) χ B Hence, χ is a natural isomorphism.Conversely, for each object A VI = (( A ~ , φ I ~ ) ~ ∈ I , A ) in VBunC ∗ Alg , we define η A : F F V ◦ G V F ( A VI ) → V ( A VI ) by η A = (id I , β A ), so that it only remains to specify the map β A :( A Γ ) → A . We define β A for each [ a δ ] ∈ ( A Γ ) by β A ([ a δ ]) = τ lu lim δ a δ . Note that β A is well-defined because if [ a δ ] = [ b δ ], then lim δ a δ = lim δ b δ . Further, since N [ a δ ] ∈ C ( I ), it follows that for a = lim δ a δ , N a ∈ C ( I ). Of course, β A is surjective becausefor each a ∈ A , the net a δ = a for all δ is such that β A ([ a δ ]) = a . Further, β A is injectivebecause lim δ a δ = lim δ b δ implies [ a δ ] = [ b δ ]. Finally, β A is linear, multiplicative, and *-preserving because addition and involution are τ lu -continuous and multiplication is jointly τ lu -continuous. Hence, we have established that (id I , β A ) is an isomorphism in VBunC ∗ Alg .Clearly, the following diagram commutes for each arrow σ : A VI → B VJ in VBunC ∗ Alg : F F V ◦ G V F (cid:0) A VI (cid:1) V (cid:0) A VI (cid:1) F F V ◦ G V F (cid:0) B VJ (cid:1) V (cid:0) B VJ (cid:1) η A G V F ◦ F F V ( σ ) 1 V ( σ ) η B Hence, η is a natural isomorphism.Next, we define a pair of functors relating FieldC ∗ Alg and
UBunC ∗ Alg that providea categorical equivalence. We define F F U : FieldC ∗ Alg → UBunC ∗ Alg on objects andmorphisms by A FI = (cid:0) ( A ~ ) ~ ∈ I , Γ A (cid:1)
7→ A UI := (cid:16)(cid:0) A ~ , φ I ~ (cid:1) ~ ∈ I , Γ U A (cid:17) ( α, β ) (cid:16) α, β | Γ U A (cid:17) , where φ ~ ( a ) := a ( ~ ) for all a ∈ A and ~ ∈ I . As previously, it follows from Lemma 3 belowthat this induces a map from Hom FieldC ∗ Alg ( A FI , B FJ ) to Hom UBunC ∗ Alg ( A UI , B UJ ).28urther, define a functor G UF : UBunC ∗ Alg → FieldC ∗ Alg on objects and morphismsby A UI = (cid:16)(cid:0) A ~ , φ I ~ (cid:1) ~ ∈ I , A (cid:17)
7→ A FI := (cid:16) ( A ~ ) ~ ∈ I , τ lu ˆ A (cid:17) ( α, β ) (cid:16) α, τ lu ˆ β (cid:17) , where ˆ β ( γ a ) = γ β ( a ) for each a ∈ A , and the overline denotes the unique linear, continuousextension of the map β to the τ lu -completion of ˆ A . Our next categorical equivalence followsthen from the same argument used for Lemma 4, given appropriate replacements of V (or0) by U . Lemma 5. F F U : FieldC ∗ Alg ⇆ UBunC ∗ Alg : G UF is a categorical equivalence. Now we define F UV = F F V ◦ G UF and G V U = F F U ◦ G V F . That these functors providean equivalence follows immediately from the preceding two lemmas. Hence, this completesthe proof of Proposition 7.
References
Aliprantis, C. and Border, K. (1999).
Infinite Dimensional Analysis: A Hitchhiker’s Guide .Springer-Verlag, Berlin.Awodey, S. (2010).
Category Theory . Oxford University Press, New York, 2nd edition.Bagarello, F., Fragoulopoulou, M., Inoue, A., and Trapani, C. (2006). The Completion of aC*-algebra with a Locally Convex Topology.
Journal of Operator Theory , 56(2):357–376.Bieliavsky, P. and Gayral, V. (2015).
Deformation Quantization for Actions of K¨ahlerianLie Groups , volume 236 of
Memoirs of the American Mathematical Society . AmericanMathematical Society, Providence, RI.Binz, E., Honegger, R., and Rieckers, A. (2004). Field-theoretic Weyl Quantization as aStrict and Continuous Deformation Quantization.
