Separable morphisms of operator Hilbert systems, Pietsch factorizations and entanglement breaking maps
aa r X i v : . [ m a t h . OA ] M a r SEPARABLE MORPHISMS OF OPERATOR HILBERT SYSTEMS, PIETSCHFACTORIZATIONS AND ENTANGLEMENT BREAKING MAPS
ANAR DOSI
Abstract.
In this paper we investigate operator Hilbert systems and their separable morphisms.We prove that the operator Hilbert space of Pisier is an operator system, which possesses the self-duality property. It is established a link between unital positive maps and Pietch factorizations,which allows us to describe all separable morphisms from an abelian C ∗ -algebra to an operatorHilbert system. Finally, we prove a key property of entanglement breaking maps that involvesoperator Hilbert systems. Introduction
The separable morphisms between operator systems play a fundamental role in many aspects ofquantum information theory. A key result proven in [26] by Paulsen, Todorov and Tomforde assertsthat a separability of a linear mapping between finite dimensional matrix algebras is equivalentto its property to be an entanglement breaking mapping. The latter in turn is equivalent to maxmatrix (or min-max matrix) positive mapping of the related operator system structures. Thus aseparable channel can be thought as a max matrix positive mapping between finite-dimensionalmatrix algebras preserving the related traces. Whether the separable morphisms characterize themax matrix positive maps of operator systems was formulated in [26, Problem 6.16] as an openproblem. How to be with the min-max matrix positive maps (see [26, Problem 6.17])? On thisconcern a possible characterization of separable morphisms between some operator systems is ofgreat importance.The operator systems are unital self-adjoint subspaces of the operator space B ( H ) of all boundedlinear operators on a Hilbert space H . They critically occurred in Paulsen’s approach [24] to thenormed quantum functional analysis [16], [28], [19]. Abstract characterization of operator systemswas proposed by Choi and Effros in [2]. They are matrix-ordered ∗ -vector spaces with theirArchimedian matrix order units. In the duality concept (see [10], [11]) they are weakly closed,unital, separated, quantum cones on a ∗ -vector space X with a unit e . Recall that a quantumcone C on X is a quantum additive subset of the hermitian matrix space M ( X ) h over X such that a ∗ C a ⊆ C for all scalar matrices a ∈ M . If C − e is an absorbent quantum set in M ( X ) h , thenwe say that C is unital, where e = { e ⊕ n : n ∈ N } . If C ∩ − C = { } , the quantum cone is called aseparated one. The operator system structures of ordered spaces were investigated in [25] and [26].They can be treated as quantizations of unital cones in a unital ∗ -vector space. Tensor productsof operator systems were considered in [21]. For the quotients, exactness and nuclearity in theoperator system category see [22]. The matrix duality and quantum polars of quantum cones wereinvestigated in [10], [12] and [13]. Based on duality of quantum cones, the classification of operatorsystem structures among the operator space structures on a unital ∗ -vector space was obtainedin [15] (see also [14]). It is proved that the operator system structures on a unital ∗ -vector space X with their unital quantum cones C are in bijection relation with the operator space structures Date : March 28, 2019.2000
Mathematics Subject Classification.
Primary 46L07; Secondary 46B40, 47L25.
Key words and phrases.
Quantum cone, multinormed W ∗ -algebra, quantum system, quantum order . on X with their hermitian unit balls B ; the latter means that B ∗ = B and e ∈ B . Thus thereare no operator column and row Hilbert systems as well as Haagerup tensor product of operatorsystems in their direct proper senses. Nonetheless the operator Hilbert space H o of Pisier turnsout to be an operator system whose matrix norm is equivalent to the orignal matrix norm of H o .That is a key missing object of the theory of operator systems, which plays an important role inthe separability problem mentioned above. Notice that in the finite dimensional case the operatorHilbert system was constructed in [23] by Ng and Paulsen.The present paper is devoted to operator Hilbert systems and their morphisms. First we describethe min and max quantizations of the related unital cone c of a unital Hilbert ∗ -space H , andthe related state space of the cone. To be precise, fix a unital Hilbert ∗ -space H with its unithermitian vector e , and define the σ (cid:0) H, H (cid:1) -closed, unital cone c e = (cid:8) ζ ∈ H h : k ζ k ≤ √ ζ , e ) (cid:9) ,where H is the conjugate Hilbert space, σ (cid:0) H, H (cid:1) is the weak topology obtained by means ofthe canonical duality h· , ·i of the pair (cid:0) H, H (cid:1) . By a quantization of c e we mean a weakly closed,separated, unital, quantum cone C ⊆ M ( H ) h such that C ∩ H = c e . So are the quantizations min c e and max c e , and max c e ⊆ C ⊆ min c e for every quantization C of c e . Using the matrix duality hh· , ·ii of the pair (cid:0) M ( H ) , M (cid:0) H (cid:1)(cid:1) associated with (cid:0) H, H (cid:1) , one can define the quantum polar C ⊡ = (cid:8) η ∈ M (cid:0) H (cid:1) h : hh C , η ii ≥ (cid:9) to be a quantum cone on H . We have also the conjugate cone c e and conjugate quantum cone C on H . In Section 3, we prove that the operator Hilbert space H o is an operator system whose quantum cone C o is a quantization of c e and it is self-dual in thesense of C ⊡ o = C o , that is, H o is a self-dual operator system. Moreover, (max c e ) ⊡ = min c e and(min c e ) ⊡ = max c e .In Section 4, we investigate the positive maps between operator Hilbert systems. Since the min-operator system structure on a unital ∗ -vector space is given by the standard cone C ( X ) + of theabelian C ∗ -algebra C ( X ) of all complex continuous functions on a compact Hausdorff topologicalspace X (see [25]), the characterization of separable morphisms C ( X ) → H plays a key role inthe solution of the min-max matrix positive mapping problem confirmed above. Fix a hermitianbasis F for H containing e , and a probability measure µ on X . A family of real valued Borelfunctions k = { k f : f ∈ F } ⊆ ball L ∞ ( X, µ ) with k e = 1 is said to be an H - support on X if k f ⊥ k e , f = e and P f = e ( v, k f ) ≤ ( v, k e ) in L ( X, µ ) for all v ∈ C ( X ) + . If k is an H -supporton X then T : C ( X ) → H , T v = P f ( v, k f ) f is a unital positive mapping, that is, T (1) = e and T (cid:0) C ( X ) + (cid:1) ⊆ c e . There is a bijection between unital positive maps T : C ( X ) → H and H -supports k on X (see below Theorem 4.1). In this case, T is an absolutely summable mapping,which admits a unique bounded linear extension T k : L ( X, µ ) → H being a unital positivemapping of operator Hilbert systems. Moreover, T k coincides with the Pietsch extension of anabsolutely summable mapping T [29]. The present theory can be treated as an ordered version ofPietsch factorizations for absolutely summable maps.Recall that a positive mapping φ : V → W of operator systems is called separable if φ = P l p l ⊙ q l is a sum of 1-rank operators made from positive functionals q l on V and positive elements p l in W in the sense of φ ( v ) = lim k P kl =1 q l ( v ) p l in W for every v ∈ V . We obtain the followingcharacterization of the separable morphisms from C ( X ) to H . A morphism C ( X ) → H isseparable iff its support k on X is maximal, in the sense of P f = e k f ≤ L ∞ ( X, µ ). Thus theseparable morphisms C ( X ) → H are in bijection relation with the maximal supports on X . Inthis case, all extensions T k : L ( X, µ ) → H are Hilbert-Schmidt operators, whereas the originalseparable morphisms T : C ( X ) → H are nuclear operators.It is known [26] that a linear mapping φ : M n → ( M m , max M + m ) is matrix positive iff φ : M n → M m is separable. A problem of Paulsen-Todorov-Tomforde from [26] asks that whether the latter EPARABLE MORPHISMS OF OPERATOR HILBERT SYSTEMS 3 statement characterizes the matrix positive maps φ : V → ( W , max W + ) of operator systems. Weprovide an example of a morphism between operator Hilbert systems which is not separable.Namely, let H be an infinite dimensional Hilbert space with its hermitian basis F . Fix e, u ∈ F ,which in turn define the unital cones c e and c u in H , respectively. If T ∈ B ( H ) is a unitary given by T = u ⊙ e + e ⊙ u + P f = u,e f ⊙ f , then the matrix positive mapping T : ( H, max c u ) → ( H, max c e )given by T is not separable.Finally, we consider the finite dimensional case, and prove that the unital cone c e of the 2-dimensional Hilbert space ℓ admits only one quantization, that is, min c e = max c e . As anapplication to quantum information theory we prove the following key property of the operatorHilbert systems. Let H be an operator Hilbert system, M either a finite-dimensional von Neumannalgebra or another operator Hilbert system, and let ϕ : H → M be a linear mapping. Then ϕ is an entanglement breaking mapping iff ϕ ∗ : M ∗ → (cid:0) H, max c e (cid:1) is matrix positive. Similarly, ϕ ∗ : H → M ∗ is an entanglement breaking mapping iff ϕ : M → ( H, max c e ) is matrix positive.2. Preliminaries
In this section we introduce some preliminary notions and results. The vector space of all m × n -matrices v = [ v ij ] i,j over a complex vector space V is denoted by M m,n ( V ), and we set M m ( V ) = M m,m ( V ) and M m,n = M m,n ( C ). Further, M ( V ) (respectively, M ) denotes the vectorspace of all infinite (respectively, scalar) matrices over V with only finitely many non-zero entries.A linear mapping ϕ : V → W admits the canonical linear extensions ϕ ( n ) : M n ( V ) → M n ( W )(respectively, ϕ ( ∞ ) : M ( V ) → M ( W )) over all matrix spaces defined as ϕ ( n ) (cid:16) [ v ij ] i,j (cid:17) = [ ϕ ( v ij )] i,j (respectively, ϕ ( ∞ ) | M n ( V ) = ϕ ( n ) ). Notice that ϕ ( ∞ ) preserves the standard matrix operations.2.1. The quantum duality.
By a quantum set B on V we mean a collection B = ( B n ) ofsubsets B n ⊆ M n ( V ), n ≥
1. Sometimes we write B ∩ M n ( V ) instead of B n . If B and C are quantum sets on V then we put B ⊆ C whenever B n ⊆ C n , n ≥
1. In a similar way, all set-theoretic operations and basic algebraic operations can be defined over all quantum sets on V . TheMinkowski functional of an absorbent (in M ( V )) absolutely matrix convex set (see [17]) is called a matrix seminorm on V . A polynormed (or locally convex) topology defined by a separatingfamily of matrix seminorms is called a quantum topology , and the vector space V equipped with aquantum topology is called a quantum space. Thus a quantum topology t on V can be identifiedwith a filter base of absorbent, absolutely matrix convex sets on V such that { ε U : U ∈ t , ε > } is a neighborhood filter base of the origin with respect to the relevant polynormed topology in M ( V ). In particular, it inherits a polynormed topology t | M n ( V ) in each M n ( V ). Note that t | M n ( V ) = ( t | V ) n [7] (see also [5]), where ( t | V ) n indicates the direct product topology in V n generated by t | V . Conversely, each polynormed topology t in V is a trace of a certain quantumtopology t in M ( V ) called its quantization , that is, t = t | V . All these quantizations are runningwithin min and max quantizations [17], that is, if t is a quantum topology on V with t = t | V ,then min t ⊆ t ⊆ max t . A quantum space whose quantum topology is given by a matrix normis a called an operator (or quantum normed ) space. By a morphism between quantum spaces wemean a matrix continuous linear mapping.
A linear mapping ϕ : V → W between quantum spacesis matrix continuous iff ϕ ( ∞ ) : M ( V ) → M ( W ) is a continuous linear mapping of the relevantpolynormed spaces.Let ( V, W ) be a dual pair of vector spaces with the pairing h· , ·i : V × W → C . This pairingdefines a quantum (or matrix ) pairing hh· , ·ii : M m ( V ) × M n ( W ) → M mn , hh v, w ii = [ h v ij , w st i ] ( i,s ) , ( j,t ) = w ( m ) ( v ) = v ( n ) ( w ) , ANAR DOSI where v = [ v ij ] i,j ∈ M m ( V ), w = [ w st ] s,t ∈ M n ( W ), which are identified with the canonical linearmaps v : W → M m , v ( y ) = [ h v ij , y i ] i,j , and w : V → M n , w ( x ) = [ h x, w st i ] s,t , respectively. Thesame size matrix spaces M n ( V ) and M n ( W ) are also in the canonical duality determined by thescalar pairing h· , ·i : M n ( V ) × M n ( W ) → C , h v, w i = X i,j h v ij , w ij i . So, we have the weak and Mackey topologies σ ( M n ( V ) , M n ( W )) and κ ( M n ( V ) , M n ( W )), re-spectively. Actually, σ ( M n ( V ) , M n ( W )) = σ ( V, W ) n and κ ( M n ( V ) , M n ( W )) = κ ( V, W ) n (see [30, 4.4.2, 4.4.3] and [6]). If V = W = C then the scalar pairing h· , ·i : M n × M n → C is given by h a, b i = P i,j a ij b ji = τ ( ab t ) = τ ( a t b ), where τ is the trace on M n and a t (or b t )indicates to the transpose matrix. This duality defines the trace class norm k a k = τ ( | a | ) =sup {|h a, b i| : b ∈ ball M n } , a ∈ M n . The space M n equipped with the norm k·k is denoted by T n ,which is the predual of the von Neumann algebra M n . The following assertion was proved in [8]. Theorem 2.1.
Let ( V, W ) be a dual pair. The weak topology σ ( V, W ) admits only one quantization s ( V, W ) called the weak quantum topology of the dual pair ( V, W ) . The quantum topology s ( V, W ) has the defining family { p w : w ∈ M ( W ) } of matrix seminorms,where p w ( v ) = khh v, w iik (see [8]). Thus min σ ( V, W ) = max σ ( V, W ) = s ( V, W ). Moreover, s ( V, W ) | V n = ( s ( V, W ) | V ) n = σ ( V, W ) n = σ ( M n ( V ) , M n ( W )) for all n (see [7], [8]). Aquantum topology t on V is said to be compatible with the duality ( V, W ) if ( V, t | V ) ′ = W .In this case, t has a neighborhood filter base of the origin, which consists of s ( V, W )-closed,absorbent, absolutely matrix convex sets in M ( V ). Moreover, s ( V, W ) (cid:22) t (cid:22) r ( V, W ) [7, Lemma5.1], where r ( V, W ) = max κ ( V, W ).Given a quantum set B in M ( V ) we have its weak closure B − with respect to the weakquantum topology s ( V, W ), and the absolute matrix (or operator ) polar B ⊙ in M ( W ) definedas the quantum set B ⊙ = { w ∈ M ( W ) : sup khh B , w iik ≤ } . One can easily verify that B ⊙ is s ( W, V )-closed, absolutely matrix convex set in M ( W ) (see [7]). Similarly, it is defined theabsolute matrix polar M ⊙ ⊆ M ( V ) of a quantum set M ⊆ M ( W ). If t is a quantum topology on V compatible with the duality ( V, W ) then t ⊙ = { n B ⊙ : B ∈ t , n ∈ N } is a quantum bornologybase, which consists of s ( W, V )-compact quantum sets on W (see [9]). The following quantumversion of the classical bipolar theorem was proved in [17] (see also [18]) by Effros and Webster. Theorem 2.2.
Let ( V, W ) be a dual pair and let B be an absolutely matrix convex set in M ( V ) .Then B ⊙⊙ = B − , where B − is the s ( V, W ) -closure of B . In [10] we found a new proof of the Bipolar Theorem 2.2 based on the duality theory of quantumcones. Thus the method of quantum cones is an alternative tool to investigate quantum spaces.2.2.
Involution and quantum cones. By an involution on a vector space X we mean a con-jugate linear (or ∗ -linear) mapping x x ∗ on X such that x ∗∗ = x for all x ∈ X . A vector spaceequipped with an involution is called a ∗ -vector space . An element x ∈ X is called hermitian if x ∗ = x . The set of all hermitian elements is denoted by X h , which is a real linear subspace in X .It is easy to see that each x ∈ X has a unique decomposition x = Re ( x ) + i Im ( x ) with hermitiansRe ( x ) and Im ( x ). Now assume that X is a ∗ -vector space and ( X , Y ) is a dual pair such thatthe involution on X is σ ( X , Y )-continuous. Then Y possesses the canonical involution y y ∗ , h x, y ∗ i = h x ∗ , y i ∗ . Indeed, the linear functional y ∗ being a composition of weakly continuous map-pings x x ∗ and x
7→ h x, y i turns out to be weakly continuous. Hence y ∗ ∈ Y . In this case( X , Y ) is called a dual ∗ -pair. The involution on X is naturally extended to an involution over EPARABLE MORPHISMS OF OPERATOR HILBERT SYSTEMS 5 the matrix space M ( X ). Namely, if x = [ x ij ] i,j ∈ M n ( X ) then we set x ∗ = (cid:2) x ∗ ji (cid:3) i,j ∈ M n ( X ),whereas x t = [ x ji ] i,j indicates to the transpose of x . Thus M ( X ) turns out to be a ∗ -vector spacetoo. Note that hh x, y ∗ ii = hh x ∗ , y ii ∗ for all x ∈ M ( X ) and y ∈ M ( Y ) (see [12, Lemma 4.1]). If a, b ∈ M and y ∈ M ( Y ) then ( ayb ) ∗ = b ∗ y ∗ a ∗ . Indeed, hh x, ( ayb ) ∗ ii = hh x ∗ , ayb ii ∗ = (( I ⊗ a ) hh x ∗ , y ii ( I ⊗ b )) ∗ = ( I ⊗ b ∗ ) hh x ∗ , y ii ∗ ( I ⊗ a ∗ )= ( I ⊗ b ∗ ) hh x, y ∗ ii ( I ⊗ a ∗ ) = hh x, b ∗ y ∗ a ∗ ii for all x ∈ M ( X ). Further, if x ∈ M n ( X ) and y ∈ M n ( Y ) then h x, y ∗ i = h x ∗ , y i ∗ (see [12] for thedetails), that is, ( M n ( X ) , M n ( Y )) equipped with the scalar pairing is a ∗ -dual pair as well.Let X be a ∗ -vector space. Then M ( X ) h = { x ∈ M ( X ) : x ∗ = x } is a real subspace of M ( X ).If B ⊆ M ( X ) h is a quantum set then we say that B is a hermitian quantum set on X . A hermitianquantum set C over X is said to be a quantum cone on X if C + C ⊆ C and a ∗ C a ⊆ C for all a ∈ M .A quantum cone C is said to be a quantum ∗ -cone if M ( X ) h = C − C . A quantum cone C on X iscalled a separated quantum cone on X if C ∩ − C = { } . Any C ∗ -algebra A possesses the quantum ∗ -cone M ( A ) + = (cid:0) M n ( A ) + (cid:1) , which is the set of all positive elements in M ( A ). Obviously, anyintersection of quantum cones is a quantum cone. In particular, the quantum cone U c generatedby a quantum set U is well defined.Now let ( X , Y ) be a dual ∗ -pair. If C is a quantum set on X then its quantum polar C ⊡ in M ( Y ) is defined as the quantum set C ⊡ = { y ∈ M ( Y ) h : hh C , y ii ≥ } . The latter is s ( Y , X )-closed quantum cone on Y . If C is a quantum ∗ -cone on X then C ⊡ = { y ∈ M ( Y ) : hh C , y ii ≥ } and it is a separated quantum cone on Y (see [12]). The following bipolar theorem for quantumcones was proved in [12]. Theorem 2.3.
Let ( X , Y ) be a dual ∗ -pair and let C be a s ( X , Y ) -closed, quantum cone on X .Then C = C ⊡⊡ . In this case, for y ∈ M n ( Y ) h we have y ∈ C ⊡ iff h C n , y i ≥ . Let ( X , Y ) and ( X , Y ) be dual ∗ -pairs, and let ϕ : X → X be a weakly continuous ∗ -linearmapping with its algebraic dual mapping ϕ ∗ , that is, h ϕ ( x ) , y i = h x , ϕ ∗ ( y ) i for all x ∈ X and y ∈ Y . Then(2.1) ϕ ∗ ( Y ) ⊆ Y and (cid:10)(cid:10) ϕ ( ∞ ) ( x ) , y (cid:11)(cid:11) = DD x , ( ϕ ∗ ) ( ∞ ) ( y ) EE for all x ∈ M ( X ) and y ∈ M ( Y ). Indeed, ϕ ∗ ( y ) being a composition of the weakly con-tinuous mapping ϕ and σ ( X , Y )-continuous functional y turns out to be σ ( X , Y )-continuous.Therefore ϕ ∗ ( y ) ∈ Y . Moreover, (cid:10)(cid:10) ϕ ( ∞ ) ( x ) , y (cid:11)(cid:11) = [ h ϕ ( x ,i,j ) , y ,s,t i ] ( i,s ) , ( j,t ) = [ h x ,i,j , ϕ ∗ ( y ,s,t ) i ] ( i,s ) , ( j,t ) = DD x , ( ϕ ∗ ) ( ∞ ) ( y ) EE . Notice that ϕ ∗ : Y → Y is a (weakly continuous) ∗ -linear mapping, for h x , ϕ ∗ ( y ∗ ) i = h ϕ ( x ) , y ∗ i = h ϕ ( x ) ∗ , y i ∗ = h ϕ ( x ∗ ) , y i ∗ = h x ∗ , ϕ ∗ ( y ) i ∗ = h x , ϕ ∗ ( y ) ∗ i , x ∈ X . Note also that ϕ ( ∞ ) : M ( X ) → M ( X ) is a ∗ -linear mapping. Indeed, ϕ ( ∞ ) ( x ∗ ) = ϕ ( ∞ ) (cid:16)(cid:2) x ∗ ji (cid:3) i,j (cid:17) = (cid:2) ϕ (cid:0) x ∗ ji (cid:1)(cid:3) i,j =[ ϕ ( x ji ) ∗ ] i,j = [ ϕ ( x ij )] ∗ i,j = ϕ ( ∞ ) ( x ) ∗ for all x ∈ M ( X ). Lemma 2.1.
Let ( X , Y ) and ( X , Y ) be dual ∗ -pairs, C and C quantum sets on X and X ,respectively, and let ϕ : X → X be a weakly continuous ∗ -linear mapping such that ϕ ( ∞ ) ( C ) ⊆ C . Then ( ϕ ∗ ) ( ∞ ) ( y ∗ ) = ( ϕ ∗ ) ( ∞ ) ( y ) ∗ for all y ∈ M ( Y ) , and ( ϕ ∗ ) ( ∞ ) (cid:0) C ⊡ (cid:1) ⊆ C ⊡ . Similarly, if ( ϕ ∗ ) ( ∞ ) ( K ) ⊆ K for quantum sets K and K on Y and Y , respectively, then ϕ ( ∞ ) (cid:0) K ⊡ (cid:1) ⊆ K ⊡ . ANAR DOSI
Proof.
Take y ∈ M ( Y ). For every x ∈ M ( X ) we have DD x, ( ϕ ∗ ) ( ∞ ) ( y ) ∗ EE = DD x ∗ , ( ϕ ∗ ) ( ∞ ) ( y ) EE ∗ = (cid:10)(cid:10) ϕ ( ∞ ) ( x ∗ ) , y (cid:11)(cid:11) ∗ = (cid:10)(cid:10) ϕ ( ∞ ) ( x ) ∗ , y (cid:11)(cid:11) ∗ = (cid:10)(cid:10) ϕ ( ∞ ) ( x ) , y ∗ (cid:11)(cid:11) = DD x, ( ϕ ∗ ) ( ∞ ) ( y ∗ ) EE thanks to (2.1). Hence ( ϕ ∗ ) ( ∞ ) ( y ) ∗ = ( ϕ ∗ ) ( ∞ ) ( y ∗ ). Finally, if y ∈ C ⊡ then ( ϕ ∗ ) ( ∞ ) ( y ) ∈ M ( Y ) h and DD C , ( ϕ ∗ ) ( ∞ ) ( y ) EE = (cid:10)(cid:10) ϕ ( ∞ ) ( C ) , y (cid:11)(cid:11) ⊆ hh C , y ii ≥
0, which means that ( ϕ ∗ ) ( ∞ ) ( y ) ∈ C ⊡ ,that is, ( ϕ ∗ ) ( ∞ ) (cid:0) C ⊡ (cid:1) ⊆ C ⊡ . The rest follows from the symmetry and (2.1). (cid:3) The unital quantum cones.
Let X be a ∗ -vector space with its fixed hermitian element e . We say that ( X , e ) or just X is a unital space. The quantum set ( { e n } ) on X is denoted by e ,where e n = e ⊕ n ∈ M n ( X ) h . A quantum cone C on the unital space ( X , e ) is said to be a unitalquantum cone if C − e is absorbent in M ( X ) h . Note that e ⊆ C and C turns out to be a quantum ∗ -cone if C is a unital quantum cone. Moreover, C − e is a matrix convex set in M ( X ) containingthe origin (see [12] for the details). The quantum set ∩ r> r ( C − e ) is called the algebraic closureof C and it is denoted by C − . Note that C ⊆ C − whenever e ⊆ C . We say that C is a closed ( oran Archimedian ) quantum cone if it coincides with its algebraic closure, that is, C = C − . Noticethat C − is smaller than any (polynormed) topological closure of C . By analogy, a cone c in X issaid to be unital if c − e is absorbent in X h , and it is closed if c − = c , where c − = ∩ r> r ( c − e ) isthe algebraic closure of c . In particular, X h = c − c and e ∈ c . Lemma 2.2.
Let X be a unital ∗ -vector space with its unit e , and let C be a quantum cone on X .If C m is unital in the sense that C m − e ⊕ m is absorbent in M m ( X ) h for some m then C is a unitalquantum cone. In particular, if c is a unital cone in X then c c is a unital quantum cone on X .Proof. Take x ∈ X h . Then x ⊕ m ∈ M m ( X ) h and x ⊕ m + re ⊕ m ∈ C m for some r >
0. Since C is aquantum cone, we deduce that x + re = ε ( x ⊕ m + re ⊕ m ) ε ∗ ∈ c , where ε = (cid:2) . . . (cid:3) ∈ M ,m and c = C . Hence c is a unital cone. In particular, e ∈ c and X h = c − c .Now take x ∈ M n ( X ) h and prove that x + re ⊕ n ∈ C for some r >
0. If x = av ⊕ n for some a ∈ M + n and v ∈ c then x = a / v ⊕ n a / ∈ c c ⊆ C . But if x = − av ⊕ n then − a + rI n ≥ − v + se ∈ c for some real r, s ≥
0. It follows that x + rse ⊕ n = ( − a + rI n ) v ⊕ n + r ( − v ) ⊕ n + rse ⊕ n = ( − a + rI n ) v ⊕ n + r ( − v + se ) ⊕ n ∈ c c .Taking into account that X h = c − c , we conclude that x + re ⊕ n ∈ c c for some r > x = av ⊕ n with a ∈ ( M n ) h and v ∈ X h . Thus c c − e ⊕ n absorbs all hermitians from ( M n ) h ⊗ X h .But M n ( X ) h = ( M n ) h ⊗ X h (see [26, Lemma 3.7]). Hence c c is unital, which in turn implies that C is a unital quantum cone on X . (cid:3) Now let X be a unital ∗ -vector space with its unit e and let ( X , Y ) be a dual ∗ -pair. Consider thefollowing quantum subset M ( Y ) e = { y ∈ M ( Y ) : hh e, y ii = I } in M ( Y ), which is s ( Y , X )-closedand matrix additive set. The following unital bipolar theorem was proved in [13]. Theorem 2.4.
Let ( X , Y ) be a dual ∗ -pair with the unital space X , and let C be a s ( X , Y ) -closed,unital quantum cone on X . Then C = (cid:0) C ⊡ ∩ M ( Y ) e (cid:1) ⊡ . The quantum set C ⊡ ∩ M ( Y ) e from Theorem 2.4 is called a matricial state space of C , andit is denoted by S ( C ). Notice that S ( C ) is a matrix additive subset in M ( Y ) h . In the case, ofa C ∗ -algebra A we write S ( A + ) instead of S (cid:0) M ( A ) + (cid:1) keeping in mind the canonical quantumcone M ( A ) + of positive elements in M ( A ). EPARABLE MORPHISMS OF OPERATOR HILBERT SYSTEMS 7
Finally, let c be a separated, (algebraically) closed, unital cone in X . Recall that a linearfunctional σ : X → C is said to be a state of the cone c if σ ( e ) = 1 and σ ( c ) ≥ σ ispositive). If S ( c ) is the set of all states of the cone c , then k x k e = sup | S ( c ) ( x ) | , x ∈ X is an order ∗ -norm on X in the sense of k x ∗ k e = k x k e , x ∈ X , and k x k e = inf { r > − re ≤ x ≤ re } for all x ∈ X h (see [25]). Put Y to be the normed dual of X equipped with the norm k·k e . Then ( X , Y )is a dual ∗ -pair, S ( c ) ⊆ Y , and c = S ( c ) ⊡ ∩ X [25]. The unital quantum cone S ( c ) ⊡ (with respectto ( X , Y )) is called the minimal quantization min c of the cone c , whereas c ⊡⊡ is the maximalquantization max c of the cone c (see [15]). Thus for every separated, closed, unital quantum cone C with c = C ∩X we have max c ⊆ C ⊆ min c . Notice that max c is the s ( X , Y )-closure of the unitalquantum cone c c generated by c (see Lemma 2.2).2.4. The lattice ideal generated by a Radon measure.
Now let X be a compact Hausdorfftopological space, C ( X ) is the abelian C ∗ -algebra of all complex continuous functions on X equipped with the uniform norm k v k ∞ = sup | v ( X ) | , v ∈ C ( X ), whose topological dual C ( X ) ∗ is reduced to the Banach space M ( X ) of all Radon charges on X . Note that M ( X ) is a ∗ -vectorspace with the natural involution µ µ ∗ , h v, µ ∗ i = h v ∗ , µ i ∗ for all v ∈ C ( X ). The real vectorspace of all hermitian charges is denoted by M ( X ) h , which is equipped with the cone M ( X ) + ofpositive measures on X . It is well known that M ( X ) h is a complete vector lattice with respect tothe vector order induced by means of cone M ( X ) + . The related lattice operations are denotedby ∨ and ∧ , respectively. A real vector subspace V ⊆ M ( X ) h is said to be a closed subspaceif it contains ∨ S (sup) and ∧ S (inf) whenever S ⊆ V . A vector subspace I ⊆ M ( X ) h is saidto be an ideal of M ( X ) h if | λ | ≤ | µ | for λ ∈ M ( X ) h and µ ∈ I implies that λ ∈ I . In thiscase, | µ | , µ + , µ − ∈ I whenever µ ∈ I , and I turns out to be a vector sublattice. Any intersectionof ideals turns out to be an ideal automatically, therefore each subset S ⊆ M ( X ) h generates anideal to be the smallest ideal of M ( X ) h containing S . An ideal which in turn is a closed subspaceis called a closed ideal. Similarly, one can define the closed ideal in M ( X ) h generated by S . Theclosed ideal in M ( X ) h generated by a singleton { µ } is denoted by I µ ( X ). One can prove that I µ ( X ) = I | µ | ( X ), and λ ∈ M ( X ) + belongs to I µ ( X ) iff λ = ∨ { λ ∧ n | µ | : n ∈ N } .Let µ ∈ M ( X ) + . The L p -spaces corresponding to a Radon measure µ ∈ M ( X ) + are denotedby L p ( X, µ ), 1 ≤ p ≤ ∞ , which are ∗ -vector spaces. The Banach space L ( X, µ ) is identified witha closed subspace of M ( X ) up to an isometrical isomorphism such that L ( X, µ ) h = I µ ( X ). Theidentification is given by the mapping L ( X, µ ) → M ( X ), η ηµ , where h h, ηµ i = R h ( t ) η ( t ) dµ for all h ∈ C ( X ). The present result is well known as Lebesgue-Nikodym theorem [1, Ch.V, 5.5,Theorem 2].Further, notice that µ ∈ M ( X ) iff h v, | µ |i = sup { µ ( w ) : w ∈ C ( X ) , | w | ≤ v } < ∞ for every v ∈ C ( X ) + . In this case, | µ | ∈ M ( X ) + and µ = u | µ | for a Borel function u on X such that | u | = 1 almost everywhere with respect to | µ | . The space of all probability measures on X isdenoted by P ( X ), which is a w ∗ -compact subspace in the space M ( X ). Notice that P ( X ) is the w ∗ -closure of the convex hull of its extremal boundary ∂ P ( X ) which consists of Dirac measures δ t , t ∈ X thanks to Krein-Milman theorem.Fix µ ∈ M ( X ) + . Recall that a point s ∈ X is said to be a µ -mass if µ ( s ) >
0. Notice that s is a unique mass with respect to δ s . Lemma 2.3.
A point s ∈ X is a µ -mass iff δ s ∈ I µ ( X ) . In this case δ s = s ′ µ with s ′ ∈ L ( X, µ ) .In this case, s ′ = µ ( s ) − [ s ] .Proof. First assume that s is a µ -mass. Take a Borel set N ⊆ X such that µ ( N ) = 0. Then s / ∈ N , which in turn implies that δ s ( N ) = 0. Hence δ s is absolutely continuous with respectto µ . By Lebesgue-Nikodym theorem, δ s = s ′ µ ∈ I µ ( X ), s ′ ( t ) ≥ µ -almost all t ∈ X , and ANAR DOSI s ′ ∈ L ( X, µ ). But s ′ ( s ) = h s ′ , δ s i = h s ′ , s ′ µ i = R ( s ′ ) dµ , that is, s ′ ∈ L ( X, µ ). Conversely,suppose that δ s ∈ I µ ( X ). Since { s } is a Borel set, the condition µ ( s ) = 0 would imply that δ s ( s ) = 0, a contradiction.Actually, s ′ = µ ( s ) − [ s ]. Indeed, µ ( s ) − [ s ] is a Borel function from L ( X, µ ) and (cid:10) v, (cid:0) µ ( s ) − [ s ] (cid:1) µ (cid:11) = µ ( s ) − Z v [ s ] dµ = µ ( s ) − Z v ( s ) dµ = v ( s ) = h v, δ s i for all v ∈ C ( X ), which means that s ′ = µ ( s ) − [ s ]. (cid:3) Remark 2.1.
Notice that s ′ = dδ s dµ is the Radon-Nikodym derivative of δ s with respect to µ . Pietsch factorization.
Let V be a (Hausdorff) polynormed space. A family ( v i ) i ∈ I in V is said to be an absolutely summable if P i ∈ I k v i k < ∞ for every continuous seminorm k·k on V . A continuous mapping T : V → W of polynormed spaces is called an absolutely summable if( T v i ) i ∈ I is absolutely summable in W for every summable family ( v i ) i ∈ I in V , that is, T transformssummable families from V to absolutely summable ones in W . A linear mapping T : V → W between normed spaces V and W is absolutely summable iff there exists a positive ρ such that P n k T v n k ≤ ρ sup { P n |h v n , a i| : a ∈ ball V ∗ } for all finite families ( v n ) n ∈ n in V [29, Proposition2.2.1]. Put π ( T ) = inf { ρ } , which is a norm in the space A ( V, W ) of all absolutely summablemaps between V and W . If W is complete then A ( V, W ) equipped with the π -norm is a Banachspace. Now let T ∈ B ( V, W ) (the space of all bounded linear operators from V to W ) andlet X ⊆ ball V ∗ be an essential subset in the sense of k v k = sup |h v, X i| for all v ∈ V , thatis, the canonical representation V → C ( X ), v
7→ h v, ·i is an isometry. The known result ofPietsch [29, Theorem 2.3.3] asserts that T ∈ A ( V, W ) iff there exists µ ∈ M ( X ) + such that k T v k ≤ R X |h v, t i| dµ ( t ) for all v ∈ V . In this case, π ( T ) = min { µ ( X ) } over all µ ∈ M ( X ) + with the just indicated property. In particular, T ∈ A ( C ( X ) , W ) iff k T v k ≤ R X | v ( t ) | dµ ( t ), v ∈ C ( X ) for a certain µ ∈ M ( X ) + , where X is a compact Hausdorff topological space. Forthe Hilbert spaces K and H we have A ( K, H ) = B ( K, H ) and k T k ≤ π ( T ) ≤ √ k T k , where B ( K, H ) is the space of all Hilbert-Schmidt operators from K to H . The idea of the proof of thefollowing key lemma of Pietsch will be used later on. For the completeness we provide its proof. Lemma 2.4.
Let X be a compact Hausdorff topological space, µ ∈ M ( X ) + , ι : C ( X ) → L ( X, µ ) the canonical representation, H a Hilbert space and let T : L ( X, µ ) → H , T = P mr =1 ζ r ⊙ η r bea finite-rank operator given by a finite family ( ζ r ) r ⊆ H and µ -step functions ( η r ) r ⊆ L ( X, µ ) .Then T ι : C ( X ) → H is a nuclear operator with k T ι k ≤ k T k .Proof. One can choose a partition X = X ∪ . . . ∪ X n of X into µ -measurable subsets X r ⊆ X suchthat T = P nr =1 η r ⊙ χ r for a new family ( η r ) r ⊆ H and an orthogonal family ( χ r ) r ⊆ L ( X, µ ),where χ r is the characteristic function of X r . Put b χ r = µ ( X r ) − / χ r and µ r = χ r µ ∈ I µ ( X ) + .Notice that ( b χ r ) r is a finite orthonormal family in L ( X, µ ), T ( b χ r ) = µ ( X r ) − / ( χ r , χ r ) η r = µ ( X r ) / η r and P r k T ( b χ r ) k ≤ k T k . Moreover, k µ r k = h , µ r i = R χ r dµ = µ ( X r ) for all r ,and T ( ι ( v )) = P r ( v, χ r ) η r = P r (cid:0)R v ( t ) χ r ( t ) dµ (cid:1) η r = P r h v, µ r i η r for all v ∈ C ( X ). Thus EPARABLE MORPHISMS OF OPERATOR HILBERT SYSTEMS 9
T ι = P nr =1 η r ⊙ µ r is a nuclear operator and k T ι k ≤ X r k η r k k µ r k = X r µ ( X r ) / µ ( X r ) / k η r k ≤ X r µ ( X r ) ! / X r µ ( X r ) k η r k ! / = X r µ ( X r ) ( η r , η r ) ! / = X r (cid:16) µ ( X r ) / η r , µ ( X r ) / η r (cid:17)! / = X r ( T ( b χ r ) , T ( b χ r )) ! / = X r k T ( b χ r ) k ! / ≤ k T k , that is, k T ι k ≤ k T k . (cid:3) Actually, the assertion proven in Lemma 2.4 is true for every T ∈ B ( L ( X, µ ) , H ) [29, 3.3.3Proposition 2]. As a result we obtain the following factorization [29, 3.3.4] of an absolutelysummable mapping. Proposition 2.1.
Let T ∈ A ( V, W ) and let X ⊆ ball V ∗ be an essential subset. There exists a µ ∈ M ( X ) + such that T can be factorized as T = T ιT , that is, the following diagram C ( X ) ι −→ L ( X, µ ) T ↑ ↓ T V T −→ W commutes with k T k ≤ and k T k ≤ π ( T ) . The factorization from Proposition 2.1 is known as the Pietsch factorization.
Remark 2.2. If T ∈ B ( H, C ( X )) then ιT ∈ B ( H, L ( X, µ )) for every µ ∈ M ( X ) + . Indeed,for a Hilbert basis F for H we have P f ∈ F | ( T f ) ( t ) | = P f ∈ F |h T f, δ t i| = P f ∈ F |h f, T ∗ δ t i| = P f ∈ F | ( f, T ∗ δ t ) | ≤ k T ∗ δ t k ≤ k T ∗ k = k T k , t ∈ X . It follows that k ιT k = P f k ( ιT ) f k = P f R | ( T f ) ( t ) | dµ ≤ k T k R < ∞ . Based on these results one can prove that a superposition of two absolutely summable mapsturns out to be a nuclear operator (see [29, 3.3.5]).3.
Quantum cones on a Hilbert space
In this section we introduce unital cones in a Hilbert space and classify their quantizations.3.1.
Hilbert ∗ -space. Let H be a Hilbert space. By an involution on H we mean a ∗ -linearmapping H → H , ζ ζ ∗ such that ζ ∗∗ = ζ and ( ζ ∗ , η ∗ ) = ( ζ , η ) ∗ for all ζ , η ∈ H . In thecase of H = ℓ ( F ) the mapping ζ ζ ∗ with ζ ∗ = P f ∈ F ζ ∗ f f for ζ = P f ∈ F ζ f f is a naturalinvolution on H , where ζ f = ( ζ , f ), f ∈ F . The set of all hermitian vectors from a Hilbert ∗ -space H is denoted by H h . Notice that H h is a real Hilbert space, for ( ζ , η ) ∈ R whenever ζ , η ∈ H h . For every ζ ∈ H we have a unique expansion ζ = Re ζ + i Im ζ into its hermitian parts,( i Im ζ , Re ζ ) = (Re ζ , i Im ζ ) ∗ = ((Re ζ ) ∗ , ( i Im ζ ) ∗ ) = − (Re ζ , i Im ζ ), and k ζ k = k Re ζ k + k Im ζ k + ( i Im ζ , Re ζ ) + (Re ζ , i Im ζ ) = k Re ζ k + k Im ζ k . Take a (real) Hilbert basis F for H h ,which turns out to be a (complex) basis for H . For every ζ ∈ H with ζ = P f ∈ F ζ f f we haveRe ζ = P f (Re ζ f ) f and Im ζ = P f (Im ζ f ) f , which in turn implies that ζ ∗ = Re ζ − i Im ζ = P f ζ ∗ f f . Thus every involution on H is reduced to the above considered example of ℓ ( F ) withrespect to a suitable basis for H . The conjugate Hilbert space to H is denoted by H , whose vectors are denoted by ζ , ζ ∈ H .Thus λζ = λ ∗ ζ and (cid:0) ζ , η (cid:1) = ( ζ , η ) ∗ = ( ζ ∗ , η ∗ ) for all ζ , η ∈ H and λ ∈ C . Notice that thecanonical mapping ψ : H → H ∗ , ψ ( η ) = ( · , η ) is an isometric isomorphism. Thus (cid:0) H, H (cid:1) isa dual pair with the canonical duality h ζ , η i = ( ζ , η ), ζ , η ∈ H . Moreover, it is a dual ∗ -pair,for h ζ ∗ , η i = ( ζ ∗ , η ) = ( ζ , η ∗ ) ∗ = h ζ , η ∗ i ∗ , ζ , η ∈ H , which means that the involution is weaklycontinuous. In particular, H possesses the involution η η ∗ , h ζ , η ∗ i = h ζ ∗ , η i ∗ (see Subsection2.2). Thus h ζ , η ∗ i = h ζ , η ∗ i for all ζ ∈ H , which in turn implies that η ∗ = η ∗ . In particular, (cid:16) ζ ∗ , η ∗ (cid:17) = (cid:0) ζ ∗ , η ∗ (cid:1) = ( ζ ∗ , η ∗ ) ∗ = ( ζ , η ) = (cid:0) ζ, η (cid:1) ∗ , ζ , η ∈ H , which means that H is a Hilbert ∗ -space as well.Later on we fix a hermitian unit vector e from H , which can be extended up to a basis F for H h .Thus ( H, e ) is a unital space. Since e ∗ = e ∗ = e , it follows that (cid:0) H, e (cid:1) is a unital Hilbert ∗ -spaceeither. As above in Subsection 2.1, the duality h· , ·i of the dual ∗ -pair (cid:0) H, H (cid:1) can be extended up toa matrix duality hh· , ·ii : M ( H ) × M (cid:0) H (cid:1) → M by hh ζ , η ii = [ h ζ ik , η jl i ] ( i,j ) , ( k,l ) = [( ζ ik , η jl )] ( i,j ) , ( k,l ) ,and each (cid:0) M n ( H ) , M n (cid:0) H (cid:1)(cid:1) is a dual ∗ -pair (see Subsection 2.2). In this case, for a ∈ M n,m , η ∈ M m (cid:0) H (cid:1) and b ∈ M m,n we have(3.1) aηb = " m X k,l =1 a ik η kl b lj i,j = " m X k,l =1 a ∗ ik η kl b ∗ lj i,j = ( a ∗ ) t η ( b ∗ ) t . Note also that η ∗ = [ η ij ] ∗ i,j = [ η ji ∗ ] i,j = (cid:2) η ∗ ji (cid:3) i,j = η ∗ and hh ζ ∗ , η ∗ ii = [ h ζ ∗ ki , η lj ∗ i ] ( i,j ) , ( k,l ) = (cid:2)(cid:0) ζ ∗ ki , η ∗ lj (cid:1)(cid:3) ( i,j ) , ( k,l ) = [( ζ ki , η lj ) ∗ ] ( i,j ) , ( k,l ) = [ h ζ ki , η lj i ∗ ] ( i,j ) , ( k,l ) = [ h ζ ik , η jl i ] ∗ ( i,j ) , ( k,l ) = hh ζ , η ii ∗ for all ζ ∈ M ( H ) and η ∈ M (cid:0) H (cid:1) . Recall that the matrix norm k·k o of an operator Hilbertspace H o is given by k ζ k o = (cid:13)(cid:13)(cid:10)(cid:10) ζ , ζ (cid:11)(cid:11)(cid:13)(cid:13) / , ζ ∈ M ( H o ). Thus k ζ ∗ k o = (cid:13)(cid:13)(cid:10)(cid:10) ζ ∗ , ζ ∗ (cid:11)(cid:11)(cid:13)(cid:13) / = (cid:13)(cid:13)(cid:13)DD ζ ∗ , ζ ∗ EE(cid:13)(cid:13)(cid:13) / = (cid:13)(cid:13)(cid:13)(cid:10)(cid:10) ζ , ζ (cid:11)(cid:11) ∗ (cid:13)(cid:13)(cid:13) / = (cid:13)(cid:13)(cid:10)(cid:10) ζ , ζ (cid:11)(cid:11)(cid:13)(cid:13) / = k ζ k o for all ζ ∈ M ( H o ), and k e k o = k e k = 1.3.2. Hilbert space norm on M ( H ) . As above let (
H, e ) be a unital Hilbert ∗ -space and let F be a basis for H h which contains e . Along with the matrix pairing hh· , ·ii we have the scalarpairing h· , ·i : M n ( H ) × M n (cid:0) H (cid:1) → C given by h ζ , η i = P i,j h ζ ij , η ij i (see Subsection 2.1). Notethat M n ( H ) is a Hilbert space with ( ζ , η ) = P i,j ( ζ ij , η ij ) = P i,j h ζ ij , η ij i = h ζ , η i for all ζ , η ∈ M n ( H ). Moreover, every ζ ∈ M n ( H ) admits a unique expansion ζ = P f ∈ F (cid:10)(cid:10) ζ , f (cid:11)(cid:11) f ⊕ n . Indeed, ζ = [ ζ ij ] i,j = hP f ( ζ ij , f ) f i i,j = P f [( ζ ij , f )] i,j ⊗ f = P f ∈ F (cid:10)(cid:10) ζ , f (cid:11)(cid:11) f ⊕ n . Note that (cid:10) ζ , a ⊗ f (cid:11) = D ζ , af ⊕ n E = X (cid:0) ζ ij , a ∗ ij f (cid:1) = X a ij ( ζ ij , f ) = τ (cid:16) a (cid:10)(cid:10) ζ , f (cid:11)(cid:11) t (cid:17) (3.2) = τ (cid:0) a t (cid:10)(cid:10) ζ , f (cid:11)(cid:11)(cid:1) for all ζ ∈ M n ( H ) and a ∈ M n , where τ indicates to the standard trace of a matrix. On thematrix space M n ( H ) we have the Hilbert space norm k ζ k = (cid:10) ζ , ζ (cid:11) / , ζ ∈ M n ( H ). The familyof unit balls ball k·k is an absolutely convex quantum set H in M ( H ) whereas B = ball k·k o is an absolutely matrix convex set in M ( H ). The self-dual property of H o asserts [16, 3.5.2]that B ⊙ = B with respect to the duality (cid:0) H, H (cid:1) . In particular, k ζ k o = sup (cid:13)(cid:13)(cid:10)(cid:10) ζ , B (cid:11)(cid:11)(cid:13)(cid:13) for all ζ ∈ M ( H o ). For the hermitian parts H ∩ M ( H ) h and B ∩ M ( H ) h we use the notations H h and B h , respectively. EPARABLE MORPHISMS OF OPERATOR HILBERT SYSTEMS 11
Lemma 3.1. If ζ , η ∈ M n ( H ) then h ζ , η i = P f τ (cid:16)(cid:10)(cid:10) ζ , f (cid:11)(cid:11) (cid:10)(cid:10) η, f (cid:11)(cid:11) ∗ (cid:17) and k ζ k o ≤ k ζ k ≤√ n k ζ k o . In particular, k ζ k = (cid:16)P f τ (cid:16)(cid:12)(cid:12)(cid:10)(cid:10) ζ , f (cid:11)(cid:11)(cid:12)(cid:12) (cid:17)(cid:17) / and H ∩ M n ( H ) ⊆ B ∩ M n ( H ) ⊆√ n H ∩ M n ( H ) for all n ∈ N .Proof. Take ζ ∈ M n ( H ). Using (3.1) and (3.2), we derive that h ζ , η i = X f (cid:28) ζ , (cid:16)(cid:10)(cid:10) η, f (cid:11)(cid:11) ∗ (cid:17) t f ⊕ n (cid:29) = X f (cid:28) ζ , (cid:16)(cid:10)(cid:10) η, f (cid:11)(cid:11) ∗ (cid:17) t ⊗ f (cid:29) = X f τ (cid:16)(cid:10)(cid:10) η, f (cid:11)(cid:11) ∗ (cid:10)(cid:10) ζ , f (cid:11)(cid:11)(cid:17) = X f τ (cid:16)(cid:10)(cid:10) ζ , f (cid:11)(cid:11) (cid:10)(cid:10) η, f (cid:11)(cid:11) ∗ (cid:17) . In particular, k ζ k = (cid:10) ζ , ζ (cid:11) / = X f τ (cid:16)(cid:10)(cid:10) ζ , f (cid:11)(cid:11) ∗ (cid:10)(cid:10) ζ , f (cid:11)(cid:11)(cid:17)! / = X f τ (cid:16)(cid:12)(cid:12)(cid:10)(cid:10) ζ , f (cid:11)(cid:11)(cid:12)(cid:12) (cid:17)! / , that is, k ζ k = (cid:16)P f τ (cid:16)(cid:12)(cid:12)(cid:10)(cid:10) ζ , f (cid:11)(cid:11)(cid:12)(cid:12) (cid:17)(cid:17) / . It follows that k ζ k o = (cid:13)(cid:13)(cid:10)(cid:10) ζ , ζ (cid:11)(cid:11)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X f,g (cid:16)(cid:10)(cid:10) ζ , f (cid:11)(cid:11) ( hh ζ , g ii ∗ ) t ⊗ I n (cid:17) (cid:10)(cid:10) f ⊕ n , g ⊕ n (cid:11)(cid:11)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ X f (cid:13)(cid:13)(cid:13)(cid:13)(cid:10)(cid:10) ζ , f (cid:11)(cid:11) (cid:16)(cid:10)(cid:10) ζ , f (cid:11)(cid:11) ∗ (cid:17) t ⊗ I n (cid:13)(cid:13)(cid:13)(cid:13) ≤ X f (cid:13)(cid:13)(cid:10)(cid:10) ζ , f (cid:11)(cid:11)(cid:13)(cid:13) = X f (cid:13)(cid:13)(cid:13)(cid:12)(cid:12)(cid:10)(cid:10) ζ , f (cid:11)(cid:11)(cid:12)(cid:12) (cid:13)(cid:13)(cid:13) ≤ X f τ (cid:16)(cid:12)(cid:12)(cid:10)(cid:10) ζ , f (cid:11)(cid:11)(cid:12)(cid:12) (cid:17) = k ζ k , that is, k ζ k o ≤ k ζ k . Furthermore, k ζ k = (cid:10) ζ , ζ (cid:11) = (cid:0)(cid:10)(cid:10) ζ , ζ (cid:11)(cid:11) I n , I n (cid:1) ≤ (cid:13)(cid:13)(cid:10)(cid:10) ζ , ζ (cid:11)(cid:11)(cid:13)(cid:13) k I n k = n k ζ k o ,that is, k ζ k ≤ √ n k ζ k o . The rest is clear. (cid:3) Remark 3.1.