Annales de l’Institut Henri Poincar´e ,5:327–346.Bratteli, O. and Robinson, D. (1987).
Operator Algebras and Quantum Statistical Mechanics ,volume 1. Springer, New York.Bratteli, O. and Robinson, D. (1996).
Operator Algebras and Quantum Statistical Mechanics ,volume 2. Springer, New York.Buchholz, D. (1996a). Phase space properties of local observables and structure of scalinglimits.
Annales de l’Institut Henri Poincar´e , 64(4):433–459.Buchholz, D. (1996b). Quarks, gluons, colour: facts or fiction.
Nuclear Physics B , 469:333–353. 29uchholz, D. and Verch, R. (1995). Scaling algebras and renormalization group in algebraicquantum field theory.
Reviews in Mathematical Physics , 7(8):1195.Buchholz, D. and Verch, R. (1998). Scaling algebras and renormalization group in alge-braic quantum field theory. ii. instructive examples.
Reviews in Mathematical Physics ,10(6):775–800.Dixmier, J. (1977).
C*-Algebras . North Holland, New York.Engelking, R. (1989).
General Topology . Heldermann Verlag, Berlin.Haag, R. (1992).
Local Quantum Physics . Springer, Berlin.Honegger, R. and Rieckers, A. (2005). Some Continuous Field Quantizations, Equivalentto the C*-Weyl Quantization.
Publications of the Research Institute for MathematicalSciences, Kyoto University , 41(113-138).Honegger, R., Rieckers, A., and Schlafer, L. (2008). Field-Theoretic Weyl Deformation Quan-tization of Enlarged Poisson Algebras.
Symmetry, Integrability and Geometry: Methodsand Applications , 4:047–084.Kadison, R. and Ringrose, J. (1997).
Fundamentals of the Theory of Operator Algebras .American Mathematical Society, Providence, RI.Kirchberg, E. and Wasserman, S. (1995). Operations on continuous bundles of C*-algebras.
Mathematische Annalen , 303:677–697.Landsman, N. P. (1993a). Deformations of Algebras of Observables and the Classical Limitof Quantum Mechanics.
Reviews in Mathematical Physics , 5(4).Landsman, N. P. (1993b). Strict deformation quantization of a particle in external gravita-tional and Yang-Mills fields.
Journal of Geometry and Physics , 12:93–132.Landsman, N. P. (1998a).
Mathematical Topics Between Classical and Quantum Mechanics .Springer, New York.Landsman, N. P. (1998b). Twisted Lie Group C*-Algebras as Strict Quantizations.
Lettersin Mathematical Physics , 46:181–188.Landsman, N. P. (2003a). Functorial quantization and the Guillemin–Sternberg conjecture.In Ali, S., editor,
Proc. XXth Workshop on Geometric Methods in Physics, Bialowieza .Springer.Landsman, N. P. (2003b). Quantization as a functor. In Voronov, T., editor,
Quantization,Poisson Brackets and beyond , pages 9–24. Contemp. Math., 315, AMS.Landsman, N. P. (2007). Between Classical and Quantum. In Butterfield, J. and Earman, J.,editors,
Handbook of the Philosophy of Physics , volume 1, pages 417–553. Elsevier, NewYork. 30andsman, N. P. (2017).
Foundations of Quantum Theory: From Classical Concepts toOperator Algebras . Springer.Munkres, J. (2000).
Topology . Prentice Hall, Upper Saddle River, NJ, 2nd edition.Nilsen, M. (1996). C*-bundles and C ( X )-algebras. Indiana University Mathematics Journal ,45(2):463–477.nLab authors (2021). one-point compactification. http://ncatlab.org/nlab/show/one-point%20compactification .Revision 39.Rieffel, M. (1989). Deformation Quantization of Heisenberg manifolds.
Communications inMathematical Physics , 122:531–562.Rieffel, M. (1993).
Deformation quantization for actions of R d . Memoirs of the AmericanMathematical Society. American Mathematical Society, Providence, RI.Sakai, S. (1971). C*-algebras and W*-algebras . Springer, New York.van Nuland, T. D. (2019). Quantization and the resolvent algebra.