Notice that Schwarz inequality for the scalar pairing on (cid:0)
H, H (cid:1) follows from thematrix and then classical Schwarz inequalities in the following way h ζ , η i = X f τ (cid:16)(cid:10)(cid:10) ζ , f (cid:11)(cid:11) (cid:10)(cid:10) η, f (cid:11)(cid:11) ∗ (cid:17) ≤ X f = e τ (cid:16)(cid:12)(cid:12)(cid:10)(cid:10) ζ , f (cid:11)(cid:11)(cid:12)(cid:12) (cid:17)! / X f = e τ (cid:16)(cid:12)(cid:12)(cid:10)(cid:10) η, f (cid:11)(cid:11)(cid:12)(cid:12) (cid:17)! / = k ζ k k η k for all ζ , η ∈ M n ( H ) (see Lemma 3.1). Using the matrix ball B and the scalar pairing h· , ·i , we can define the norm (not a matrix one) k ζ k so = sup (cid:12)(cid:12)(cid:10) ζ , B (cid:11)(cid:12)(cid:12) on M ( H ). Corollary 3.1. If ζ ∈ M n ( H ) then k ζ k so ≤ √ n k ζ k and k ζ k o ≤ k ζ k so ≤ n k ζ k o .Proof. For all η ∈ B and a, b ∈ ball HS n (Hilbert-Schmidt operators) we have ( hh ζ , η ii a, b ) = h ζ , b ∗ ηa i and b ∗ ηa ∈ B . Then khh ζ , η iik = sup | ( hh ζ , η ii ball HS n , ball HS n ) | ≤ sup (cid:12)(cid:12)(cid:10) ζ , B (cid:11)(cid:12)(cid:12) = k ζ k so , which in turn implies that k ζ k o = sup (cid:13)(cid:13)(cid:10)(cid:10) ζ , B (cid:11)(cid:11)(cid:13)(cid:13) ≤ k ζ k so . Further, using Lemma 3.1,we derive that k ζ k so ≤ sup (cid:12)(cid:12)(cid:10) ζ , √ n H (cid:11)(cid:12)(cid:12) = √ n sup (cid:12)(cid:12)(cid:10) ζ , H (cid:11)(cid:12)(cid:12) . Take η ∈ H . But (see Remark 3.1) h ζ , η i ≤ k ζ k k η k ≤ k ζ k , that is, k ζ k so ≤ √ n k ζ k . Finally, k ζ k so ≤ n k ζ k o by to Lemma 3.1. (cid:3) Notice that k ζ k = k ζ k = k ζ k o = k ζ k so for all ζ ∈ H = M ( H ). The unital cone c in ( H, e ) . As above let (
H, e ) be a unital Hilbert ∗ -space and let (cid:0) H, H (cid:1) be the related dual ∗ -pair. We define the following closed (or σ (cid:0) H, H (cid:1) -closed) cone c = n ζ ∈ H h : k ζ k ≤ √ ζ , e ) o in H . Note that e ∈ c , and ( ζ , e ) ≥ ζ ∈ c . Take ζ ∈ H h with ζ = ζ + ( ζ , e ) e , where ζ = P f = e ( ζ , f ) f ∈ H eh , H eh = H h ∩ H e and H e = { e } ⊥ . Since k ζ k = k ζ k + ( ζ , e ) , we concludethat ζ ∈ c iff k ζ k ≤ ( ζ , e ). Thus ζ = ζ + λe ∈ c whenever λ ≥ k ζ k . The set of all states of thecone c is denoted by S ( c ). Since h ζ , e i ≥ ζ ∈ c , and h e, e i = 1, we obtain that e ∈ S ( c ).We write ζ ≤ η for ζ , η ∈ H whenever η − ζ ∈ c . Lemma 3.2.
The cone c is a separated, unital cone such that − e ≤ ball ( H eh ) ≤ e and c ∩ H eh = { } .Thus c ∩ − c = { } and c − e is an absorbent set in H h . In particular, − e ≤ F ≤ e , ( F \ { e } ) ∩ c = ∅ and H h = c − c . Moreover, S ( c ) ⊆ H and S ( c ) = ball (cid:0) H eh (cid:1) + e .Proof. Take ζ ∈ c ∩ − c . Since ( ± ζ , e ) ≥ k ζ k ≤ √ ζ , e ), it follows that k ζ k = 0 or ζ = 0.Note that k e − ζ k = q k ζ k ≤ √ √ e − ζ , e ) for all ζ ∈ ball ( H eh ), which means that e ≥ ball ( H eh ). But ball ( H eh ) + e ⊆ c as well. Hence − e ≤ ball ( H eh ) ≤ e . Taking into account that( H eh , e ) = { } , we deduce that c ∩ H eh = { } . Further, c − e is an absorbent set in H h . Indeed, for ζ ∈ H h choose a real r with k ζ k− ( ζ , e ) ≤ r . Then ζ + re = ζ +(( ζ , e ) + ( re, e )) e = ζ +( ζ + re, e ) e and k ζ k ≤ ( ζ + re, e ), which means that ζ + re ∈ c . In particular, ζ = ζ + re − re and ζ + re, re ∈ c ,thereby H h = c − c .Further, prove that S ( c ) = ball (cid:0) H eh (cid:1) + e . Take η = η + e with η ∈ ball ( H eh ). Then h e, η i =( e, η ) = 1. If ζ ∈ c then ζ = ζ + ( ζ , e ) e with k ζ k ≤ ( ζ , e ). Note that h ζ , η i = ( ζ , η ) + ( ζ , e )and | ( ζ , η ) | ≤ k ζ k k η k ≤ k ζ k ≤ ( ζ , e ). But ( ζ , η ) is real, therefore h ζ , η i ≥
0. Consequently, η ∈ S ( c ). Conversely, take σ ∈ S ( c ). Using the fact H h = c − c , we deduce that σ is a ∗ -linearfunctional. Take ζ ∈ H h with ζ = ζ + ( ζ , e ) e , where ζ ∈ H eh . Since − e ≤ k ζ k − ζ ≤ e ,we derive that | σ ( ζ ) | ≤ k ζ k , which in turn implies that | σ ( ζ ) | = | σ ( ζ ) + ( ζ , e ) | ≤ k ζ k + | ( ζ , e ) | ≤ k ζ k . In the general case of ζ ∈ H we derive that | σ ( ζ ) | ≤ | σ (Re ζ ) + if (Im ζ ) | = (cid:0) σ (Re ζ ) + σ (Im ζ ) (cid:1) / ≤ (cid:0) k Re ζ k + k Im ζ k (cid:1) / ≤ √ k ζ k , which means that σ is a boundedlinear functional on H , that is, σ = η for a certain η ∈ H . But η = η +( η, e ) e = η + σ ( e ) e = η + e ,where η ∈ H e . Prove that η ∈ ball ( H eh ). Take any ζ ∈ H eh , and put ζ = ζ + k ζ k e ∈ c . Then σ ( ζ ) = h ζ , η i = ( ζ , η ) + k ζ k ≥
0, which in turn implies that ( ζ , η ) ∈ R . But ( ζ , η ) =( ζ , Re η ) − i ( ζ , Im η ), therefore Im η ⊥ H eh . Since H eh is a real Hilbert space and Im η ∈ H eh ,we conclude that Im η = 0. Thus η ∈ H eh and ( ζ , η ) ≥ − k ζ k for all ζ ∈ H eh . In particular,( − ζ , η ) ≥ − k ζ k or | ( ζ , η ) | ≤ k ζ k . Consequently, k η k = sup | (ball ( H eh ) , η ) | ≤
1, whichmeans that η ∈ ball ( H eh ). Thus S ( c ) = ball (cid:0) H eh (cid:1) + e . (cid:3) By symmetry we have the cone c = (cid:8) ζ ∈ H h : (cid:13)(cid:13) ζ (cid:13)(cid:13) ≤ √ (cid:0) ζ, e (cid:1)(cid:9) in (cid:0) H, e (cid:1) , and S ( c ) = ball ( H eh )+ e thanks to Lemma 3.2. Note that S ( c ) ⊆ c and S ( c ) ⊆ c . Remark 3.2.
Note that c = { ζ ∈ H h : h ζ , S ( c ) i ≥ } . Indeed, take ζ ∈ H h with h ζ , S ( c ) i ≥ . Then ( ζ , e ) = h ζ , e i ≥ and ( ζ , − k ζ k ζ + e ) ≥ , which in turn implies that k ζ k = (cid:0) ζ , k ζ k − ζ (cid:1) ≤ ( ζ , e ) . Hence ζ ∈ c . The inclusion c ⊆ { ζ ∈ H h : h ζ , S ( c ) i ≥ } is obvious. Consider the norm k ζ k e = sup |h ζ , S ( c ) i| , ζ ∈ H associated with the unital cone c (see Subsec-tion 2.3). EPARABLE MORPHISMS OF OPERATOR HILBERT SYSTEMS 13
Proposition 3.1.
The norm k·k e on H is a unital ∗ -norm, which is equivalent to the originalnorm of H . Moreover, k·k e is an order norm in the sense of k ζ k e = inf { r > − re ≤ ζ ≤ re } forall ζ ∈ H h . In particular, min c = S ( c ) ⊡ and max c = c ⊡⊡ with respect to the dual ∗ -pair (cid:0) H, H (cid:1) .Proof.
Using Lemma 3.2, we obtain that k ζ k e ≤ k ζ k sup k S ( c ) k = k ζ k sup (cid:13)(cid:13) ball (cid:0) H eh (cid:1) + e (cid:13)(cid:13) =2 k ζ k , ζ ∈ H and k e k e = sup |h e, S ( c ) i| = 1. Moreover, k ζ ∗ k e = sup |h ζ ∗ , S ( c ) i| = sup |h ζ , S ( c ) i ∗ | = k ζ k e , k Re ζ k e = sup |h Re ζ , S ( c ) i| = sup | Re h ζ , S ( c ) i| ≤ k ζ k e . Similarly, k Im ζ k e ≤ k ζ k e . Nowtake a nonzero ζ ∈ H h with its expansion ζ = ζ + λe , where λ = ( ζ , e ). Then k ζ k = k ζ k − ( ζ , ζ + λe ) = k ζ k − (cid:0) ζ , k ζ k (cid:0) k ζ k − ζ + e (cid:1) + ( λ − k ζ k ) e (cid:1) = k ζ k k ζ k − (cid:0) ζ , k ζ k − ζ + e (cid:1) + ( λ − k ζ k ) k ζ k − ( ζ , e ) ≤ k ζ k k ζ k − sup (cid:10) ζ , ball (cid:0) H eh (cid:1) + e (cid:11) + | λ − k ζ k| k ζ k − h ζ , e i≤ k ζ k e + ( | λ | + k ζ k ) k ζ k − h ζ , e i ≤ k ζ k e + 2 h ζ , e i ≤ k ζ k e ,which in turn implies that k ζ k = (cid:0) k Re ζ k + k Im ζ k (cid:1) / ≤ k ζ k e . Notice that if ζ ∈ H eh then(3.3) k ζ k = (cid:0) ζ , k ζ k − ζ (cid:1) = (cid:0) ζ , k ζ k − ζ + e (cid:1) = sup (cid:12)(cid:12)(cid:10) ζ , ball (cid:0) H eh (cid:1) + e (cid:11)(cid:12)(cid:12) = k ζ k e . Now put α = inf { r > − re ≤ ζ ≤ re } for ζ ∈ H h . Then (see Remark 3.2) α = inf { r > re ± ζ ≥ } = inf { r > h re ± ζ , η i ≥ , η ∈ S ( c ) } = inf { r > r ± h ζ , η i ≥ , η ∈ S ( c ) } = inf { r > |h ζ , S ( c ) i| ≤ r } = sup |h ζ , S ( c ) i| = k ζ k e ,that is, k·k e is an order norm on H . Consequently, H is the normed dual of H equipped with thenorm k·k e . It follows that min c = S ( c ) ⊡ and max c = c ⊡⊡ with respect to the dual ∗ -pair (cid:0) H, H (cid:1) (see Subsection 2.3). (cid:3)
Remark 3.3.
As we have seen from the proof of Proposition 3.1 that − k ζ k ≤ k ζ k e ≤ k ζ k forall ζ ∈ H . A similar estimations are obtained below in Lemma 3.4 for the related matrix norms. Corollary 3.2. If S ( c ) c ∩ H is the cone in H generated by S ( c ) then c = S ( c ) c ∩ H = R + S ( c ) .Thus min c = c ⊡ in M ( H ) h , min c = c ⊡ in M (cid:0) H (cid:1) h , and both S ( c ) and c generate the same closedquantum cone c ⊡⊡ in M ( H ) h . Thus max c = S ( c ) ⊡⊡ in M ( H ) h , and max c = S ( c ) ⊡⊡ in M (cid:0) H (cid:1) h .In particular, every ζ ∈ M n ( H ) h with P f = e (cid:12)(cid:12)(cid:10)(cid:10) ζ , f (cid:11)(cid:11)(cid:12)(cid:12) ≤ hh ζ , e ii in M n belongs to max c .Proof. If ζ = ζ + λe ∈ c with k ζ k ≤ λ then ζ = λ (cid:0) λ − ζ + e (cid:1) ∈ λ (ball ( H eh ) + e ) ⊆ λS ( c ) ⊆ R + S ( c ) ⊆ S ( c ) c ∩ H by Lemma 3.2. Since S ( c ) c ∩ H ⊆ c , the equalities S ( c ) c ∩ H = R + S ( c ) = c follow. UsingProposition 3.1, we deduce that min c = S ( c ) ⊡ = ( R + S ( c )) ⊡ = c ⊡ , and max c = c ⊡⊡ = S ( c ) ⊡⊡ ,which is the closed quantum cone in M ( H ) h generated by S ( c ) or c . By symmetry, we havemin c = c ⊡ and max c = S ( c ) ⊡⊡ .Finally, take ζ ∈ M n ( H ) h with P f = e (cid:12)(cid:12)(cid:10)(cid:10) ζ , f (cid:11)(cid:11)(cid:12)(cid:12) ≤ hh ζ , e ii in M n . Each (cid:10)(cid:10) ζ , f (cid:11)(cid:11) is diagonalizablebeing a hermitian matrix from M n , that is, (cid:10)(cid:10) ζ , f (cid:11)(cid:11) = µ ∗ f v f | d f | µ f , where d f is a diagonal realmatrix with its polar decomposition d f = v f | d f | , k v f k ≤ v f is a diagonal matrix) and a unitary µ f . Then (cid:10)(cid:10) ζ , f (cid:11)(cid:11) f ⊕ n = µ ∗ f | d f | (cid:0) v f f ⊕ n + e ⊕ n (cid:1) µ f − µ ∗ f | d f | e ⊕ n µ f = µ ∗ f | d f | / (cid:0) v f f ⊕ n + e ⊕ n (cid:1) | d f | / µ f − (cid:12)(cid:12)(cid:10)(cid:10) ζ , f (cid:11)(cid:11)(cid:12)(cid:12) e ⊕ n and v f f ⊕ n + e ⊕ n ∈ (ball ( H eh ) + e ) ⊕ n = S ( c ) ⊕ n . It follows that ζ = X f = e (cid:10)(cid:10) ζ , f (cid:11)(cid:11) f ⊕ n + hh ζ , e ii e ⊕ n = lim λ X λ (cid:10)(cid:10) ζ , f (cid:11)(cid:11) f ⊕ n + hh ζ , e ii e ⊕ n = lim λ X λ µ ∗ f | d f | / (cid:0) v f f ⊕ n + e ⊕ n (cid:1) | d f | / µ f + hh ζ , e ii − X λ (cid:12)(cid:12)(cid:10)(cid:10) ζ , f (cid:11)(cid:11)(cid:12)(cid:12)! e ⊕ n ∈ ( S ( c ) c ) − = S ( c ) ⊡⊡ = c ⊡⊡ = max c , where λ is running over all finite subsets of F \ { e } . Whence ζ ∈ max c . (cid:3) Corollary 3.3. If S ( c ) ◦ is the polar of S ( c ) with respect to the duality (cid:0) H, H (cid:1) then S ( c ) ◦ ∩ H h =abc {{ e } ∪ ball ( H eh ) } in the real Hilbert space H h , and abc {{ e } ∪ ball ( H e ) } ⊆ S ( c ) ◦ ⊆ {{ e } ∪ ball ( H eh ) } in the Hilbert space H , where abc indicates to the absolutely convex hull of a given set.Proof. First note that h e, S ( c ) i = (cid:10) e, ball (cid:0) H eh (cid:1) + e (cid:11) = h e, e i = 1 andsup |h ball ( H e ) , S ( c ) i| = sup (cid:12)(cid:12)(cid:10) ball ( H e ) , ball (cid:0) H eh (cid:1)(cid:11)(cid:12)(cid:12) ≤ , therefore abc {{ e } ∪ ball ( H e ) } ⊆ S ( c ) ◦ . Note also that abc {{ e } ∪ ball ( H e ) } is closed. Indeed,if ζ = lim n ( λ n ζ n + µ n e ) for ( ζ n ) n ⊆ ball ( H e ), λ n , µ n ∈ C with | λ n | + | µ n | ≤
1, then µ = lim n µ n and ζ − µe = lim n λ n ζ n = η and k η k = lim n | λ n | k ζ n k ≤
1, Thus η ∈ ball ( H e ) and ζ = η + µe = k η k (cid:0) k η k − η (cid:1) + µe with k η k + | µ | = lim n ( | λ n | k ζ n k + | µ n | ) ≤ lim n ( | λ n | + | µ n | ) ≤
1, that is, ζ ∈ abc {{ e } ∪ ball ( H e ) } .Take ζ ∈ S ( c ) ◦ ∩ H h with ζ = ζ + ( ζ , e ) e , ζ ∈ H eh and ( ζ , e ) ∈ R . Then s = sup |h ζ , S ( c ) i| ≤ ζ = 0 then ζ = λe with | λ | = |h ζ , e i| ≤ s , therefore ζ ∈ abc {{ e } ∪ ball ( H e ) } . Assume that ζ =0. Then | ( ζ , e ) − ( ζ , η ) | = |h ζ , − η + e i| ≤ s for all η ∈ ball ( H eh ). In particular, | ( ζ , e ) − r k ζ k| = (cid:12)(cid:12) ( ζ , e ) − (cid:0) ζ , r k ζ k − ζ (cid:1)(cid:12)(cid:12) ≤ s for all r ∈ R , | r | ≤
1. It follows that | ( ζ , e ) | ≤ s (for r = 0) and k ζ k ≤ s − | ( ζ , e ) | (for r = ± ζ = k ζ k (cid:0) k ζ k − ζ (cid:1) + ( ζ , e ) e with k ζ k + | ( ζ , e ) | ≤ s ≤ ζ ∈ abc {{ e } ∪ ball ( H eh ) } in H h . Hence S ( c ) ◦ ∩ H h = abc {{ e } ∪ ball ( H eh ) } .In the case of a non-hermitian ζ ∈ S ( c ) ◦ we have sup |h Re ζ , S ( c ) i| = sup | Re h ζ , S ( c ) i| ≤ sup |h ζ , S ( c ) i| ≤
1, which means that Re ζ ∈ S ( c ) ◦ ∩ H h . Similarly, Im ζ ∈ S ( c ) ◦ ∩ H h . Finally, ζ = k ζ k (cid:0) k ζ k − ζ (cid:1) + ( ζ , e ) e and k ζ k + | ( ζ , e ) | ≤ k Re ζ k + k Im ζ k + | (Re ζ , e ) | + | (Im ζ , e ) | ≤
2, that is, ζ ∈ {{ e } ∪ ball ( H e ) } . Actually, ζ = Re ζ + i Im ζ ∈ { Re ζ , Im ζ } ∈ {{ e } ∪ ball ( H eh ) } in the Hilbert space H . (cid:3) Note that S ( c ) ◦ is the unit ball of the norm k·k e . Based on Corollary 3.3, we conclude that k·k e is equivalent to the Minkowski functional of the closed set abc {{ e } ∪ ball ( H e ) } .3.4. The min and max quantizations of c . As above let H be a Hilbert ∗ -space H with aunit e and related unital cone c . Take ζ ∈ M n ( H ) h , which is identified with the bounded ∗ -linearmapping ζ : H → M n such that ζ ( η ) = hh ζ , η ii for all η ∈ H . Put b = ζ ( e ) ∈ M n , which is ahermitian matrix. We say that e is a dominant point for ζ if b ≥ ζ ( η ) = α ( η ) bα ( η ) ∗ , η = η + e ∈ S ( c ) for a certain continuous mapping α : ball (cid:0) H eh (cid:1) → ball M n such that α (0) = I n . Proposition 3.2.
For each n we have (min c ) ∩ M n ( H ) h = n ζ ∈ M n ( H ) h : k a ∗ ζ a k ≤ √ a ∗ ζ a, e ) , a ∈ M n, o . Moreover, for ζ ∈ M n ( H ) h we have ζ ∈ min c iff e is a dominant point for ζ . EPARABLE MORPHISMS OF OPERATOR HILBERT SYSTEMS 15
Proof.
Take ζ ∈ M n ( H ) h . By Proposition 3.1, min c = S ( c ) ⊡ , therefore ζ ∈ min c iff hh ζ , η + e ii ≥ M n for all η ∈ ball ( H eh ). Since ( hh ζ , η + e ii a, a ) = h a ∗ ζ a, η + e i , a ∈ M n, , we derive that ζ ∈ min c iff a ∗ ζ a ∈ S ( c ) ⊡ ∩ H for all a ∈ M n, . But S ( c ) ⊡ ∩ H = c , which is a classical versionof the Unital Bipolar Theorem 2.4 (see also Remark 3.2).Now take ζ ∈ (min c ) ∩ M n ( H ) h with its canonical expansion ζ = ζ + be ⊕ n with b = hh ζ , e ii and ζ ∈ M n ( H ) h , and hh ζ , e ii = 0. Note that ( ba, a ) = ( hh ζ , e ii a, a ) = h a ∗ ζ a, e i = ( a ∗ ζ a, e ) ≥ a ∈ M n, , which means that b ≥
0. Moreover, ζ defines a ∗ -linear mapping ζ : H e → M n with ζ ( η ) = hh ζ , η ii for all η ∈ H e . Since a ∗ ζ a = ( a ∗ ζ a ) and a ∗ ζ a ∈ c , it follows that | ( ζ ( η ) a, a ) | = |h a ∗ ζ a, η i| ≤ k a ∗ ζ a k k η k ≤ ( a ∗ ζ a, e ) k η k = ( k η k ba, a )for all a , which in turn implies that − b ≤ ζ ( η ) ≤ b whenever η ∈ ball H eh . In particular,0 ≤ ζ ( η ) + b ≤ b . Hence ζ ( η ) + b = α ( η ) bα ( η ) ∗ for a unique α ( η ) ∈ M n such that | α ( η ) | = lim k (cid:0) b + k − (cid:1) − / ( ζ ( η ) + b ) (cid:0) b + k − (cid:1) − / (see [4, 1.6. Lemma 2]). Note that α ( η ) : b ( C n ) → C n is a well defined linear mapping such that α ( η ) b / = ( ζ ( η ) + b ) / , k α ( η ) k ≤ √ α ( η ) (ker ( b )) = { } . Notice that b / ( C n ) ⊥ = b ( C n ) ⊥ = ker ( b ). For the fixed a = a + b / ( a ) with a ∈ ker ( b ) and b / ( a ) ∈ im (cid:0) b / (cid:1) , weobtain that k α ( η ) a k = (cid:13)(cid:13) α ( η ) (cid:0) b / ( a ) (cid:1)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) ( ζ ( η ) + b ) / a (cid:13)(cid:13)(cid:13) ,which means (see below Remark 3.4) that α : ball (cid:0) H eh (cid:1) → ball M n is a continuous mapping (onecan equip ball M n with the strong operator topology SOT). Since α ( η ) is uniquely defined and ζ is linear, we derive that α (0) = I n . Moreover, ζ ( η ) + b = ζ ( η ) + ζ ( e ) = (cid:10)(cid:10) ζ + be ⊕ n , η + e (cid:11)(cid:11) = hh ζ , η + e ii = ζ ( η + e )for all η ∈ ball H eh . Thus ζ ( η ) = α ( η ) bα ( η ) ∗ for all η = η + e ∈ S ( c ), which means that e isa dominant point for ζ .Conversely, suppose ζ ( η ) = α ( η ) bα ( η ) ∗ , η = η + e ∈ S ( c ) for a certain continuous mapping α : ball (cid:0) H eh (cid:1) → ball M n such that α (0) = I n and b ≥
0. Then( hh ζ , η ii a, a ) = ( α ( η ) bα ( η ) ∗ a, a ) = ( bα ( η ) ∗ a, α ( η ) ∗ a ) ≥ η = η + e ∈ S ( c ) and a ∈ M n, . It follows that hh ζ , η ii ≥ η ∈ S ( c ), which meansthat ζ ∈ min c . (cid:3) Remark 3.4.
Let ( a γ ) γ be a net of positive operators from B ( H ) such that r ≤ a γ ≤ s for some r, s > . If a = lim γ a γ in B ( H ) then a / = lim γ a / γ in B ( H ) . Indeed, let us surround the interval [ r, s ] by a circle C in Re > , and put d to be the distance from C to [ r, s ] . The resolvent functions R γ ( z ) = ( z − a γ ) − and R ( z ) = ( z − a ) − are holomorphic on C \ [ r, s ] for all γ . Since R γ ( z ) and R ( z ) are normal operators, it follows that kR γ ( z ) k ≤ sup (cid:8) | z − t | − : r ≤ t ≤ s (cid:9) ≤ d − forall z ∈ C and γ . Similarly, kR ( z ) k ≤ d − , z ∈ C . If √ z is the principal branch of the rootfunction then (cid:13)(cid:13) √ z R ( z ) − √ z R γ ( z ) (cid:13)(cid:13) = (cid:12)(cid:12) √ z (cid:12)(cid:12) kR ( z ) ( a − a γ ) R γ ( z ) k ≤ sup (cid:12)(cid:12)(cid:12) √ C (cid:12)(cid:12)(cid:12) d − k a − a γ k for all z ∈ C , that is, lim γ √ z R γ ( z ) = √ z R ( z ) uniformly on C . Using the holomorphic functionalcalculus (see [20, 2.2.15] ) on the interior of C , we conclude that lim γ a / γ = lim γ Z C √ z R γ ( z ) dz = Z C √ z R ( z ) dz = a / , that is, a / = lim γ a / γ in B ( H ) . The assertion just proven is valid still in the case of r = 0 [3] (see also [31] ). Now let us prove a duality result for min and max quantizations of the cone c . Theorem 3.1.
The equalities (max c ) ⊡ = min c and (min c ) ⊡ = max c hold.Proof. By Proposition 3.1, min c = S ( c ) ⊡ is a closed, quantum cone on H . Using the Bipolar The-orem 2.3, we derive that min c = (min c ) ⊡⊡ = S ( c ) ⊡⊡⊡ . By Corollary 3.2, we have that (max c ) ⊡ = S ( c ) ⊡⊡⊡ = min c . Hence (max c ) ⊡ = min c . By symmetry, we also have (max c ) ⊡ = min c , whichin turn implies that (min c ) ⊡ = (max c ) ⊡⊡ = max c by the Bipolar Theorem 2.3. (cid:3) The unital quantum cones on ( H, e ) . Now we introduce new quantizations of the sepa-rated, unital cone c in a Hilbert ∗ -space H . Since the functional e : H → C is a contraction, it turnsout to be a matrix contraction on the operator Hilbert space H o . The projection φ e : H o → H o , φ e ζ = h ζ , e i e is a matrix contraction as well, for (cid:13)(cid:13)(cid:13) φ ( n ) e ( ζ ) (cid:13)(cid:13)(cid:13) o = khh ζ , e ii e ⊕ n k o ≤ khh ζ , e iik k e ⊕ n k o ≤k ζ k o , ζ ∈ M n ( H o ), n ∈ N . In particular, ϕ e : H o → H o , ϕ e ( ζ ) = ζ − φ e ( ζ ) = ζ is a ma-trix bounded mapping and (cid:13)(cid:13)(cid:13) ϕ ( n ) e ( ζ ) (cid:13)(cid:13)(cid:13) o = (cid:13)(cid:13)(cid:13) ζ − φ ( n ) e ( ζ ) (cid:13)(cid:13)(cid:13) o ≤ k ζ k o , ζ ∈ M n ( H o ), n ∈ N . Thus ζ = ϕ ( n ) e ( ζ ) = P f = e (cid:10)(cid:10) ζ , f (cid:11)(cid:11) f ⊕ n , k ζ k o ≤ k ζ k o and k ζ k ≤ k ζ k (see Lemma 3.1) whenever ζ = P f ∈ F (cid:10)(cid:10) ζ , f (cid:11)(cid:11) f ⊕ n . Moreover, a ∗ ζ a = P f ∈ F (cid:10)(cid:10) a ∗ ζ a, f (cid:11)(cid:11) f ⊕ m for all a ∈ M n,m , which in turnimplies that ( a ∗ ζ a ) = ϕ ( m ) e ( a ∗ ζ a ) = a ∗ ϕ ( n ) e ( ζ ) a = a ∗ ζ a . On the unital space ( H, e ) consider thefollowing quantum cones C l , C o and C u whose slices given by C l ∩ M n ( H ) = (cid:8) ζ ∈ M n ( H ) h : k a ∗ ζ a k ≤ m − / τ ( hh a ∗ ζ a, e ii ) , a ∈ M n,m , m ∈ N (cid:9) , C o ∩ M n ( H )= { ζ ∈ M n ( H ) h : k a ∗ ζ a k so ≤ τ ( hh a ∗ ζ a, e ii ) , a ∈ M n,m , m ∈ N } , C u ∩ M n ( H )= { ζ ∈ M n ( H ) h : k a ∗ ζ a k ≤ τ ( hh a ∗ ζ a, e ii ) , a ∈ M n,m , m ∈ N } . The fact that these quantum cones are unital will be verified below. Note that for every ζ from each of these cones we have τ ( hh a ∗ ζ a, e ii ) ≥ a ∈ M . Taking into account that( hh ζ , e ⊕ n ii a, a ) = h a ∗ ζ a, e ⊕ n i = τ ( hh a ∗ ζ a, e ii ), we conclude that hh ζ , e ⊕ n ii ≥
0. Further, notethat C l ∩ H = C o ∩ H = C u ∩ H = { ζ ∈ H h : k ζ k ≤ ( ζ , e ) } = c . As above for each ζ ∈ M n ( H ) h there corresponds a unique expansion ζ = P f ∈ F (cid:10)(cid:10) ζ , f (cid:11)(cid:11) f ⊕ n withhermitian (cid:10)(cid:10) ζ , f (cid:11)(cid:11) ∈ M n . Put M n ( H ) e = { ζ ∈ M n ( H ) : hh ζ , e ii = 0 } , M n ( H ) eh = M n ( H ) h ∩ M n ( H ) e , and H eh = H h ∩ M ( H ) eh , which is the unit ball of M n ( H ) eh relative to the norm k·k .Similarly, B eh = B h ∩ M ( H ) eh is the unit ball of M n ( H ) eh relative to the matrix norm k·k o . Wealso put K eh = (cid:0) n / H eh (cid:1) n , which is a convex quantum set. Thus K eh + e = (cid:0) n / H eh + e ⊕ n (cid:1) n , B eh + e = ( B eh + e ⊕ n ) n and H eh + e = ( H eh + e ⊕ n ) n are quantum sets on H . Similarly, we have thequantum sets K eh + e , B eh + e and H eh + e on H . Proposition 3.3.
The following equalities C l = (cid:0) K eh + e (cid:1) ⊡ , C o = (cid:0) B eh + e (cid:1) ⊡ and C u = (cid:0) H eh + e (cid:1) ⊡ hold with respect to the dual ∗ -pair (cid:0) H, H (cid:1) . In particular, C l ⊆ C o ⊆ C u are the inclusions of theseparated, closed, unital, quantum cones on H , which are quantizations of c .Proof. First take ζ ∈ C l ∩ M n ( H ). Then ζ ∈ M n ( H ) h and k ζ k ≤ n − / τ ( hh ζ , e ii ), where ζ = P f = e (cid:10)(cid:10) ζ , f (cid:11)(cid:11) f ⊕ n = ϕ ( n ) e ( ζ ) ∈ M n ( H ) eh . If η ∈ H eh then η = P f = e (cid:10)(cid:10) η, f (cid:11)(cid:11) f ⊕ n , k η k ≤ h ζ , η i = X f = e D ζ , (cid:10)(cid:10) η, f (cid:11)(cid:11) t f ⊕ n E = X f = e τ (cid:0)(cid:10)(cid:10) ζ , f (cid:11)(cid:11) (cid:10)(cid:10) η, f (cid:11)(cid:11)(cid:1) = h ζ , η i EPARABLE MORPHISMS OF OPERATOR HILBERT SYSTEMS 17 (see (3.1) and (3.2)). Note that h ζ , η i = h ζ ∗ , η i = P i,j (cid:10) ζ ∗ ji , η ij (cid:11) = P i,j (cid:10) ζ ji , η ∗ ij (cid:11) ∗ = h ζ , η ∗ i ∗ = h ζ , η i ∗ , which means that h ζ , η i ∈ R . Note also that all matrices (cid:10)(cid:10) ζ , f (cid:11)(cid:11) and (cid:10)(cid:10) η, f (cid:11)(cid:11) , f = e are hermitians, therefore τ (cid:0)(cid:10)(cid:10) ζ , f (cid:11)(cid:11) (cid:10)(cid:10) η, f (cid:11)(cid:11)(cid:1) ∈ R as well. Since h ζ , η i ≤ k ζ k k η k (see Remark3.1), we deduce that sup (cid:12)(cid:12)(cid:10) ζ , H eh (cid:11)(cid:12)(cid:12) ≤ k ζ k ≤ n − / τ ( hh ζ , e ii ) = n − / h ζ , e ⊕ n i or − h ζ , e ⊕ n i ≤ (cid:10) ζ , n / H eh (cid:11) ≤ h ζ , e ⊕ n i . Hence (cid:10) ζ , n / H eh + e ⊕ n (cid:11) ≥
0. Conversely, suppose the latter holds for ζ ∈ M n ( H ) h . Since − H eh = H eh , it follows that sup (cid:12)(cid:12)(cid:10) ζ , H eh (cid:11)(cid:12)(cid:12) ≤ n − / h ζ , e ⊕ n i . But k ζ k = sup (cid:12)(cid:12)(cid:10) ζ , H e (cid:11)(cid:12)(cid:12) = sup (cid:12)(cid:12) Re (cid:10) ζ , H e (cid:11)(cid:12)(cid:12) = sup (cid:12)(cid:12)(cid:10) ζ , Re H e (cid:11)(cid:12)(cid:12) = sup (cid:12)(cid:12)(cid:10) ζ , H eh (cid:11)(cid:12)(cid:12) = sup (cid:12)(cid:12)(cid:10) ζ , H eh (cid:11)(cid:12)(cid:12) , therefore k ζ k ≤ n − / h ζ , e ⊕ n i . Consequently, C l ∩ M n ( H ) = (cid:8) ζ ∈ M n ( H ) h : (cid:10) a ∗ ζ a, m / H eh + e ⊕ m (cid:11) ≥ , a ∈ M n,m , m ∈ N (cid:9) . Taking into account that (cid:10) a ∗ ζ a, m / H eh + e ⊕ m (cid:11) = (cid:0)(cid:10)(cid:10) ζ , m / H eh + e ⊕ m (cid:11)(cid:11) a, a (cid:1) , we derive that ζ ∈ C l iff (cid:10)(cid:10) ζ , m / H eh + e ⊕ m (cid:11)(cid:11) ≥ ζ ∈ (cid:0) K eh + e (cid:1) ⊡ . Similarly, ζ ∈ C u iff (cid:10)(cid:10) ζ , H eh + e ⊕ n (cid:11)(cid:11) ≥ ζ ∈ (cid:0) H eh + e (cid:1) ⊡ .Further, k ζ k so ≤ τ ( hh ζ , e ii ) means that sup (cid:12)(cid:12)(cid:10) ζ , B eh (cid:11)(cid:12)(cid:12) ≤ τ ( hh ζ , e ii ) = h ζ , e ⊕ n i . The latter inturn is equivalent to (cid:10) ζ , B eh + e ⊕ n (cid:11) ≥
0. Thus C o ∩ M n ( H ) = (cid:8) ζ ∈ M n ( H ) h : (cid:10) a ∗ ζ a, B eh + e ⊕ m (cid:11) ≥ , a ∈ M n,m , m ∈ N (cid:9) . As above, we derive that ζ ∈ C o iff (cid:10)(cid:10) ζ , B eh + e ⊕ n (cid:11)(cid:11) ≥
0, that is, C o = (cid:0) B eh + e (cid:1) ⊡ . Based on Lemma3.1, we obtain that H eh + e ⊕ n ⊆ B eh + e ⊕ n ⊆ √ n H eh + e ⊕ n in M n ( H ) h , or H eh + e ⊆ B eh + e ⊆ K eh + e are inclusions of the quantum sets on H . Therefore C l = (cid:0) K eh + e (cid:1) ⊡ ⊆ C o = (cid:0) B eh + e (cid:1) ⊡ ⊆ C u = (cid:0) H eh + e (cid:1) ⊡ .Further, prove that all these quantum cones are separated. Take ζ ∈ M n ( H ) h . Suppose ζ ∈ C u ∩ − C u with ζ = ζ + hh ζ , e ii e ⊕ n . Since (cid:10)(cid:10) ± ζ , H eh + e ⊕ n (cid:11)(cid:11) ≥
0, it follows that hh ζ , e ii =0 and (cid:10)(cid:10) ζ , H eh (cid:11)(cid:11) = { } . In particular, (cid:10)(cid:10) ζ , B eh (cid:11)(cid:11) = { } . Since k η ∗ k o = k η k o for all η ∈ M ( H o ), it follows that Re η , Im η ∈ B whenever η ∈ B . Hence (cid:10)(cid:10) ζ , B (cid:11)(cid:11) = { } and k ζ k =sup (cid:13)(cid:13)(cid:10)(cid:10) ζ , B (cid:11)(cid:11)(cid:13)(cid:13) = 0, that is, ζ = 0. Thus all quantum cones are separated.Finally prove that C l is unital. Since C l is a topologically closed quantization of the unital cone c , it follows that max c = c ⊡⊡ = ( c c ) − ⊆ C l thanks to Proposition 3.1. But max c is unital (seeLemma 2.2), therefore so is C l . In particular, so are both C o and C u . (cid:3) Remark 3.5.
The fact that C o (in turn C u ) is unital also follows from the following argument.Take ζ ∈ M n ( H ) h . Prove that (cid:10)(cid:10) ζ + re ⊕ n , B eh + e ⊕ m (cid:11)(cid:11) ≥ for large positive r . But (cid:10)(cid:10) ζ , B eh + e (cid:11)(cid:11) is a bounded set of hermitian matrices in M . Then − rI nm ≤ (cid:10)(cid:10) ζ , B eh + e ⊕ m (cid:11)(cid:11) ≤ rI nm for all m ,which in turn implies that (cid:10)(cid:10) ζ + re ⊕ n , B eh + e ⊕ m (cid:11)(cid:11) ⊆ (cid:10)(cid:10) ζ , B eh + e ⊕ m (cid:11)(cid:11) + (cid:10)(cid:10) re ⊕ n , B eh + e ⊕ m (cid:11)(cid:11) = (cid:10)(cid:10) ζ , B eh + e ⊕ m (cid:11)(cid:11) + rI nm ≥ , that is, C o is unital. In particular, so is C u . The quantum polars.
Now consider the quantum set B eh + e = (cid:0) B eh + e ⊕ n (cid:1) n on H . If η ∈ B eh then hh e, η ii = hh e, η ii ∗ = [ h e, η ij i ] ∗ i,j = [ h e, η ji i ∗ ] i,j = [( e, η ji ) ∗ ] i,j = [( η ji , e )] i,j = hh η, e ii t = 0.Thus B eh + e ⊆ M (cid:0) H (cid:1) h ∩ M (cid:0) H (cid:1) e = M (cid:0) H (cid:1) he (see Subsection 2.3), and put B e = (cid:0) B eh + e (cid:1) ⊙ , which is a closed, absolutely matrix convex subset on H . Note that B eh + e is a matrix convexsubset of M (cid:0) H (cid:1) h . Lemma 3.3.
The equality B ⊙ e ∩ M (cid:0) H (cid:1) he = B eh + e holds.Proof. Take z ∈ amc (cid:0) B eh + e (cid:1) − ∩ M n (cid:0) H (cid:1) he , where amc (cid:0) B eh + e (cid:1) − is the closed absolutely matrixconvex hull of B eh + e in M (cid:0) H (cid:1) . Then z ∈ M n (cid:0) H (cid:1) h , hh e, z ii = I n and z = lim k a k ( η k + e ⊕ n k ) b k with a k , b k ∈ ball M and η k ∈ B eh . It follows that I n = hh e, z ii = lim k (cid:10)(cid:10) e, a k (cid:0) η k + e ⊕ n k (cid:1) b k (cid:11)(cid:11) = lim k a k hh e, η k ii b k + a k (cid:10)(cid:10) e, e ⊕ n k (cid:11)(cid:11) b k = lim k a k b k , which in turn implies that lim k a k e ⊕ n k b k = lim k a k b k e ⊕ n = e ⊕ n . In particular, we have the limit η = lim k a k η k b k = z − e ⊕ n ∈ M n (cid:0) H (cid:1) h and k η k o ≤ lim sup k k a k k k η k k o k b k k ≤
1, that is, η ∈ B eh .Hence z = η + e ⊕ n ∈ B eh + e . Thus amc (cid:0) B eh + e (cid:1) − ∩ M (cid:0) H (cid:1) he = B eh + e .Finally, using the Bipolar Theorem 2.2, we deduce that B ⊙ e ∩ M (cid:0) H (cid:1) he = (cid:0) B eh + e (cid:1) ⊙⊙ ∩ M (cid:0) H (cid:1) he = amc (cid:0) B eh + e (cid:1) − ∩ M (cid:0) H (cid:1) he = B eh + e . (cid:3) Lemma 3.4.
The following inclusions − B e ⊆ B ⊆ B e of quantum balls on H hold. Inparticular, the Minkowski functional k·k e of B e is a matrix norm which is equivalent to k·k o .Proof. First note that B eh + e ⊆ B h + B h ⊆ B . Therefore 2 − B = (cid:0) B (cid:1) ⊙ ⊆ (cid:0) B eh + e (cid:1) ⊙ = B e ,that is, the second inclusion follows. To prove the first one, take z ∈ − B h ∩ M n ( H ) with itsexpansion z = w + ae ⊕ n , where w = P f = e (cid:10)(cid:10) z, f (cid:11)(cid:11) f ⊕ n ∈ M n ( H ) eh and a = hh z, e ii is hermitian.Moreover, k w k o = (cid:13)(cid:13)(cid:13) ϕ ( n ) e ( z ) (cid:13)(cid:13)(cid:13) o ≤ k z k o ≤ w ∈ B eh ) and k a k ≤ k z k o ≤ − . Thus w + e ⊕ n ∈ B eh + e and ( a − I n ) e ⊕ n ∈ (3 /
2) amc ( e ) ⊆ (3 /
2) amc ( B eh + e ), which in turn impliesthat z = w + e ⊕ n + ( a − I n ) e ⊕ n ∈ B eh + e + (3 /
2) amc ( B eh + e ) ⊆ (5 /
2) amc ( B eh + e ). Hence2 − B h ⊆ (5 /
2) amc (cid:0) B eh + e (cid:1) ⊆ (5 / (cid:0) B eh + e (cid:1) ⊙⊙ = (5 / B ⊙ e , and B ⊆ (cid:0) B h (cid:1) ⊆ B ⊙ e . Bypassing to the quantum polars, we obtain that 10 − B e ⊆ B ⊙ = B .Finally, for the matrix norm k ζ k e = sup (cid:13)(cid:13)(cid:10)(cid:10) ζ , B eh + e (cid:11)(cid:11)(cid:13)(cid:13) defined by means of B e , we obtainthat 2 − k ζ k e ≤ k ζ k ≤ k ζ k e for all ζ ∈ M ( H ), which means that k·k e and k·k o are equivalentmatrix norms. (cid:3) Theorem 3.2.
Let H be a Hilbert space. The operator Hilbert space H o is an operator systemwhose unital quantum cone of positive elements is given by C o with S ( C o ) = B eh + e . Moreover, C ⊡ o = C o , where C o is the related quantum cone on H o with S (cid:0) C o (cid:1) = B eh + e . Thus H o is a self-dualoperator system.Proof. As above B denotes the unit ball of the matrix norm k·k o . By Lemma 3.4, B e is anabsorbent, s (cid:0) H, H (cid:1) -closed, absolutely matrix convex set on H . As in [12, Lemma 4.3] (see also[15]) consider the Paulsen’s power P H of H and related s ( P H , P H )-closed (see Theorem 2.1),unital, quantum cone C B e on P H obtained by means of B e . For brevity we write C ( B e ) insteadof C B e . Notice that C ( B e ) is a cone on H . The s (cid:0) H, H (cid:1) -closed, quantum cone on H generatedby C ( B e ) is denoted by C e . Actually, C e = C ( B e ) ⊡⊡ , where C ( B e ) ⊡ is the quantum polar ofthe cone C ( B e ) with respect to the dual ∗ -pair (cid:0) H, H (cid:1) . The quantum cone C e is unital and S ( C e ) = C ( B e ) ⊡ ∩ M (cid:0) H (cid:1) e . Since B e is an absorbent, s (cid:0) H, H (cid:1) -closed, absolutely matrix convexset on H , we derive that S ( C e ) = B ⊙ e ∩ M (cid:0) H (cid:1) he = B eh + e EPARABLE MORPHISMS OF OPERATOR HILBERT SYSTEMS 19 by virtue of Lemma 3.3. Using Proposition 3.3, we deduce that C e = (cid:0) B eh + e (cid:1) ⊡ = C o . The matrix normed topology on H of the unital quantum cone C o is given by the absolutelymatrix convex set b C o = h − H ( C o − e ) on H (see [12, Corollary 5.1]). Namely, if p o is the Minkowskifunctional of b C o then p o is a matrix norm by Proposition 3.3. Moreover, p o ( ζ ) = sup (cid:13)(cid:13)(cid:10)(cid:10) ζ , C ⊡ o ∩ M (cid:0) H (cid:1) e (cid:11)(cid:11)(cid:13)(cid:13) = sup (cid:13)(cid:13)(cid:10)(cid:10) ζ , C ⊡ e ∩ M (cid:0) H (cid:1) e (cid:11)(cid:11)(cid:13)(cid:13) = sup (cid:13)(cid:13)(cid:13)DD ζ , C ( B e ) ⊡ ∩ M (cid:0) H (cid:1) e EE(cid:13)(cid:13)(cid:13) = sup (cid:13)(cid:13)(cid:10)(cid:10) ζ , B ⊙ e ∩ M (cid:0) H (cid:1) he (cid:11)(cid:11)(cid:13)(cid:13) = sup (cid:13)(cid:13)(cid:10)(cid:10) ζ , B eh + e (cid:11)(cid:11)(cid:13)(cid:13) = k ζ k e for all ζ ∈ M ( H ), that is, b C o = b C e = S ( C e ) ⊙ = (cid:0) B eh + e (cid:1) ⊙ = B e . Thus p o = k·k e , which isequivalent to the matrix norm k·k o thanks to Lemma 3.4. Consequently, H o is an operator systemwith the related separated, closed, unital, quantum cone C o .By symmetry we have a separated, closed, unital, quantum cone C o on H with S (cid:0) C o (cid:1) = B eh + e .By Proposition 3.3, C o is a quantization of the unital cone c in H . Therefore max c ⊆ C o ⊆ min c .By passing to the quantum polars and using Theorem 3.1, we deduce that max c ⊆ C ⊡ o ⊆ min c .In particular, C ⊡ o is a unital quantum cone. By Unital Bipolar Theorem 2.4 and Bipolar Theorem2.3, we derive that C ⊡ o = (cid:16) C ⊡⊡ o ∩ M (cid:0) H (cid:1) he (cid:17) ⊡ = (cid:16) ( B eh + e ) ⊡⊡⊡ ∩ M (cid:0) H (cid:1) he (cid:17) ⊡ = (cid:16) ( B eh + e ) ⊡ ∩ M (cid:0) H (cid:1) he (cid:17) ⊡ . If η ∈ ( B eh + e ) ⊡ ∩ M n (cid:0) H (cid:1) he then η = η + e ⊕ n and hh B eh , η ii + I ≥
0, which in turn impliesthat sup khh B eh , η iik ≤
1. In particular, k η k o = (cid:13)(cid:13)(cid:10)(cid:10) k η k − o η , η (cid:11)(cid:11)(cid:13)(cid:13) ≤ η ∈ B eh . Thus( B eh + e ) ⊡ ∩ M (cid:0) H (cid:1) he ⊆ B eh + e . Conversely, take η = η + e ⊕ n ∈ B eh + e . Using the matrix Schwarzinequality [16, 3.5.1], we obtain that sup khh B eh , η iik ≤ sup k B eh k o k η k o ≤
1, which in turn impliesthat hh B eh + e , η ii = hh B eh , η ii + I ≥
0. The latter means that η ∈ ( B eh + e ) ⊡ ∩ M (cid:0) H (cid:1) he . Hence( B eh + e ) ⊡ ∩ M (cid:0) H (cid:1) he = B eh + e and C ⊡ o = (cid:0) B eh + e (cid:1) ⊡ = C o by Proposition 3.3. (cid:3) Remark 3.6.
The unital, quantum cone C o on H in Theorem 3.2 can be replaced by S ⊡ for S = (cid:0) B (cid:1) ∩ M (cid:0) H (cid:1) he . Namely, note that B eh + e ⊆ (cid:0) B (cid:1) ∩ M (cid:0) H (cid:1) he ⊆ (cid:0) B eh + e (cid:1) . Thefirst inclusion is immediate. Further, take η = η + e ⊕ n ∈ (cid:0) B (cid:1) ∩ M n (cid:0) H (cid:1) he . Then k η k o ≤k η k o + k e ⊕ n k o ≤ , θ = k η k − o η + e ⊕ n ∈ B eh + e and η = k η k o θ +(1 − k η k o ) e ⊕ n ∈ (cid:0) B eh + e (cid:1) .In particular, − B e ⊆ S ⊙ ⊆ B e . As above S ⊙ responds to a unique closed, unital, separated,quantum cone C on H such that b C = S ⊙ and S ( C ) = S ⊙⊙ ∩ M (cid:0) H (cid:1) he . Since S ⊆ S ( C ) and S ⊙ = b C , it follows that C = S ⊡ , that is, S is a prematricial state space of C [15] , and the relatednormed quantum topology coincides with the original one of H o . Remark 3.7.
The matricial state space B eh + e can not be replaced by B ⊙ ∩ M (cid:0) H (cid:1) he . Indeed, firstnote that B ⊙ ∩ M (cid:0) H (cid:1) he = B ∩ M (cid:0) H (cid:1) he ⊆ B eh + e . Take η = η + e ⊕ n ∈ B ∩ M n (cid:0) H (cid:1) he . Since B eh + e is a matrix convex set, it follows that η i,i = η ,i,i + e ∈ B ∩ H he = (cid:0) ball H (cid:1) ∩ H he for all i . Takinginto account that k η i,i k = k η ,i,i k + 1 , we conclude that η ,i,i = 0 for all i , that is, the diagonalof η consists of zeros. In particular, every diagonal entry of hh η , η ii = [ h η ,i,k , η ,j,l i ] ( i,j ) , ( k,l ) iszero. Hence hh η, η ii = I + hh η , η ii in M n and the hermitian matrix hh η , η ii admits a positiveeigenvalue λ . It follows that λ is an eigenvalue of hh η, η ii . But khh η, η iik = k η k o ≤ ,therefore η = 0 . Consequently, B ⊙ ∩ M (cid:0) H (cid:1) he = e and M ( H ) e ⊆ e ⊙ = (cid:0) B ⊙ ∩ M (cid:0) H (cid:1) he (cid:1) ⊙ , that is, (cid:0) B ⊙ ∩ M (cid:0) H (cid:1) he (cid:1) ⊙ is an unbounded quantum set, which can not generate the original normedquantum topology of H o . Remark 3.8.
In the case of a finite dimensional Hilbert space H of dimension n the quantumcone C o is reduced to one from [23] , that is, ( H, C o ) = SOH ( n ) . Namely, let us prove that if C is the quantum cone of the operator system SOH ( n ) then S ( C ) = B eh + e . First notice that if ζ = ζ + ae ⊕ m ∈ C ∩ M m ( H ) then a ≥ , ζ ∗ = ζ and − ae ⊕ m ≤ ζ ≤ ae ⊕ m in the operator system M m (SOH ( n )) (see [23, Proposition 3.3] ). If a = I the latter is equivalent to k ζ k o ≤ . Thus B eh + e ⊆ C . Further, take η = η + ce ⊕ k ∈ S ( C ) then c = hh e, η ii = I and hh B eh + e ,η ii ≥ .In particular, hh B eh ,η ii + I ≥ , which means that sup khh B eh ,η iik ≤ . Taking into accountthat η / k η k o ∈ B eh , we derive that k η k o = khh η / k η k o ,η iik ≤ sup khh B eh ,η iik ≤ , thatis, η ∈ B eh . Thus S ( C ) ⊆ B eh + e . Conversely, SOH ( n ) is a self-dual operator system [23,Theorem 3.4] , therefore B eh + e ⊆ C = C ⊡ , which in turn implies that B eh + e ⊆ C ⊡ ∩ M (cid:0) H (cid:1) e = S ( C ) . Consequently, C = S ( C ) ⊡ = (cid:0) B eh + e (cid:1) ⊡ = C o thanks to the Unital Bipolar Theorem 2.4 andProposition 3.3. The positive maps of operator Hilbert systems
In this section we analyze the positive maps between ordered Hilbert spaces. Everywhere below X denotes a Hausdorff compact topological space, C ( X ) the abelian C ∗ -algebra of all complexcontinuous functions on X with the norm k v k ∞ = sup | v ( X ) | , v ∈ C ( X ), and the unital quantumcone M ( C ( X )) + of all positive matrix valued functions on X , which is a quantization of the cone C ( X ) + .4.1. Positive maps between unital Hilbert spaces.
Now let (
K, u ) and (
H, e ) be unitalHilbert spaces with the related unital cones c u and c e , respectively, and let F be a hermitian basisfor H , which contains e . A bounded family k = { k f : f ∈ F } ⊆ K h is said to be an H -support in K if k e ∈ c u and X f = e ( η, k f ) ≤ ( η, k e ) for all η ∈ c u . In this case, ( η, k e ) ≥ η ∈ c u . Indeed, put k f = k uf + r f u with k uf ∈ K uh , r f ∈ R . Notethat r e ≥ k k ue k ≤ r e , for k e = k ue + r e u ∈ c u . It follows that ( η, k e ) = ( η , k ue ) + r e ( η, u )and | ( η , k ue ) | ≤ k η k k k ue k ≤ r e k η k . In particular, if η ∈ c u then k η k ≤ ( η, u ), and | ( η, k ue ) | = | ( η , k ue ) | ≤ r e ( η, u ), which in turn implies that ( η, k e ) = ( η , k ue ) + r e ( η, u ) ≥
0. Note also that P f = e r f = P f = e ( u, k f ) ≤ ( u, k e ) = r e , for u ∈ c u . If r e = 1 and r f = 0 for all f = e then wesay that k is a unital H - support. Remark 4.1.
Let k = { k f : f ∈ F } ⊆ K h be a bounded family with k e ∈ ball K uh + u and k f ⊥ u , f = e . Then k is a unital H -support iff P f = e ( η , k f ) ≤ (( η , k e ) + k η k ) for all η ∈ K uh . Indeed,since η = η + k η k u ∈ c u for every η ∈ K uh , it follows that P f = e ( η , k f ) = P f = e ( η, k f ) ≤ ( η, k e ) = (( η , k e ) + k η k ) . Note also that ( η , k e ) = 0 whenever k e = u (see below Subsection4.3). If additionally P f k k f k p < ∞ we say that k is of type p , where p = 1 ,
2. An H -support k in K defines a linear operator T k : K → H, T k η = X f ( η, k f ) f, EPARABLE MORPHISMS OF OPERATOR HILBERT SYSTEMS 21 which is positive in the sense of T k ( c u ) ⊆ c e . In particular, T k is a ∗ -linear mapping. Note that | ( T k η, e ) | = | ( η, k e ) | ≤ sup k k k for all η ∈ ball K . It follows that k T k η k ≤ k T k η k + | ( η, u ) | k T k u k ≤ k T k η k + k T k u k ≤ k T k ( η + u ) k + 2 k T k u k≤ √ T k ( η + 3 u ) , e ) ≤ √ k k k for all η = η + ( η, u ) u ∈ ball K h . For η ∈ ball K we derive that k T k η k ≤ k T k Re η k + k T k Im η k ≤ √ k k k , that is, T k ∈ B ( K, H ) with k T k k ≤ √ k k k . Remark 4.2.
Let k be an H -support in K . Then T k ∈ B ( K, H ) iff k is of type . Indeed,take a Hilbert basis ( η i ) i ∈ I for K . Since T k η = P f ( η, k f ) f , η ∈ K , we deduce that P f k k f k = P f P i ∈ I | ( k f , η i ) | = P i ∈ I P f | ( η i , k f ) | = P i ∈ I k T η i k = k T k . If k is of type then T k ∈B ( K, H ) , k T k ≤ P f k f k k k f k ≤ P f k k f k < ∞ . Proposition 4.1. If T : ( K, u ) → ( H, e ) is a (untal) positive mapping then T = T k for a certain(unital) H -support k in K .Proof. First note that
T u = ζ + re ∈ c e , that is, ζ = P f = e r f f ∈ H eh , r ≥ P f = e r f = k ζ k ≤ r . For η ∈ K we have T η = T η + ( η, u ) T u = X f ( T η , f ) f + ( η, u ) ( ζ + re ) = Sη + ( η, u ) ζ + ( γ ( η ) + ( η, ru )) e, where S : K u → H e , Sη = P f = e ( T η , f ) f , and γ : K u → C , γ ( η ) = ( T η , e ). Take η ∈ K uh .Then η = η + k η k u ∈ c u and Sη + k η k ζ + ( γ ( η ) + r k η k ) e = T η ∈ c e . It follows that Sη ∈ H eh , γ ( η ) ≥ − r k η k and k Sη + k η k ζ k ≤ γ ( η ) + r k η k . In particular, S and γ are ∗ -linear maps, | γ ( η ) | ≤ r k η k and k Sη k ≤ k Sη + k η k ζ k + kk η k ζ k ≤ | γ ( η ) | + 2 r k η k ≤ r k η k for all η ∈ K uh . Thus γ ∈ ( K u ) ∗ , | γ ( η ) | ≤ | γ (Re η ) | + | γ (Im η ) | ≤ r k η k and k Sη k ≤k S Re η k + k S Im η k ≤ r k η k for all η ∈ K u . But K u is a Hilbert space, therefore γ ( η ) =( η , γ ) for a certain γ ∈ K u . Since ( η , γ ∗ ) = ( η ∗ , γ ) ∗ = γ ( η ∗ ) ∗ = γ ( η ) = ( η , γ ), η ∈ K u ,it follows that γ ∈ K uh and k γ k = ( γ , γ ) = γ ( γ ) ≤ r k γ k , that is, γ ∈ r ball K uh . Put k e = γ + ru ∈ r ball K uh + ru ⊆ c u . Thus T η = Sη + ( η, u ) ζ + (( η, γ ) + ( η, ru )) e = Sη + ( η, u ) ζ + ( η, k e ) e and both S and T are bounded ∗ -linear operators. It follows that Sη = P f = e ( Sη , f ) f = P f = e (cid:0) η , k uf (cid:1) f for uniquely defined k uf = S ∗ f ∈ K uh , (cid:13)(cid:13) k uf (cid:13)(cid:13) ≤ k S ∗ k ≤ r , f = e . Put k = { k f : f ∈ F } with k f = k uf + r f u . Note that k k f k = (cid:13)(cid:13) k uf (cid:13)(cid:13) + r f ≤ r for all f = e , and k k e k = k γ k + r ≤ r , that is, sup k k k ≤ r . Moreover, T η = X f = e (cid:0) η , k uf (cid:1) f + X f = e ( η, r f u ) f + ( η, k e ) e = X f = e ( η, k f ) f + ( η, k e ) e = X f ( η, k f ) f. Finally, for η ∈ c u we have T η = Sη + ( η, u ) ζ + ( γ ( η ) + ( η, ru )) e ∈ c e and X f = e ( η, k f ) = k Sη + ( η, u ) ζ k ≤ ( γ ( η ) + ( η, ru )) = (( η, γ ) + ( η, ru )) = ( η, k e ) , which means that k is an H -support in K and T η = T k η for all η ∈ K . If T is a unital positivemapping then ζ = 0, that is, r f = 0 for all f = e , and r = 1. The latter means that k is a unital H -support (see Remark 4.1). (cid:3) The unital cone L ( X, µ ) + . The matrix algebra M n ( C ( X )) is identified with the algebra C ( X, M n ) of all M n -valued continuous functions on X . The following result is known (see [26,Theorem 3.2]). For the sake of a reader we provide its detailed proof within the duality context,which is a bit different than its original one. Proposition 4.2.
The equality holds M ( C ( X )) + = min C ( X ) + .Proof. By its very definition S (cid:0) C ( X ) + (cid:1) = P ( X ) is the space of all probability measures on X .Note that P ( X ) is a w ∗ -compact subset of the space M ( X ) = C ( X ) ∗ of all finite Radon chargeson X . Based on Krein-Milman theorem, we conclude that P ( X ) is the w ∗ -closure of the convexhull of its extremal boundary ∂ P ( X ) which consists of Dirac measures δ t , t ∈ X . For every v ∈ C ( X ) we have k v k ∞ = sup {| v ( t ) | : t ∈ X } = sup {|h v, δ t i| : t ∈ X } = sup |h v, ∂ P ( X ) i| ≤ sup |h v, P ( X ) i| = k v k e ≤ sup |h v, ball M ( X ) i| = k v k ∞ , that is, k v k ∞ = k v k e (see Subsection2.3). It follows that min C ( X ) + = P ( X ) ⊡ = ( ∂ P ( X )) ⊡ . Take v ∈ M n ( C ( X )). Then v ∈ min C ( X ) + iff hh v, ∂ P ( X ) ii ≥
0. The latter means (see Proposition 3.2) that ( v ( t ) a, a ) = a ∗ v ( t ) a = ( a ∗ va ) ( t ) = h a ∗ va, δ t i = ( hh v, δ t ii a, a ) ≥ t ∈ X and a ∈ M n, , that is, v ∈ M n ( C ( X )) + . Whence M n ( C ( X )) ∩ min C ( X ) + = M n ( C ( X )) + for all n . (cid:3) Now fix µ ∈ M ( X ) + and consider the Hilbert ∗ -space H = L ( X, µ ) with the canonicalrepresentation mapping ι : C ( X ) → L ( X, µ ). Put ι (1) = u . Note that ι a ∗ -linear map-ping and µ ( X ) / = (cid:0)R (cid:1) / = k u k = k ι (1) k ≤ k ι k k k ∞ = k ι k = sup {k ι (ball C ( X )) k } ≤ sup n k ball C ( X ) k ∞ (cid:0)R (cid:1) / o ≤ µ ( X ) / , that is, k ι k = µ ( X ) / . Recall that each element η ∼ ∈ L ( X, µ ) being an equivalence class has a Borel representative η . We use the same notation η for the class η ∼ either. If µ ∈ P ( X ) then u takes place the role of a unit in L ( X, µ ), and therelated cone c consists of those real-valued Borel functions η on X such that η = η + ru with η ⊥ u , r = ( η, u ) = R ηdµ ≥ R η ( t ) dµ ≤ r . We use the notation L ( X, µ ) + instead of c . Recall that L ( X, µ ) possesses another conventional cone lifted from the cone C ( X ) + . Thusa hermitian class η ∼ ∈ L ( X, µ ) is positive iff η ( t ) ≥ µ -almost all t ∈ X . These cones areessentially distinct. A real-valued Borel representative of a class from the cone L ( X, µ ) + couldtake an highly negative values being far to be positive in the ordinary sense. Example 4.1.
Let us equip the compact interval X = [ − , ⊆ R with Lebesgue’s measure − dt .Put η = χ [ − , − /n ] + (1 − √ n ) χ [ − /n, + (1 + √ n ) χ [0 , /n ] + χ [1 /n, , where χ M indicates to thecharacteristic function of a subset M from X . Note that η = η + 1 with η = −√ nχ [ − /n, + √ nχ [0 , /n ] . Since R η = 2 − ( −√ n/n + √ n/n ) = 0 , we conclude that η = η + 1 is an orthogonalexpansion in L ( X, µ ) + and R η ( t ) dµ = 1 , that is, η ∈ L ( X, µ ) + . But η ([ − /n, −√ n < for n > . A very similar example can be constructed with a continuous function (orrepresentative) η . Corollary 4.1.
Let η ∈ M n ( L ( X, µ )) h with its expansion η = η + au ⊕ n , a = hh η, u ii ∈ M n .Then η ∈ min L ( X, µ ) + iff a ≥ and R ( η ( t ) β, β ) dµ ≤ ( aβ, β ) for all β ∈ M n, . In the case ofan atomic measure µ concentrated on a countable subset S ⊆ X we have − a ≤ µ ( s ) / η ( s ) ≤ a in M n , s ∈ S whenever η ∈ min L ( X, µ ) + .Proof. Using Proposition 3.2, we deduce that η ∈ min L ( X, µ ) + iff β ∗ ηβ ∈ L ( X, µ ) + for all β ∈ M n, . Since β ∗ ηβ = β ∗ η β + β ∗ aβu , it follows that β ∗ aβ ≥ k β ∗ η β k ≤ β ∗ aβ . But η is identified with a Borel function η : X → M n and ( β ∗ η β ) ( t ) = β ∗ η ( t ) β = ( η ( t ) β, β ) for all β ∈ M n, . Similarly, β ∗ aβ = ( aβ, β ) ≥
0, which means that a ≥
0. Thus η ∈ min L ( X, µ ) + iff a ≥ R ( η ( t ) β, β ) dµ = R ( β ∗ η β ) ( t ) dµ = k β ∗ η β k ≤ ( aβ, β ) . EPARABLE MORPHISMS OF OPERATOR HILBERT SYSTEMS 23
Finally, assume that µ is an atomic measure concentrated on S , and η ∈ min L ( X, µ ) + . Forevery s ∈ S we have µ ( s ) ( η ( s ) β, β ) ≤ R ( η ( t ) β, β ) dµ ≤ ( aβ, β ) , which in turn implies that µ ( s ) / | ( η ( s ) β, β ) | ≤ ( aβ, β ) for all β ∈ M n, , that is, − a ≤ µ ( s ) / η ( s ) ≤ a in M n for all s ∈ S . (cid:3) Remark 4.3. If v ∈ C ( X ) + with its orthogonal expansion v = v + ru in L ( X, µ ) satisfies anextra positivity condition − v + ru ≥ in C ( X ) then v ∈ L ( X, µ ) + . Indeed, since v ≥ , itfollows that r = ( v, u ) = R v ≥ . Moreover, − ru ≤ v ≤ ru or v ≤ r u , which in turn impliesthat k v k = (cid:0)R v (cid:1) / ≤ r (cid:0)R u (cid:1) = r , that is, v ∈ L ( X, µ ) + . Conversely, if v ∈ L ( X, µ ) + forsome v ∈ C ( X ) , and µ is atomic measure concentrated on S , then using Corollary 4.1, we derivethat − rµ ( s ) − / ≤ v ( s ) ≤ rµ ( s ) − / for all s ∈ S . Thus ± v + rµ − / ≥ . Thus the canonical, unital ∗ -linear mapping ι : C ( X ) → L ( X, µ ) is not positive is the senseof Subsection 4.1.
Proposition 4.3.
Let A ⊆ X be a µ -measurable subset with µ ( A ) > . Then χ A ∈ L ( X, µ ) + iff µ ( A ) ≥ / .Proof. First notice that ( χ A , u ) = µ ( A ) and χ A − µ ( A ) u ∈ L ( X, µ ) uh . Thus χ A = ( χ A − µ ( A ) u )+ µ ( A ) u is the orthogonal decomposition of χ A in L ( X, µ ). It follows that χ A ∈ L ( X, µ ) + iff k χ A − µ ( A ) u k ≤ µ ( A ). But k χ A − µ ( A ) u k = Z ( χ A ( t ) − µ ( A )) dµ = Z A ( χ A ( t ) − µ ( A )) dµ + Z X \ A ( χ A ( t ) − µ ( A )) dµ = (1 − µ ( A )) µ ( A ) + µ ( A ) µ ( X \ A ) = (1 − µ ( A )) µ ( A ) . Thus (1 − µ ( A )) µ ( A ) ≤ µ ( A ) iff µ ( A ) ≥ / (cid:3) Finally, suppose that µ ∼ µ ′ in P ( X ), that is, I µ ( X ) = I µ ′ ( X ). By Lebesgue-NikodymTheorem, µ ′ = kµ for some k ∈ L ( X, µ ) such that k ( t ) > µ -almost all t ∈ X and R k ( t ) dµ = 1. In this case, k − ∈ L ( X, µ ′ ) or k − / ∈ L ( X, µ ′ ). Moreover, L ( X, µ ) is identifiedwith L ( X, µ ′ ) along with the ∗ -linear unitary U : L ( X, µ ) → L ( X, µ ′ ), U ( η ) = η/ √ k . Namely,( U η , U η ) ′ = Z η ( t ) η ∗ ( t ) k ( t ) − dµ ′ = Z η ( t ) η ∗ ( t ) dµ = ( η , η )for all η i ∈ L ( X, µ ). Note that u ′ = u/ √ k is a unit vector in L ( X, µ ′ ), and we have the relatedunital cone L ( X, µ ′ ) + . If η ∈ L ( X, µ ) + then U ( η ) ∈ L ( X, µ ′ ) h and k U ( η ) k ′ = k η k ≤ √ η, u ) = √ Z η ( t ) dµ = √ Z (cid:16) η/ p k e (cid:17) ( t ) (cid:16) / p k e (cid:17) ( t ) dµ ′ = √ U ( η ) , u ′ ) ′ , which means that U ( η ) ∈ L ( X, µ ′ ). Thus U L ( X, µ ) + = L ( X, µ ′ ) + or U is an order isomor-phism of the related unital Hilbert spaces. In this case, U ι : C ( X ) → L ( X, µ ′ ), ( U ι ) (1) = 1 / √ k is not the canonical mapping that responds to µ ′ .4.3. A unital positive mapping from C ( X ) to ( H, e ) . For brevity we focus on unital positivemaps instead of positive maps. As above we fix a Hilbert space H with its hermitian basis F ,the unital cone c , and fix also a probability measure µ (or integral R ) on a compact Hausdorff topological space X . A family of real valued Borel functions k = { k f : f ∈ F } ⊆ ball L ∞ ( X, µ ) h with k e = u is said to be an H - support on X if k f ⊥ k e , f = e and X f = e ( v, k f ) ≤ ( v, k e ) in L ( X, µ ) for all v ∈ C ( X ) + . Note that ( v, k e ) = R v ≥ v ≥
0. If additionally, P f = e k k f k p < ∞ in L ( X, µ ) for p = 1 ,
2, we say that k is an H - support on X of type p . But if P f = e k f ≤ k e in L ∞ ( X, µ ) we saythat k is a maximal H - support on X . Note that a maximal support if of type 2 automatically.Indeed, P f ∈ λ k f ≤ u in L ∞ ( X, µ ) implies that P f ∈ λ R k f ≤ λ ⊆ F \ { e } ,therefore P f = e k k f k = P f = e R k f ≤ Lemma 4.1. If k is an H -support on X then T : C ( X ) → ( H, e ) , T v = P f ( v, k f ) f is a unitalpositive mapping, that is, T (1) = e and T (cid:0) C ( X ) + (cid:1) ⊆ c . Moreover, if k is of type p then T admitsa unique bounded linear extension T k : L ( X, µ ) → ( H, e ) , T k = P f f ⊙ k f , which is a nuclearoperator if p = 1 and Hilbert-Schmidt operator if p = 2 .Proof. If v ∈ C ( X ) h with − ≤ v ≤
1, then v ± k e ≥ | ( v, k e ) | ≤ R | v | ≤ R P f = e ( v, k f ) = P f = e ( v ± k e , k f ) ≤ ( v ± k e , k e ) = (( v, k e ) ± . In particular, P f = e ( v, k f ) ≤ (1 − | ( v, k e ) | ) , which in turn implies that k T v k = X f ( v, k f ) ! / ≤ | ( v, k e ) | + X f = e ( v, k f ) ! / ≤ . Hence k T | ball C ( X ) h k ≤
1. In the case of any v ∈ ball C ( X ), we have Re v, Im v ∈ ball C ( X ) h and k T v k ≤ k T Re v k + k T Im v k ≤
2, that is, T is a well defined bounded linear mapping.Further, take v ∈ C ( X ) + . Taking into account that k is an H -support on X , we deduce that k T v k = P f ( v, k f ) ≤ v, k e ) = 2 ( T v, e ) or k T v k ≤ √
T v, e ), that is, T (cid:0) C ( X ) + (cid:1) ⊆ c .Moreover, T u = P f ( k e , k f ) f = ( k e , k e ) e = (cid:0)R (cid:1) e = e . Thus T is a unital positive mapping.Finally, assume that k is of type 2. For every v ∈ C ( X ) we have k T v k = P f | ( v, k f ) | ≤k v k P f k k f k . By continuity argument T admits a unique extension T k : L ( X, µ ) → ( H, e ), T k ι = T such that T k = P f f ⊙ k f and k T k k = P f k T ∗ k f k = P f k k f k < ∞ . Hence T k is aHilbert-Schmidt operator. If k is of type 1 then k T k k ≤ P f k f k k k f k = P f k k f k < ∞ , whichmeans that T k is a nuclear operator. (cid:3) Below in Theorem 4.1, we prove that the bounded linear extension T k : L ( X, µ ) → ( H, e )exists for every H -support k on X . Proposition 4.4.
Let T : C ( X ) → ( H, e ) be a unital positive mapping. There is a unique proba-bility measure µ on X and an H -support k ⊆ ball L ∞ ( X, µ ) h on X such that T v = P f ( v, k f ) f , v ∈ C ( X ) . The functions k f , f = e are uniquely determined modulo µ -null functions, and T v = lim λ Z v ( t ) X f ∈ λ k f ( t ) f + e ! dµ, where λ is running over all finite subsets in F \ { e } , and we used the related Radon integral for H -valued measurable functions on X . Thus there is a one to one correspondence between unitalpositive maps C ( X ) → ( H, e ) and H -supports on X . EPARABLE MORPHISMS OF OPERATOR HILBERT SYSTEMS 25
Proof. If v ∈ C ( X ) + then T v ∈ c . In particular, ( T v, e ) ≥
0, which means that v ( T ( v ) , e )is a positive Radon integral, that is, ( T v, e ) = h v, µ i for a certain µ ∈ M ( X ) + . Note that R dµ = ( T , e ) = k e k = 1, that is, µ ∈ P ( X ). Moreover, P f = e ( T v, f ) ≤ ( T v, e ) for all v ∈ C ( X ) + . Since f + e ∈ S ( c ) (see Lemma 3.2), it follows that ( T v, f + e ) = h v, µ f i forsome µ f ∈ M ( X ) + . But ( T v, f + e ) = ( T v, f ) + (
T v, e ) ≤ T v, e ) for all v ∈ C ( X ) + , whichmeans that µ f ≤ µ in M ( X ) h for all f = e . Thus { µ f } ⊆ I µ ( X ), where I µ ( X ) is the closed(lattice) ideal of the complete lattice M ( X ) h generated by µ (see Subsection 2.4). Using Lebesgue-Nikodym Theorem, we deduce that µ f = m f µ for some (real) Borel function m f ∈ L ( X, µ ) h suchthat 0 ≤ m f ≤
2. The functions { m f : f = e } are uniquely determined modulo µ -null functions.It follows that ( T v, f ) = (
T v, f + e ) − ( T v, e ) = h v, m f µ i − h v, µ i = h v, k f µ i for all v ∈ C ( X ), where k f = m f − L ( X, µ ) h . Since T e ,we obtain that h , k f µ i = ( T , f ) = 0, that is, k f ⊥ u in L ( X, µ ) for all f = e .Thus T v = P f ( v, k f ) f = P f = e (cid:0)R v ( t ) k f ( t ) dµ (cid:1) f + (cid:0)R v ( t ) dµ (cid:1) e . In particular, T v = lim λ X f ∈ λ (cid:18)Z v ( t ) k f ( t ) dµ (cid:19) f + (cid:18)Z v ( t ) dµ (cid:19) e = lim λ Z X f ∈ λ v ( t ) k f ( t ) f + v ( t ) e ! dµ, where λ is running over all finite subsets in F \ { e } . Notice that we used the canonical extensionof the Radon integral to H -valued functions on X (see below Remark 4.4).Finally, prove that k = { k f } ⊆ ball L ∞ ( X, µ ) h . Since P f = e h v, k f µ i ≤ h v, µ i for all v ∈ C ( X ) + , we conclude that |h v, k f µ i| ≤ h v, µ i , v ∈ C ( X ) + , which means that − µ ≤ k f µ ≤ µ in M ( X ) h . It follows that | k f | µ = | k f µ | = ( k f µ ) ∨ ( − k f µ ) ≤ µ (see [1, Ch. V, 5.4]), that is, | k f | ≤ µ -almost everywhere on X . Thus k ⊆ ball L ∞ ( X, µ ) h and it is an H -support on X . The restfollows from Lemma 4.1. (cid:3) Remark 4.4.
Let µ be a Radon measure on a Hausdorff compact space X , H a Hilbert space andlet v : X → H be a weakly (or weak ∗ ) measurable mapping with µ -integrable norm. Thus h v ( · ) , η i is measurable for every η ∈ H , and R k v ( t ) k dµ < ∞ . There is a unique element R v ( t ) dµ ∈ H such that (cid:10)R v ( t ) dµ, η (cid:11) = R h v ( t ) , η i dµ for all η ∈ H (see [27, 2.5.14] ). If v is continuous then R v ( t ) dµ is a limit of Riemann sums P Nm =1 µ ( E m ) v ( t m ) taken over all partitions { E m } of X into disjoint Borel subsets (see [27, E 2.5.8] ). In particular, if v ( X ) ⊆ c for a certain closed cone c then R v ( t ) dµ ∈ c . Now we can prove that all unital positive maps C ( X ) → ( H, e ) admit unique extensions up topositive maps between Hilbert spaces.
Theorem 4.1.
Let T : C ( X ) → ( H, e ) be a unital positive mapping with its H -support k ⊆ ball L ∞ ( X, µ ) on X . Then T is an absolutely summable mapping, k is a unital H -support in L ( X, µ ) , and T admits a unique bounded linear extension T k : ( L ( X, µ ) , u ) → ( H, e ) , which isa unital positive mapping of Hilbert spaces.Proof. By Proposition 4.4, there is a unique probability measure µ on X and an H -support k ⊆ ball L ∞ ( X, µ ) h on X such that T v = P f ( v, k f ) f , v ∈ C ( X ). The functions k f are uniquelydetermined modulo µ -null functions. Prove that T : ( C ( X ) , k·k ) → H is bounded. If v ∈ C ( X ) h then v = v + − v − with v + , v − ∈ C ( X ) + and | v | = v + ∨ v − = v + + v − . Moreover, X f = e ( v, k f ) = X f = e ( v + , k f ) + X f = e ( v − , k f ) − X f = e ( v + , k f ) ( v − , k f ) ≤ ( v + , k e ) + ( v − , k e ) + 2 X f = e | ( v + , k f ) ( v − , k f ) |≤ ( v + , k e ) + ( v − , k e ) + 2 X f = e ( v + , k f ) ! / X f = e ( v − , k f ) ! / ≤ ( v + , k e ) + ( v − , k e ) + 2 ( v + , k e ) ( v − , k e ) = ( v + + v − , k e ) = ( | v | , k e ) , which in turn implies that k T v k = X f = e ( v, k f ) + ( v, k e ) ≤ ( | v | , k e ) + ( v, k e ) ≤ (cid:18)Z | v | dµ (cid:19) , that is, k T v k ≤ √ R | v | dµ . In the case of any v ∈ C ( X ) we derive that k T v k ≤ k T Re v k + k T Im v k ≤ √ R ( | Re v | + | Im v | ) dµ ≤ √ R | v | dµ . By the known result of Pietsch [29, 2.3.3],we deduce that T is an absolutely summable mapping with k T k ≤ π ( T ) ≤ √ µ ( X ) = 2 √
2. Itfollows that T is factorized throughout the Hilbert space L ( X, µ ) [29, 3.3.4]. Namely, k T v k ≤ √ (cid:0)R | v | dµ (cid:1) / (cid:0)R dµ (cid:1) / = 2 √ k v k for all v ∈ C ( X ), and taking into account the density of ι ( C ( X )) in L ( X, µ ), we obtain a unique bounded linear extension T k : L ( X, µ ) → H , T k ι = T .Moreover, T k η = P f ( η, k f ) f for all η ∈ L ( X, µ ) due to the density of ι ( C ( X )) in L ( X, µ ).It remains to prove that k is a unital H -support in the unital Hilbert space ( L ( X, µ ) , u ). If v ∈ ι ( C ( X )) ∩ L ( X, µ ) uh then as above we have P f = e ( v , k f ) ≤ ( | v | , k e ) ≤ k| v |k (cid:0)R dµ (cid:1) = k v k = (( v , k e ) + k v k ) . Notice that ( v , k e ) = ( v , u ) = 0. Take η ∈ L ( X, µ ) uh . Then η = lim n v ,n in L ( X, µ ) for a certain sequence ( v ,n ) n from ι ( C ( X )) ∩ L ( X, µ ) uh . For everyfinite subset λ ⊆ F \ { e } we have X f ∈ λ ( η , k f ) = lim n X f ∈ λ ( v ,n , k f ) ≤ lim n k v ,n k = k η k = (( η , k e ) + k η k ) , which in turn implies that P f = e ( η , k f ) ≤ (( η , k e ) + k η k ) . Consequently, k is a unital H -support in ( L ( X, µ ) , u ) (see Remark 4.1), and T = T k in the sense of Proposition 4.1. (cid:3) Notice that T (cid:0) C ( X ) + (cid:1) ⊆ c implies that T ∗ ( S ( c )) ⊆ P ( X ). Using Lemma 2.1 and Proposition4.2, we obtain that T ( ∞ ) (cid:0) M ( C ( X )) + (cid:1) = T ( ∞ ) (cid:0) min C ( X ) + (cid:1) = T ( ∞ ) (cid:16) P ( X ) ⊡ (cid:17) ⊆ S ( c ) ⊡ = min c .4.4. Separable and nuclear morphisms.
Recall that a positive mapping φ : V → W of op-erator systems is called a separable if φ = P l p l ⊙ q l for some positive functionals q l on V and positive elements p l from W , where ( p l ⊙ q l ) v = q l ( v ) p l for all v ∈ V . Thus φ ( v ) =lim k P kl =1 q l ( v ) p l in W for every v ∈ V . Notice that a separable mapping φ defines a matrixpositive mapping φ : V → ( W , max W + ) automatically. Indeed, take v ∈ M n ( V ) + . Since thepositive functionals q l on V are matrix positive, we deduce that φ ( n ) ( v ) = lim k P kl =1 q ( n ) l ( v ) p ⊕ nl =lim k P kl =1 q ( n ) l ( v ) / p ⊕ nl q ( n ) l ( v ) / . But q ( n ) l ( v ) / p ⊕ nl q ( n ) l ( v ) / ∈ W c + , therefore φ ( n ) ( v ) ∈ W ⊡⊡ + = (cid:0) W c + (cid:1) − = max W + . EPARABLE MORPHISMS OF OPERATOR HILBERT SYSTEMS 27
Now let T : C ( X ) → ( H, e ) be a unital positive mapping. By Proposition 4.4, T is given byan H -support k ⊆ ball L ∞ ( X, µ ) h on X . Suppose T is a nuclear mapping, that is, T = P l γ l ⊙ q l for some ( γ l ) l ⊆ H and ( q l ) l ⊆ M ( X ) such that P l k γ l k k q l k < ∞ . Taking into account that T isa ∗ -linear mapping and both ( C ( X ) , M ( X )) and (cid:0) H, H (cid:1) are dual ∗ -pairs, we can assume that( γ l ) l ⊆ ball H h and ( q l ) l ⊆ M ( X ) h with P l k q l k < ∞ . We say that T is a nuclear morphism if T = P l γ l ⊙ q l for some ( γ l ) l ⊆ ball H h and ( q l ) l ⊆ I µ ( X ) with P l k q l k < ∞ . Lemma 4.2.
Let T : C ( X ) → ( H, e ) be a unital positive mapping given by an H -support k ⊆ ball L ∞ ( X, µ ) h on X . Then T is a nuclear morphism if and only if T + e ⊙ q is separable for acertain q ∈ I µ ( X ) . In this case, one can assume that q ∈ I µ ( X ) + .Proof. First assume that T + e ⊙ q is separable for a certain q ∈ I µ ( X ). Then T = P l p l ⊙ q l − e ⊙ q for some ( p l ) l ⊆ c and ( q l ) l ⊆ M ( X ) + . We have p l = η l + r l e , η l ∈ H eh , k η l k ≤ r l . Put ζ l = r − l η l ∈ ball H eh , and µ l = r l q l . Then T ( v ) = X l h v, q l i ( η l + r l e ) − h v, q i e = X l h v, µ l i ( ζ l + e ) − h v, q i e = X l h v, µ l i ζ l + X l h v, µ l i − h v, q i ! e ∈ H e ⊕ C e = H for all v ∈ C ( X ). In particular, P l h , µ l i ζ l = 0 and P l h , µ l i = 1 + h , q i . The latter means that τ = P l µ l ∈ M ( X ) + with P l k µ l k = P l h , µ l i = 1+ k q k < ∞ . By Proposition 4.4, we obtain theequality µ = τ − q . But q ∈ I µ ( X ), therefore τ = µ + q ∈ I µ ( X ) + . Since { µ l } ≤ τ and I µ ( X ) is alattice ideal, it follows that { µ l } ⊆ I µ ( X ) + . Moreover, P l k ζ l k k µ l k ≤ P l k µ l k = k τ k = 1 + k q k ,which means that T is a nuclear morphism given by T = P l ζ l ⊙ µ l + e ⊙ µ , { µ l } ⊆ I µ ( X ), and k T k ≤ k q k .Conversely, suppose that T is a nuclear morphism. Then T = P l γ l ⊙ q l for some ( γ l ) l ⊆ ball H h and ( q l ) l ⊆ I µ ( X ) with P l k q l k < ∞ . Thus γ l = ζ l + r l e with ζ l ∈ ball H eh , r l ∈ R and k ζ l k + r l ≤ v ∈ C ( X ) then T v = X l h v, q l i ( ζ l + r l e ) = X l h v, q l i ζ l + X l h v, r l q l i e = X l h v, q l i ζ l + h v, µ i e, where µ = P l r l q l , P l k r l q l k = P l | r l | k q l k ≤ P l k q l k < ∞ . Thus T = P l ζ l ⊙ q l + e ⊙ µ with P l k ζ l k k q l k ≤ P l k q l k < ∞ . Using the Jordan decompositions q l = q l, + − q l, − with q l, + , q l, − ∈ M ( X ) + and k q l k = k q l, + k + k q l, − k [1, Ch. 3, 2.6], we obtain that T = P l ζ l ⊙ q l, + + P l ( − ζ l ) ⊙ q l, − + e ⊙ µ and P l k ζ l k k q l, + k + P l k− ζ l k k q l, − k ≤ P l k ζ l k k q l k < ∞ . Taking intoaccount that { q l } ⊆ I µ ( X ), we deduce that { q l, + , q l, − } ⊆ I µ ( X ) either. Thus we can assume that T = P l ζ l ⊙ µ l + e ⊙ µ with ζ l ∈ ball H eh , µ l ∈ I µ ( X ) + and P l k µ l k < ∞ . It follows that T = X l ( ζ l + e ) ⊙ µ l + e ⊙ µ − e ⊙ τ = X l η l ⊙ µ l + e ⊙ µ − e ⊙ τ, where η l = ζ l + e ∈ c and τ = P l µ l ∈ I µ ( X ) + . Consequently, we can assume that T = P l η l ⊙ µ l − e ⊙ τ for some ( η l ) l ⊆ c , ( µ l ) l ⊆ M ( X ) + , τ ∈ I µ ( X ) + such that P l k η l k k µ l k < ∞ ,which means that T + e ⊙ τ is separable. (cid:3) Corollary 4.2. If T : C ( X ) → ( H, e ) is a separable morphism then T is a nuclear morphismautomatically.Proof. One needs to use Lemma 4.2 with q = 0. (cid:3) Theorem 4.2.
Let T : C ( X ) → ( H, e ) be a unital positive mapping with its H -support k ⊆ ball L ∞ ( X, µ ) on X . If T is a nuclear-morphism then its bounded linear extension T k : L ( X, µ ) → ( H, e ) is a Hilbert-Schmidt operator. In this case, the H -support k on X is maximal whenever T is separable.Proof. Assume that T is a nuclear morphism. By Lemma 4.2, T + e ⊙ q is separable for a certain q ∈ I µ ( X ) + . Thus T + e ⊙ q = P l ( ζ l + e ) ⊙ µ l with ( ζ l ) l ⊆ ball H eh and ( µ l ) l ⊆ M ( X ) + . Put τ = P l µ l ∈ M ( X ) + . Notice that (1 + k q k ) e = T (1) + h , q i e = P l h , µ l i ( ζ l + e ) = P l h , µ l i e and h , τ i = P l h , µ l i = P l k µ l k = 1 + k q k < ∞ . Then T = P l ζ l ⊙ µ l + e ⊙ ( τ − q ), which inturn implies that µ = τ − q . Since q ∈ I µ ( X ) + , we obtain that τ = µ + q ∈ I µ ( X ) + and µ ≤ τ .Hence I µ ( X ) = I τ ( X ). By Lebesgue-Nikodym Theorem, µ = mτ for a Borel function m suchthat 0 < m ( t ) ≤ µ -almost all t ∈ X (see [1, Ch. V, 5.6, Proposition 10]). Since { µ l } ≤ τ ,we deduce also that { µ l } ⊆ I τ ( X ) and there are (unique) positive bounded Borel function { n l } on X such that µ l = n l τ for all l . Notice that τ = X l µ l = X l n l τ = ∨ ( k X l =1 n l τ ) = ∨ ( k X l =1 n l ! τ ) = ∨ k X l =1 n l ! τ = X l n l ! τ thanks to [1, Ch. V, 5.4, Proposition 6]. Hence P l n l = 1 for τ -almost (or µ -almost) everywhereon X . Put m l = n l m for all l . Thus m l are µ -almost everywhere finite Borel functions on X , and µ l = n l τ = m l mτ = m l µ . Moreover,(4.1) X l k ζ l k k m l k ≤ X l k m l k ≤ X l h , µ l i = h , τ i = 1 + k q k , thereby m − = P l n l m − = P l m l ∈ L ( X, µ ) being an absolutely summable series in L ( X, µ ),and τ = P l m l µ = m − µ . Actually, m − ∈ L ( X, µ ). Indeed, (cid:12)(cid:12)(cid:0) v, m − (cid:1)(cid:12)(cid:12) ≤ X l Z | v | m l dµ = X l Z | v | dµ l ≤ Z | v | dτ ≤ (cid:18)Z | v | dτ (cid:19) / (cid:18)Z dτ (cid:19) / for all v ∈ C ( X ). Take a sequence ( v r ) r ⊆ C ( X ) with lim r ι ( v r ) = 0 in L ( X, µ ). Thenlim r R | v | dτ = 0 by Lebesgue-Nikodym Theorem, and lim r ( v r , m − ) = 0, which means that( · , m − ) is a bounded linear functional on L ( X, µ ), or m − ∈ L ( X, µ ). In particular, { m l } ⊆ L ( X, µ ). If m l : L ( X, µ ) → C is the related bounded linear functional then h v, m l i = ( v, m l ) = Z v ( t ) m l ( t ) dµ = Z v ( t ) m l ( t ) m ( t ) dτ = Z v ( t ) dµ l = h v, µ l i for all v ∈ C ( X ), that is, µ l = ( · , m l ) for all l .By Theorem 4.1, T admits a unique bounded linear extension T k : L ( X, µ ) → ( H, e ) with therelated unital H -support k = { k f : f ∈ F } (see Proposition 4.1). Note that ( T k ι ) ( v ) = T ( v ) = P l ( v, m l ) ζ l + ( v, e and(4.2) X l k ( v, m l ) ζ l k ≤ X l Z | v | m l dµ k ζ l k = X l Z | v | dµ l k ζ l k ≤ Z | v | dτ, that is, the series P l ( v, m l ) ζ l is absolutely summable in H for every v ∈ C ( X ). Take a Borel func-tion η ∈ L ( X, µ ). Then η = lim r v r in L ( X, µ ) for some sequence ( v r ) r ⊆ C ( X ). In particular, k v r − v s k → r , s . Since τ ∼ µ , we have R | v r − v s | dτ ≤ (cid:0)R | v r − v s | dτ (cid:1) / (cid:0)R dτ (cid:1) / → EPARABLE MORPHISMS OF OPERATOR HILBERT SYSTEMS 29 r and s . Using (4.2), we obtain that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X l k ( v r , m l ) ζ l k − X l k ( v s , m l ) ζ l k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X l k ( v r − v s , m l ) ζ l k ≤ Z | v r − v s | dτ → r and s . Hence there is a limit lim r P l k ( v r , m l ) ζ l k . Using the lower semicontinuityproperty, we obtain that X l k ( η, m l ) ζ l k = X l (cid:13)(cid:13)(cid:13)(cid:16) lim r v r , m l (cid:17) ζ l (cid:13)(cid:13)(cid:13) ≤ lim inf r X l k ( v r , m l ) ζ l k = lim r X l k ( v r , m l ) ζ l k < ∞ , that is, ζ = P l ( η, m l ) ζ l ∈ H being the sum of an absolutely summable series in H . Actually, ζ = lim r P l ( v r , m l ) ζ l . Indeed, for ε > r such that P l k ( v r − v s , m l ) ζ l k ≤ ε for all r, s ≥ r . Then (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ζ − X l ( v r , m l ) ζ l (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ X l k ( η − v r , m l ) ζ l k = X l (cid:13)(cid:13)(cid:13) lim s ( v s − v r , m l ) ζ l (cid:13)(cid:13)(cid:13) ≤ lim inf s X l k ( v s − v r , m l ) ζ l k ≤ ε for all r ≥ r . Thus T k η = lim r ( T k ι ) ( v r ) = lim r P l ( v r , m l ) ζ l + ( v r , e = P l ( η, m l ) ζ l + ( η, e .Hence(4.3) T k = X l ζ l ⊙ m l + e ⊙ L ( X, µ ) . Finally take expansions ζ l = P f = e ζ l,f f in F with real ζ l,f , | ζ l,f | ≤
1, and put k ′ f = P l ζ l,f m l for all f = e . Since (cid:12)(cid:12) k ′ f (cid:12)(cid:12) ≤ P l m l = m − , it follows that k ′ f ∈ L ( X, µ ). Based on (4.3), we deduce that( T k η, f ) = ( P l ( η, m l ) ζ l , f ) = P l ( η, m l ) ζ l,f = (cid:0) η, k ′ f (cid:1) for all η ∈ L ( X, µ ) and f = e , therefore k f = k ′ f for all f = e , and k e = u . For a finite subset λ ⊆ F \ { e } we have X f ∈ λ k f = X f ∈ λ X l,t ζ l,f ζ t,f m l m t ≤ X l,t X f ∈ λ | ζ l,f | | ζ t,f | ! m l m t ≤ X l,t X f ∈ λ | ζ l,f | ! / X f ∈ λ | ζ t,f | ! / m l m t ≤ X l,t k ζ l k k ζ t k m l m t ≤ X l,t m l m t = X l m l ! = m − , that is, P f = e k f ≤ m − . Consequently, k T k k = P f k k f k = 1+ P f = e R k f dµ ≤ R m − dµ < ∞ ,which means that T k is a Hilbert-Schmidt operator. In particular, the H -support k is maximal if m = 1 or q = 0. The latter is the case of a separable T . (cid:3) Remark 4.5.
As follows from the proof of Theorem 4.2, the Radon-Nikodym derivative d ( µ + q ) dµ belongs to L ( X, µ ) and P f = e k f ≤ (cid:18) d ( µ + q ) dµ (cid:19) if T + e ⊙ q is separable for a certain q ∈ I µ ( X ) + .In particular, k is a maximal H -support on X if q = 0 (or T is separable). The maximal and Hilbert-Schmidt supports in L ( X, µ ) . As above fix µ ∈ P ( X ) ona Hausdorff compact topological space X , and let T : ( L ( X, µ ) , u ) → ( H, e ) be a unital positivemapping. By Proposition 4.1, T = T k for a unital H -support k in L ( X, µ ). Thus k = { k f : f ∈ F } is a bounded family in L ( X, µ ) h such that k f ⊥ u for all f = e , k e = k ue + u ∈ ball L ( X, µ ) uh + u ,and P f = e ( η , k f ) ≤ (( η , k ue ) + k η k ) for all η ∈ L ( X, µ ) uh (see Remark 4.1). Certainly wecan assume that k consists of real-valued Borel functions on X . We say that k is a maximal H - support in L ( X, µ ) if k e ≥ P f = e k f ≤ k e . The latter means that P f ∈ λ k f ≤ k e as theBorel functions for every finite subset λ ⊆ F \ { e } . Since P f ∈ λ R k f dµ ≤ R k e dµ , it follows that k is an H -support in L ( X, µ ) of type 2 automatically.
Proposition 4.5.
Let T k : ( L ( X, µ ) , u ) → ( H, e ) be a unital positive mapping that responds toa maximal H -support k in L ( X, µ ) , and let T = T k ι : C ( X ) → ( H, e ) be the related unital ∗ -linear mapping. Then T ( ∞ ) (cid:0) min C ( X ) + (cid:1) ⊆ max c , which means that T : (cid:0) C ( X ) , M ( C ( X )) + (cid:1) → ( H, max c ) is a morphism of the relevant operator systems, whose support k ′ ⊆ ball L ∞ ( X, µ ′ ) on X is given by the family k ′ f = k f k e , f = e and k ′ e = u , where µ ′ = k e µ ∈ I µ ( X ) + . Moreover, in thiscase T is a nuclear operator.Proof. Take v ∈ M n ( C ( X )) + . Then v ( t ) ∈ M + n for all t ∈ X . Note that T ( n ) v = X f (cid:10)(cid:10) v, k f (cid:11)(cid:11) f ⊕ n = lim λ X f ∈ λ (cid:10)(cid:10) v, k f (cid:11)(cid:11) f ⊕ n + (cid:10)(cid:10) v, k e (cid:11)(cid:11) e ⊕ n = lim λ X f ∈ λ Z v ( t ) k f ( t ) ⊕ n f ⊕ n dµ + Z v ( t ) k e ( t ) ⊕ n e ⊕ n dµ = lim λ Z v ( t ) / X f ∈ λ k f ( t ) f + k e ( t ) e ! ⊕ n v ( t ) / dµ = lim λ Z v λ ( t ) dµ, where v λ ( t ) = v ( t ) / (cid:16)P f ∈ λ k f ( t ) f + k e ( t ) e (cid:17) ⊕ n v ( t ) / and λ is running over all finite subsetsin F \ { e } . Notice that we used the canonical extension of the Radon integral to M n ( H )-valuedfunctions on X (see Remark 4.4). Fix a finite subset λ ⊆ F \ { e } . By assumption k e ( t ) ≥ P f ∈ λ k f ( t ) ≤ k e ( t ) , which means that P f ∈ λ k f ( t ) f + k e ( t ) e ∈ c , therefore v λ ( t ) ∈ c c . Inthe case of continuous k f , f ∈ λ , and k e , we derive that R v λ ( t ) dµ ∈ ( c c ) − = c ⊡⊡ = max c (seeRemark 4.4). In the general case, k f ( t ) = lim m k f,m ( t ) is a sequential limit of continuous functions { k f,m } ⊆ C ( X ), and k e ( t ) = lim m k e,m ( t ) for an increasing sequence { k e,m } ⊆ C ( X ) + for µ -almostall t ∈ X . We can assume that P f ∈ λ k f,m ≤ k e,m (just replace k e,m by k e,m ∨ (cid:16)P f ∈ λ k f,m (cid:17) / ) forall m , and put v λ,m ( t ) = v ( t ) / (cid:16)P f ∈ λ k f,m ( t ) f + k e,m ( t ) e (cid:17) ⊕ n v ( t ) / . As above v λ,m ( t ) ∈ c c and z λ,m = R v λ,m ( t ) dµ ∈ max c for all m . Then Z v λ ( t ) dµ = lim m Z v ( t ) / X f ∈ λ k f,m ( t ) f + k e,m ( t ) e ! ⊕ n v ( t ) / dµ = lim m Z v λ,m ( t ) dµ = lim m z λ,m ∈ max c , EPARABLE MORPHISMS OF OPERATOR HILBERT SYSTEMS 31 which in turn implies that T ( n ) v = lim λ R v λ ( t ) dµ ( t ) ∈ max c . In particular, T : C ( X ) → H is aunital positive mapping. By Proposition 4.4, T is given by an H -support k ′ ⊆ ball L ∞ ( X, µ ′ ) on X , where h v, µ ′ i = ( T ι ( v ) , e ) = ( ι ( v ) , k e ) = R v ( t ) k e ( t ) dµ = h v, k e µ i for all v ∈ C ( X ), that is, µ ′ = k e µ ∈ I µ ( X ) + . Similarly,( T ι ( v ) , f ) = ( ι ( v ) , k f ) = Z v ( t ) k f ( t ) dµ = Z v ( t ) k f ( t ) k e ( t ) − dµ ′ = (cid:10) v, k f k − e µ ′ (cid:11) for all v ∈ C ( X ), which means that k ′ f = k f k e for all f = e . Notice that µ ′ { k e = 0 } = 0. Finally,taking into account that T k : ( L ( X, µ ) , u ) → ( H, e ) is a Hilbert-Schmidt operator (see Remark4.2), we deduce that T : C ( X ) → ( H, e ) is a nuclear operator [29, 3.3.3]. (cid:3)
We say that k is a Hilbert-Schmidt H - support in L ( X, µ ) if k e = u and P f ∈ λ R k f dµ < ∞ . Theorem 4.3.
Let T k : ( L ( X, µ ) , u ) → ( H, e ) be a unital positive mapping that responds to aHilbert-Schmidt support k in L ( X, µ ) and let T = T k ι : C ( X ) → ( H, e ) be the related unital ∗ -linear mapping. Then T + e ⊙ q is a separable morphism for a certain q ∈ I µ ( X ) + .Proof. By assumption T k is a Hilbert-Schmidt operator given by T k η = P f ( η, k f ) f , η ∈ L ( X, µ ),and k T k k = P f k k f k = P f R k f dµ < ∞ . For every n choose a finite subset λ n ⊆ F \ { e } suchthat P F \ λ n k k f k < n . Take f ∈ λ n and a real-valued µ -step function h f,n on X such that | h f,n | ≤ | k f | and k k f − h f,n k ≤ | λ n | n , where | λ n | indicates to the cardinality of λ n . Namely,since k f = k f, + − k f, − with k f, + ≥ k f, − ≥ | k f | = k f, + + k f, − , one can choose increasingsequences h (1) f,n and h (2) f,n of positive µ -step functions such that h (1) f,n ↑ k f, + and h (2) f,n ↑ k f, − . If h f,n = h (1) f,n − h (2) f,n then k f = lim n h f,n . Note that | h f,n ( t ) | = h (1) f,n ( t ) − h (2) f,n ( t ) ≤ h (1) f,n ( t ) ≤ k f, + ( t ) ≤| k f ( t ) | if h (1) f,n ( t ) ≥ h (2) f,n ( t ), and | h f,n ( t ) | = − h (1) f,n ( t ) + h (2) f,n ( t ) ≤ h (2) f,n ( t ) ≤ k f, − ( t ) ≤ | k f ( t ) | if h (1) f,n ( t ) ≤ h (2) f,n ( t ). Define T n : L ( X, µ ) → H , T n = e ⊙ k e + P f ∈ λ n f ⊙ h f,n , which is a finite rankoperator such that T ∗ n f = h f,n , f ∈ λ n , T ∗ n e = k e = u and T ∗ n f = 0, f / ∈ { e } ∪ λ n . Moreover, k T k − T n k = X f k T ∗ k f − T ∗ n f k = X f ∈{ e }∪ λ n k T ∗ k f − T ∗ n f k + X f / ∈{ e }∪ λ n k T ∗ k f k = X f ∈ λ n k k f − h f,n k + X f / ∈{ e }∪ λ n k k f k ≤ n , that is, T k = lim n T n in B ( L ( X, µ ) , H ). Further, for every n there is a partition X = X n ∪ . . . ∪ X nm n of X into µ -measurable subsets X nr such that h f,n = P m n r =1 α f,n,r χ nr , where χ nr is thecharacteristic function of X nr . Then T n = e ⊙ k e + P f ∈ λ n P m n r =1 α f,n,r f ⊙ χ nr = e ⊙ k e + P m n r =1 ζ nr ⊙ χ nr with ζ nr = P f ∈ λ n α f,n,r f ∈ H eh . For every t ∈ X we have k ζ nr k χ nr ( t ) = X f ∈ λ n α f,n,r χ nr ( t ) = X f ∈ λ n h f,n ( t ) χ nr ( t ) = X f ∈ λ n h f,n χ nr ! ( t ) ≤ X f ∈ λ n k f χ nr ! ( t ) = X f ∈ λ n k f ! ( t ) χ nr ( t ) , that is, k ζ nr k χ nr ≤ (cid:16)P f ∈ λ n k f (cid:17) χ nr . Taking into account that P m n r =1 χ nr = 1 for all n , we obtainthat(4.4) m n X r =1 k ζ nr k χ nr ≤ X f ∈ λ n k f ! m n X r =1 χ nr ≤ X f ∈ λ n k f for all n . In particular,(4.5) m n X r =1 k ζ nr k µ ( X nr ) = m n X r =1 Z k ζ nr k χ nr dµ ≤ Z X f ∈ λ n k f dµ = X f ∈ λ n k k f k . Put µ nr = χ nr µ ∈ I µ ( X ) + . It follows that T n ι = e ⊙ k e µ + P m n r =1 ζ nr ⊙ µ nr and k T ( v ) − T n ι ( v ) k ≤k T k − T n k k v k ≤ k T k − T n k k v k for all v ∈ C ( X ), that is, T ( v ) = lim T n ι ( v ), v ∈ C ( X ). Hence T = lim n T n ι = lim n m n X r =1 ζ nr ⊙ µ nr + e ⊙ k e µ = X n,r ζ nr ⊙ µ nr + e ⊙ µ with { ζ nr } ⊆ H eh and { µ nr } ⊆ I µ ( X ) + . Put q = P n,r k ζ nr k µ nr . Using (4.5), we derive that k q k ≤ lim n m n X r =1 k ζ nr k k µ nr k = lim n m n X r =1 k ζ nr k µ ( X nr ) / µ ( X nr ) / ≤ lim n m n X r =1 k ζ nr k µ ( X nr ) ! / m n X r =1 µ ( X nr ) ! / ≤ lim n X f ∈ λ n k k f k ! / ≤ X f k k f k ! / = k T k k < ∞ , that is, q ∈ I µ ( X ) + . It follows that T = X n,r ( ζ nr + k ζ nr k e ) ⊙ µ nr + e ⊙ k e µ − e ⊙ q. But ( ζ nr + k ζ nr k e ) n,r ⊆ ball H eh + e = S ( c ) ⊆ c and P n,r ( ζ nr + k ζ nr k e ) ⊙ µ nr + e ⊙ k e µ is aseparable morphism. Whence T + e ⊙ q is separable for some q ∈ I µ ( X ) + . (cid:3) Theorem 4.4.
Let T : C ( X ) → ( H, e ) be a unital positive mapping with its H -support k on X .Then T is a nuclear morphism iff k is of type . Moreover, T is separable iff k is a maximal H -support on X . Thus there is a natural bijection between nuclear morphisms T : C ( X ) → ( H, e ) and H -supports k on X of type . In this case, separable morphisms correspond to the maximalsupports.Proof. Let T : C ( X ) → ( H, e ) be a unital positive mapping with its H -support k ⊆ ball L ∞ ( X, µ )on X . If T is a nuclear morphism then its bounded linear extension T k : L ( X, µ ) → ( H, e ) is aHilbert-Schmidt operator by virtue of Theorem 4.2. In particular, k is of type 2.Conversely, suppose k is an H -support on X of type 2. By Theorem 4.1, T k : L ( X, µ ) → ( H, e )is a unital positive mapping of the Hilbert spaces. Moreover, k turns out to be a Hilbert-Schmidt H -support in L ( X, µ ). Notice that k e = u automatically. By Theorem 4.3, T k ι + e ⊙ q is separablefor some q ∈ I µ ( X ) + . But T k ι = T is a unital positive mapping by assumption. By Lemma 4.2, T is a nuclear morphism.Further, the H -support k on X is maximal whenever T is separable thanks to Theorem4.2. Conversely, suppose P f = e k f ≤ k e . Using (4.4) from the proof of Theorem 4.3, we have EPARABLE MORPHISMS OF OPERATOR HILBERT SYSTEMS 33 P m n r =1 k ζ nr k χ nr ≤ k e = 1, which in turn implies that P m n r =1 k ζ nr k µ nr ≤ k e µ for all n . Then q = ∨ { P m n r =1 k ζ nr k µ nr } ≤ k e µ or k e µ − q ≥
0. Hence T = P n,r ( ζ nr + k ζ nr k e ) ⊙ µ nr + e ⊙ ( k e µ − q )turns out to be a separable morphism. (cid:3) Example 4.2.
Consider the Hilbert space H = ℓ with its canonical (hermitian) basis F = { f n : n ≥ } and put e = f . The cone c consists of those hermitians ζ ∈ ℓ such that k ζ k ≤√ ζ , e ) . As in Example 4.1, we equip the compact interval X = [ − , ⊆ R with Lebesgue’smeasure − dt . Put k n = k f n = n − sin ( nπt ) , n ≥ , and k = k e = 1 . The family k = { k n } is an ℓ -support on [ − , . Indeed, we know that k n ⊥ k , n ≥ , and X n> ( v, k n ) = X n> n − (cid:18)Z v ( t ) sin ( nπt ) 2 − dt (cid:19) ≤ X n> n − ! (cid:18)Z v ( t ) 2 − dt (cid:19) ≤ ( v, k ) in L [ − , for all v ∈ C [ − , + . Thus T : C [ − , → ℓ , T v = P n ≥ ( v, k n ) f n is a unitalpositive mapping. Actually, it is a separable morphism. Indeed, based on Theorem 4.3, it sufficesto prove that the support k is maximal, which can easily be detected X n> k n = X n> n − sin ( nπt ) ≤ X n> n − ≤ k e , In particular, T : L [ − , → ℓ , T = P n ≥ f n ⊙ k n is a Hilbert-Schmidt operator and k T k = P n ≥ k k n k = P n ≥ n − ≤ π / . Corollary 4.3.
Let T : C ( X ) → ( H, e ) be a unital positive mapping with its H -support k ⊆ ball L ∞ ( X, µ ) on X . If µ is an atomic measure on X of finite support then T is a separablemorphism. In particular, a unital positive mapping T : ℓ ∞ ( n ) → ( H, e ) defines a morphism T : (cid:0) ℓ ∞ ( n ) , min ℓ ∞ ( n ) + (cid:1) → ( H, max c ) of the relevant operator systems.Proof. Let S ⊆ X be a finite subset and let { c t : t ∈ S } be a family of positive real numbers with P t ∈ S c t = 1. By assumption, µ = P t ∈ S c t δ t ∈ P ( X ) is an atomic measure with the supportsupp ( µ ) = S . Using Proposition 4.5, we deduce that ( T v, e ) = h v, µ i = P t ∈ S v ( t ) c t for all v ∈ C ( X ). Moreover, ( T v, f ) = P t ∈ S v ( t ) k f ( t ) c t = h v, k f µ i for k f ∈ L ∞ ( X, µ ), k f ⊥ L ( X, µ ) (or P t ∈ S k f ( t ) c t = 0) for all f = e . Since k is an H -support on X , we obtain that X f = e X t ∈ S v ( t ) k f ( t ) c t ! ≤ X t ∈ S v ( t ) c t ! for all v ∈ C ( X ) + . Fix s ∈ S and choose its neighborhood U such that U ∩ S = { s } . Take v ∈ C ( X ) h , 0 ≤ v ≤ v ) ⊆ U and v ( s ) = 1. Then h v, µ i = P t ∈ S ∩ U v ( t ) c t = c s , h v, k f µ i = k f ( s ) c s and P f = e k f ( s ) c s = P f = e h v, k f µ i ≤ h v, µ i = c s . Thus P f = e k f ≤ L ∞ ( X, µ ), which meansthat k is a maximal H -support on X . Using Theorem 4.3, we conclude that T is a separablemorphism.Finally, if X = { , , . . . , n } is a finite set then ℓ ∞ ( X ) = C ( X ) and P ( X ) consists ofatomic measures with their finite supports. Therefore the support of every unital positive map-ping T : ℓ ∞ ( X ) → ( H, e ) is maximal. It follows that T is separable. In particular, T : (cid:0) ℓ ∞ ( X ) , min ℓ ∞ ( X ) + (cid:1) → ( H, max c ) is a morphism of the operator systems. (cid:3) Paulsen-Todorov-Tomforde problem.
Fix two basis elements u and e from a hermitianbasis F for H , and consider the related unital cones c u and c e in H , respectively. Thus we have theunital spaces ( H, u ) and (
H, e ), respectively. Since F is a basis for H , the correspondence T ( u ) = e , T ( e ) = u , T ( f ) = f , f = e, u is uniquely extended up to a unitary operator T ∈ B ( H ) such that T ζ = ( ζ , e ) u + ( ζ , u ) e + P f = u,e ( ζ , f ) f . Note that T = T k for the H -support k = { k f : f ∈ F } with k e = u , k u = e and k f = f for all f = u, e . Notice that for every ζ ∈ H uh we have P f = e ( ζ , k f ) = P f = u ( ζ , f ) ≤ k ζ k = (( ζ , k e ) + k ζ k ) , which means that k is a unital H -support in ( H, u ) (see Remark 4.1). Moreover,(
T ζ , η ) = ( ζ , e ) ( η, u ) ∗ + ( ζ , u ) ( η, e ) ∗ + X f = u,e ( ζ , f ) ( η, f ) ∗ = ( ζ , ( η, e ) u ) + ( ζ , ( η, u ) e ) + X f = u,e ( ζ , ( η, f ) f ) = ( ζ , T η )for all ζ , η ∈ H , which means that T ∗ = T = T − . In particular, h T ζ , η i = ( T ζ , η ) = ( ζ , T η ) = (cid:10) ζ , T η (cid:11) , which means that T ∈ B (cid:0) H (cid:1) , T ( η ) = T η is the dual mapping to T . Note also that T : ( H, u ) → ( H, e ) is a unital ∗ -linear mapping of unital spaces. Indeed, T ζ ∗ = ( ζ ∗ , e ) u +( ζ ∗ , u ) e + P f = u,e ( ζ ∗ , f ) f = ( ζ , e ) ∗ u + ( ζ , u ) ∗ e + P f = u,e ( ζ , f ) ∗ f = ( T ζ ) ∗ . Notice that F is ahermitian basis for H . Lemma 4.3.
For the cones c u and c e we have T ( c u ) = c e and T ( S ( c e )) = S ( c u ) . Similarly, T ( c u ) = c e and T ( S ( c e )) = S ( c u ) . In particular, T ( ∞ ) (min c u ) = min c e and T ( ∞ ) (max c u ) =max c e .Proof. Take ζ ∈ c u with ζ = ζ + ( ζ , u ) u , k ζ k ≤ ( ζ , u ), where ζ ∈ H uh . But ζ = ( ζ , e ) e + ζ ′ , ζ ′ ∈ H uh ∩ H eh , therefore T ζ = ζ ′ + ( ζ , e ) u + ( ζ , u ) e and k ζ ′ + ( ζ , e ) u k = k ζ ′ k + | ( ζ , e ) | = k ζ k ≤ ( ζ , u ) = ( T ζ , e ) . The latter means that T ζ ∈ c e , that is, T ( c u ) ⊆ c e . If ( ζ , u ) = 1then ( T ζ , e ) = 1 as well, which means that T ( S ( c u )) = T (ball H uh + u ) ⊆ ball H eh + e = S ( c e )(see Lemma 3.2). By symmetry, T ( c e ) ⊆ c u and T ( S ( c e )) ⊆ S ( c u ), therefore T ( c u ) = c e and T ( S ( c u )) = S ( c e ). Similarly, T ( c u ) = c e , T ( S ( c e )) = S ( c u ), and T ( S ( c e )) = S ( c u ).Finally, the equality T ( S ( c e )) = S ( c u ) implies that T ( ∞ ) (min c u ) = T ( ∞ ) (cid:16) S ( c u ) ⊡ (cid:17) ⊆ S ( c e ) ⊡ =min c e due to Lemma 2.1. But T ( S ( c u )) = S ( c e ) as well, thereby T ( ∞ ) (min c e ) ⊆ min c u . Hence T ( ∞ ) (min c u ) = min c e . In particular, T ( ∞ ) (min c e ) = min c u . Using again Lemma 2.1 and Theo-rem 3.1, we obtain that T ( ∞ ) (max c u ) = T ( ∞ ) (cid:16) (min c u ) ⊡ (cid:17) ⊆ (min c e ) ⊡ = max c e . By symmetry, T ( ∞ ) (max c e ) ⊆ max c u . Whence T ( ∞ ) (max c u ) = max c e . (cid:3) Thus T : ( H, max c u ) → ( H, max c e ) is a matrix positive mapping. Actually it is an isomorphismof the operator systems. Theorem 4.5.
Let H be an infinite dimensional Hilbert space and let T ∈ B ( H ) be a unitary givenby T = u ⊙ e + e ⊙ u + P f = u,e f ⊙ f . The matrix positive mapping T : ( H, max c u ) → ( H, max c e ) given by T is not separable.Proof. Suppose that T is separable, that is, T = P l p l ⊙ q l for some c u -positive functionals q l on( H, u ) and c e -positive elements p l from ( H, e ). By Corollary 3.2, q l = η l + s l u , η l ∈ H uh , k η l k ≤ s l ,and p l = ζ l + r l e , ζ l ∈ H eh , k ζ l k ≤ r l . Then T ζ = X l (( ζ , η l ) + s l ( ζ , u )) ( ζ l + r l e )= X l (( ζ , η l ) + s l ( ζ , u )) ζ l + X l r l (( ζ , η l ) + s l ( ζ , u )) e for all ζ ∈ H . In particular, T e = P l ( η l , e ) ζ l + P l r l ( η l , e ) e = u and P l ( η l , e ) ζ l ∈ H eh implythat P l r l ( η l , e ) = 0 and P l ( η l , e ) ζ l = u . Similarly, T u = P l s l ζ l + P l r l s l e = e implies that EPARABLE MORPHISMS OF OPERATOR HILBERT SYSTEMS 35 P l s l ζ l = 0 and P l r l s l = 1. Put ϕ ( ζ ) = P l r l ( ζ , η l ), ζ ∈ H . Then | ϕ ( ζ ) | ≤ P l r l | ( ζ , η l ) | ≤k ζ k P l r l s l = k ζ k , that is, ϕ ∈ ball H ∗ and T ζ = G ( ζ ) + ( ϕ ( ζ ) + ( ζ , u )) e, where G ( ζ ) = P l ( ζ , η l ) ζ l ∈ H eh for all ζ ∈ H . As we have seen above G ( e ) = u , G ( u ) = 0 and ϕ ( u ) = ϕ ( e ) = 0. Since T f = f for all f = u, e , we deduce that ϕ ( f ) = 0 and G ( f ) = f for all f = u, e . Thus ϕ ( F ) = { } , which means that ϕ = 0. Consequently, T = e ⊙ u + X l ζ l ⊙ η l with X l k ζ l k k η l k ≤ X l r l s l = 1 , which means that T is a nuclear operator. In particular, T is a compact operator. But im T contains an infinite dimensional closed subspace generated by F \ { u, e } , a contradiction. (cid:3) Thus a matrix positive mapping into max-quantization may not be separable (see [26]).4.7.
The operator Hilbert system ℓ (2) . In this subsection we analyze the 2-dimensionalcase of ℓ (2). Suppose that K = ℓ ( ǫ ) with an hermitian basis ǫ = ( ǫ , ǫ ) and unit u = ǫ .Thus c u consists of those η = η ǫ + η ǫ ∈ K h such that η ≥ | η | ≤ η . Moreover, S ( c u ) = ball K uh + u = { η ǫ + ǫ : | η | ≤ } , and ζ ∈ c u iff h ζ , S ( c u ) i ≥
0. We have the canonical ∗ -representation K → C ( S ( c u )), ζ b ζ , b ζ ( t ) = h ζ , t i , t ∈ S ( c u ) (see below Appendix Section 5).Notice that ζ ∈ c u iff b ζ ∈ C ( S ( c u )) + . Consider the algebra ℓ ∞ ( θ ) with a basis θ = ( θ , θ , θ ),and the unital linear embedding κ : K → ℓ ∞ ( θ ), κ ( η ) = κ ( η , η ) = ( η , η + η , η − η ). Notethat κ ( u ) = κ ( ǫ ) = (1 , , κ ( ǫ ) = (0 , , −
1) = θ − θ . Lemma 4.4.
Let ζ ∈ K h . Then ζ ∈ c u iff b ζ ( ǫ ) ≥ and b ζ ( ǫ ± ǫ ) ≥ . In particular, κ ( c u ) = ℓ ∞ ( θ ) + ∩ κ ( K ) , and every unital positive mapping T : K → V from K to an operatorsystem V admits a unital positive extension e T : ℓ ∞ ( θ ) → V , e T · κ = T .Proof. If ζ ∈ c u then b ζ ( ǫ ) ≥ b ζ ( ǫ ± ǫ ) ≥
0, for { ǫ , ǫ ± ǫ } ⊆ S ( c u ). Conversely, assumethat b ζ ( ǫ ) ≥ b ζ ( ǫ ± ǫ ) ≥
0. Then (cid:12)(cid:12)(cid:12)b ζ ( ǫ ) (cid:12)(cid:12)(cid:12) ≤ b ζ ( ǫ ). Take t = rǫ + ǫ ∈ S ( c u ) with | r | ≤ (cid:12)(cid:12)(cid:12)b ζ ( rǫ ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) r b ζ ( ǫ ) (cid:12)(cid:12)(cid:12) ≤ b ζ ( ǫ ), which in turn implies that b ζ ( t ) = b ζ ( ǫ + rǫ ) ≥
0. Hence b ζ ∈ C ( S ( c u )) + or ζ ∈ c u .In particular, ζ ∈ c u iff κ ( ζ ) (1) = ζ = b ζ ( ǫ ) ≥ κ ( ζ ) (2) = ζ + ζ = b ζ ( ǫ + ǫ ) ≥ κ ( ζ ) (3) = ζ − ζ = b ζ ( ǫ − ǫ ) ≥
0, that is, κ ( ζ ) ∈ ℓ ∞ ( θ ) + .Finally, suppose T : K → V is a unital positive mapping into an operator system V . Since − u ≤ ǫ ≤ u in K h , it follows that − e ≤ T ( ǫ ) ≤ e , where e = T ( u ) is the unit of V . Thus k T ( ǫ ) k ≤
1. Choose v i ∈ V + , i = 1 , v + 2 v = e + T ( ǫ ), T ( ǫ ) ≤ v . For example,one can choose v = 0 and v = 2 − ( e + T ( ǫ )), for T ( ǫ ) ≤ e implies that e + T ( ǫ ) ≥ T ( ǫ ) or v ≥ T ( ǫ ). Define e T : ℓ ∞ ( θ ) → V to be e T ( λ θ + λ θ + λ θ ) = λ v + ( λ + λ ) v − T ( λ ǫ ).Then e T κ ( η ) = e T ( η θ + ( η + η ) θ + ( η − η ) θ ) = η v + (2 η ) v − T (( η − η ) ǫ )= η ( v + 2 v ) − ( η − η ) T ( ǫ ) = η ( e + T ( ǫ )) − ( η − η ) T ( ǫ )= η e + η T ( ǫ ) = η T ( ǫ ) + η T ( ǫ ) = T η, that is, e T κ = T . In particular, e T (1) = e T ( κ (1 , T ( ǫ ) = T ( u ) = e . Note also that e T ( θ ) = v , e T ( θ ) = v and e T ( θ ) = v − T ( ǫ ), that is, e T ( θ i ) ∈ V + for all i . Consequently, e T (cid:0) ℓ ∞ ( θ ) + (cid:1) ⊆ V + , which means that e T is a unital positive extension of T . (cid:3) Now let H be an operator Hilbert space with its Hermitian basis F and a unit e ∈ F . If T : K → H is a unital positive mapping then it admits a unital positive extension e T : ℓ ∞ ( θ ) → H , e T · κ = T thanks to Lemma 4.4. Proposition 4.6.
Let K = ℓ ( ǫ ) be the operator Hilbert system with its hermitian basis ǫ = ( ǫ , ǫ ) and unit u = ǫ , ( H, e ) an operator Hilbert system, and let T : ( K, u ) → ( H, e ) be a unital positivemapping. Then T is a separable morphism automatically. In particular, T : ( K, min c u ) → ( H, max c e ) is a morphism of the operator systems.Proof. Based on Lemma 4.4, there is a unital positive extension e T : ℓ ∞ ( θ ) → H of T . UsingCorollary 4.3, we deduce that e T is a separable morphism. Since κ : K → ℓ ∞ ( θ ) is a unitalpositive mapping (see Lemma 4.4), it follows that T = e T · κ is a separable morphism either. (cid:3) Remark 4.6.
Optionally, one can use the following argument. Since e T is separable, we have e T ( ∞ ) (cid:0) min ℓ ∞ ( θ ) + (cid:1) ⊆ c ce ⊆ max c e . By Lemma 4.4, κ ( c u ) = ℓ ∞ ( θ ) + ∩ κ ( K ) , which in turnimplies that κ ∗ (cid:0) S (cid:0) ℓ ∞ ( θ ) + (cid:1)(cid:1) ⊆ S ( c u ) . Using Lemma 2.1 and Proposition 4.2, we derive that κ ( ∞ ) (min c u ) = κ ( ∞ ) (cid:16) S ( c u ) ⊡ (cid:17) ⊆ S (cid:0) ℓ ∞ ( θ ) + (cid:1) ⊡ = min ℓ ∞ ( θ ) + . Consequently, T ( ∞ ) (min c u ) = e T ( ∞ ) κ ( ∞ ) (min c u ) ⊆ e T ( ∞ ) (cid:0) min ℓ ∞ ( θ ) + (cid:1) ⊆ max c e , which means that T : ( K, min c u ) → ( H, max c e ) is matrix positive. Corollary 4.4.
The operator Hilbert system ℓ (2) admits only one quantization, that is, min c u =max c u .Proof. Put H = ℓ (2) and T = id. Using Proposition 4.6, we derive that min c u = T ( ∞ ) (min c u ) ⊆ max c u ⊆ min c u , that is, min c u = max c u . (cid:3) Corollary 4.5.
Every unital positive mapping T : C ( X ) → ℓ (2) is a separable morphism auto-matically.Proof. Based on Proposition 4.6, we conclude that the support k of T is given by Borel functions k and k from ball L ∞ ( X, µ ) such that k = 1 and | k | ≤ k ≤ k is a maximalsupport on X . By Theorem 4.3, T is a separable morphism. (cid:3) The operator Hilbert system HS n . Now consider the Hilbert space HS n of all Hilbert-Schmidt operators on ℓ ( n ). Thus HS n = M n equipped with the inner product ( x, y ) τ = τ ( xy ∗ ), x, y ∈ M n , where τ is the normalized trace on M n . In this case, k x k = ( x, x ) / τ = τ (cid:0) | x | (cid:1) / isthe Hilbert-Schmidt norm, k x k ≤ k x k ≤ √ n k x k , x ∈ M n , and τ ( e ) = 1. Moreover, ( HS n ) eh = { x ∈ ( M n ) h : τ ( x ) = ( x , e ) τ = 0 } = ( HS n ) h ∩ ker τ and every hermitian x admits a uniqueorthogonal expansion x = x + τ ( x ) e . In particular, HS + n = { x ∈ ( HS n ) h : x = x + τ ( x ) e, k x k ≤ τ ( x ) } is the unital, separated cone of the operator Hilbert system HS n called an operator Hilbert-Schmidtsystem. Proposition 4.7.
The equality M + n = HS + n holds for n < .Proof. Since the equality is trivial for n = 1 we need just to look at the case of n = 2. Take x ∈ M +2 .Then τ ( x ) ≥ x = x + τ ( x ) e with x ∈ ( HS ) eh . If λ and λ are (real) eigenvalues of x ,then λ + λ = 2 τ ( x ) = 0, that is, λ ≥ ≥ λ = − λ . But x ≥ − τ ( x ) e , thereby λ + τ ( x ) ≥ λ ≤ τ ( x ). It follows that − τ ( x ) e ≤ x ≤ τ ( x ) e and k x k ≤ k x k ≤ τ ( x ). The latter meansthat x ∈ HS +2 . Thus M +2 ⊆ HS +2 . EPARABLE MORPHISMS OF OPERATOR HILBERT SYSTEMS 37
Conversely, take x ∈ HS +2 with its expansion x = x + τ ( x ) e , x ∈ ( HS ) eh . Since τ ( x ) = 0,we obtain that x = (cid:20) a bb ∗ − a (cid:21) with a ∈ R and b ∈ C .Note that k x k = k x k / = q a + | b | and k x k = τ ( x ) / = (cid:0) − (cid:0) a + 2 | b | (cid:1)(cid:1) / = q a + | b | = k x k . It follows that x ∈ ( M n ) h and k x k ≤ τ ( x ), which means that − τ ( x ) e ≤ x ≤ τ ( x ) e . The latter in turn implies that x = x + τ ( x ) e ≥
0, that is, x ∈ M +2 . Whence M +2 = HS +2 . (cid:3) Remark 4.7.
The equality M + n = HS + n fails to be true for n ≥ . For example, take x = (cid:16) / √ (cid:17) − ∈ ( HS ) eh . Then − p / , and p / are eigenvalues of x and k x k = 1 < p / k x k . It follows that x = x + e ∈ HS +3 , whereas x / ∈ M +3 , for x admits a negative eigenvalue − p / . Operator Hilbert systems and entanglement breaking mappings.
Let V be an op-erator system and let C be a quantization of its cone V + of positive elements. For every n thequantum cone C defines a unital, closed, separated cone C ∩ M n ( V ) in M n ( V ), whose state spacein M n ( V ∗ ) is denoted by S n ( C ). Thus S n ( C ) = S ( C ∩ M n ( V )). These state spaces S n ( C ) in turndefine the state space S ( C ) of C on V ∗ .Now let W be another operator system with its unit e ′ and a quantization K of W + . Thus( V , C ) and ( W , K ) are quantum systems. Consider a matrix (or completely) positive mapping ϕ :( V , C ) → ( W , K ), that is, ϕ ( ∞ ) ( C ) ⊆ K . In particular, ϕ ( V + ) = ϕ ( C ∩V ) ⊆ K ∩W = W + . Hence ϕ is positive. It is well known [16, 5.1.1] that ϕ is completely bounded. Moreover, ( ϕ ∗ ) ( ∞ ) ( S ( K )) ⊆ R + S ( C ), where R + S ( C ) indicates to the quantum set of all positive functionals on the matrixspaces. Indeed, D C ∩ M n ( V ) , ( ϕ ∗ ) ( n ) ( s ) E = (cid:10) ϕ ( n ) ( C ∩ M n ( V )) , s (cid:11) ⊆ h K ∩ M n ( W ) , s i ≥ s ∈ S n ( K ). If ϕ is unital (that is, ϕ ( e ) = e ′ ) then D e ⊕ n , ( ϕ ∗ ) ( n ) ( s ) E = (cid:10) ϕ ( n ) ( e ⊕ n ) , s (cid:11) = h e ′⊕ n , s i = 1, which means ( ϕ ∗ ) ( n ) ( s ) ∈ S n ( C ) for every s ∈ S n ( K ). Thus ( ϕ ∗ ) ( ∞ ) ( S ( K )) ⊆ S ( C )whenever ϕ is a morphism.A linear mapping ϕ : V → W of operator systems is called an entanglement breaking if( ϕ ∗ ) ( ∞ ) (cid:0) S (cid:0) M ( W ) + (cid:1)(cid:1) ⊆ R + S ( C ) for every quantization C of V + , where ϕ ∗ indicates to thealgebraic dual mapping to ϕ . An entanglement breaking mapping ϕ : V → W is bounded au-tomatically. Moreover, if ϕ : V → ( W , max W + ) is an entanglement breaking mapping then sois ϕ : V → W . Indeed, by its very definition, ( ϕ ∗ ) ( ∞ ) ( S (max W + )) ⊆ R + S ( C ) for every quanti-zation C of V + . But max W + ⊆ M ( W ) + , therefore S (cid:0) M ( W ) + (cid:1) ⊆ S (max W + ). In particular,( ϕ ∗ ) ( ∞ ) (cid:0) S (cid:0) M ( W ) + (cid:1)(cid:1) ⊆ R + S ( C ) for every quantization C of V + , which means that ϕ : V → W is an entanglement breaking mapping.Now assume that H is an operator Hilbert system with the unit e and the related unital cone H + (the notation instead of c e ). In this case, H ∗ = H is an operator system with the unit e andthe cone H + . Proposition 4.8.
Let M be either a finite-dimensional von Neumann algebra or another operatorHilbert system, and let ϕ : H → M be a linear mapping. Then ϕ is an entanglement breaking map-ping iff ϕ ∗ : M ∗ → (cid:0) H , max H + (cid:1) is matrix positive. Similarly, ϕ ∗ : H → M ∗ is an entanglementbreaking mapping iff ϕ : M → ( H , max H + ) is matrix positive.Proof. For brevity we assume that M is a finite-dimensional von Neumann algebra. The case ofan operator Hilbert system M can be proved in a very similar way. It is known (see [26]) that ϕ is an entanglement breaking mapping iff ϕ : ( H , min H + ) → M is matrix positive, that is, ϕ ( ∞ ) (min H + ) ⊆ M ( M ) + . Using Lemma 2.1 and Theorem 3.1, we have( ϕ ∗ ) ( ∞ ) (cid:0) T ( M ∗ ) + (cid:1) = ( ϕ ∗ ) ( ∞ ) (cid:16) M ( M ) ⊡ + (cid:17) ⊆ (min H + ) ⊡ = max H + , that is, ϕ ∗ : M ∗ → (cid:0) H , max H + (cid:1) is matrix positive. Conversely, if the latter mapping is matrixpositive then ϕ ( ∞ ) (cid:16)(cid:0) max H + (cid:1) ⊡ (cid:17) ⊆ M ( M ∗ ) ⊡ + thanks to Lemma 2.1. But (cid:0) max H + (cid:1) ⊡ = min H + and M ( M ∗ ) ⊡ + = M ( M ) + again by Theorem 3.1. Hence ϕ ( ∞ ) (min H + ) ⊆ M ( M ) + , which meansthat ϕ : ( H , min H + ) → M is matrix positive.Finally, ϕ ∗ : (cid:0) H , min H + (cid:1) → M ∗ is matrix positive iff ϕ : M → ( H , max H + ) is matrix positivethanks to Lemma 2.1 and Theorem 3.1. (cid:3) Appnedix: The unital measures on the state space
In this section we analyse so called measured state space of an ordered Hilbert space H togenerate positive L -representations of H .5.1. The canonical ∗ -representation H → C ( S ( c )) . Put X = S ( c ) equipped with the weak ∗ topology σ (cid:0) H, H (cid:1) . Thus X is a compact Hausdorff topological space and there is a canonical ∗ -representation κ : H → C ( X ), κ ( ζ ) = ζ ( · ), where ζ ( t ) = h ζ , t i , t ∈ X . If ζ ( · ) = 0 then h ζ , c i = h ζ , R + S ( c ) i = R + h ζ , X i = { } by Corollary 3.2, and (cid:10) ζ , H h (cid:11) = h ζ , c i − h ζ , c i = { } ,which in turn implies that (cid:10) ζ , H (cid:11) = (cid:10) ζ , H h (cid:11) + i (cid:10) ζ , H h (cid:11) = { } , that is, ζ = 0. Moreover, ζ ∈ c iff ζ ( · ) ∈ C ( X ) + (see Remark 3.2), and κ ( e ) = e ( · ) = 1. Thus the unital ∗ -representation κ : H → C ( X ) is an order isomorphism onto its range, which means that H is realized asan operator system in C ( X ), and M ( H ) ∩ M ( C ( X )) + = M ( H ) ∩ min C ( X ) + = min c (seeProposition 4.2, and [26, Theorem 3.2]). Further, k ζ ( · ) k ∞ = sup | ζ ( X ) | = sup |h ζ , S ( c ) i| = k ζ k e for all ζ ∈ H . By Proposition 3.1, H turns out to be a complete subspace with respect to theuniform norm of C ( X ), or H is a norm-closed operator system in C ( X ). Therefore κ ∗ is an exactquotient mapping κ ∗ : M ( X ) → H, κ ∗ ( µ ) = µ · κ, ball H = κ ∗ (ball M ( X )) , ker ( κ ∗ ) = H ⊥ , where H ⊥ is the polar of H in C ( X ) ∗ . Note that h ζ , κ ∗ ( µ ∗ ) i = h κ ( ζ ) , µ ∗ i = h ζ ( · ) ∗ , µ i ∗ = h ζ ∗ ( · ) , µ i ∗ = h κ ( ζ ∗ ) , µ i ∗ = h ζ ∗ , κ ∗ ( µ ) i ∗ = h ζ , κ ∗ ( µ ) ∗ i for all ζ ∈ H , which means that κ ∗ is ∗ -linear. If µ ∈ M ( X ) + then h ζ , κ ∗ ( µ ) i = h ζ ( · ) , µ i ≥ ζ ∈ c , which in turn implies that κ ∗ ( µ ) ∈ c thanks to Corollary 3.2. Thus κ ∗ is a positivemapping in the sense of κ ∗ (cid:0) M ( X ) + (cid:1) ⊆ c . If µ ∈ P ( X ) then κ ∗ ( µ ) ∈ S ( c ), that is, κ ∗ ( µ ) = s forsome s ∈ X . We skip the bars in s for the elements of X for brevity, and we write s = s + e ∈ X (instead of s = s + e ) with uniquely defined s ∈ ball H eh .The closed subspace in M ( X ) of all atomic measures on X is denoted by ℓ ( X ). Take µ = P t ∈ S c t δ t ∈ ℓ ( X ) with P t ∈ S | c t | = k µ k < ∞ and a subset S ⊆ X . Since P t ∈ S k c t t k ≤ √ k µ k < EPARABLE MORPHISMS OF OPERATOR HILBERT SYSTEMS 39 ∞ , it follows that η = P t ∈ S c t t defines an element of H with k η k ≤ √ k µ k and h ζ , κ ∗ ( µ ) i = h ζ ( · ) , µ i = X t ∈ S c t h ζ ( · ) , δ t i = X t ∈ S c t ζ ( t ) = X t ∈ S c t h ζ , t i = h ζ , η i for all ζ ∈ H . Hence(5.1) κ ∗ X t ∈ S c t δ t ! = X t ∈ S c t t, which means that κ ∗ ( ℓ ( X )) = H . Lemma 5.1.
For each µ ∈ M ( X ) there are points s, t ∈ X , c s , c t , c e ∈ C and ν ∈ H ⊥ such that µ = c s δ s + c t δ t + c e δ e + ν. If µ ∈ M ( X ) h then µ = c s δ s + c e δ e + ν for some s ∈ X , c s , c e ∈ R and ν ∈ H ⊥ ∩ M ( X ) h . If µ ∈ P ( X ) then µ = δ s + ν for some s ∈ X and ν ∈ H ⊥ ∩ M ( X ) h .Proof. Put η = κ ∗ ( µ ). If η ∈ H h = R + c − R + c then η = c s s − c t t for some c s , c t ∈ R + and s, t ∈ S ( c ). It follows that η = κ ∗ ( c s δ s − c t δ t ) thanks to (5.1). Actually, η = η + re = k η k (cid:0) k η k − η + e (cid:1) + ( r − k η k ) e = κ ∗ ( k η k δ s + ( r − k η k ) δ e ) , where s represents the point k η k − η + e from X . In the case of any η ∈ H we have η = Re η + i Im η = κ ∗ ( c s δ s + c t δ t + c e δ e )for some s, t ∈ X and c s , c t , c e ∈ C . Thus µ = c s δ s + c t δ t + c e δ e + ν for some ν ∈ H ⊥ . In the caseof µ ∈ P ( X ) we have κ ∗ ( µ ) = s ∈ X and µ = δ s + ν with ν ∈ H ⊥ . But µ, δ s ∈ M ( X ) h , therefore ν ∈ H ⊥ ∩ M ( X ) h . (cid:3) We say that µ is a unital measure on X if µ ∈ P ( X ) and κ ∗ ( µ ) = e . By Lemma 5.1, µ isunital iff µ = δ e + ν for some ν ∈ H ⊥ ∩ M ( X ) h . In particular, δ e is a unital measure. An atomicprobability measure µ = P t ∈ S c t δ t ∈ P ( X ) with c t ≥ P t c t = 1 is unital iff P t ∈ S c t t = 0thanks to (5.1), where t ∈ ball H eh with t = t + e ∈ X . Example 5.1. If F ⊆ ball H eh is a finite subset whose convex hull contains the origin, then S = F + e ⊆ X is a finite subset and µ = P t ∈ S c t δ t is a unital measure on X with the finitesupport, where c t ≥ , P t ∈ S c t = 1 and P t ∈ F c t t = 0 in H eh . Notice that X is identified with the subset δ X = { δ t : t ∈ X } ⊆ ℓ ( X ) along with the weak ∗ continuous mapping X → M ( X ), t δ t . Thus δ X is a w ∗ -compact subset of ℓ ( X ) being ahomeomorphic copy of X . Further, the mapping κ ∗ : M ( X ) → H implements a bijection of δ X onto X , for the equality δ s = δ t over H implies that s and t are the same states of the cone c , thatis, s = t in X . Since |h ζ , t i| = |h ζ ( · ) , δ t i| for all ζ ∈ H and t ∈ X , it follows that κ ∗ | δ X : δ X → X is a weak ∗ continuous mapping of compact spaces. Thus κ ∗ | δ X is a homeomorphic inverse ofthe mapping X → δ X , t δ t . Put e X = ( κ ∗ ) − ( X ) to be a w ∗ -closed subset of M ( X ), whichcontains δ X . We say that e X is the measured state space of the cone c , and we also use the notation e S ( c ) instead of e X . Taking into account that κ ∗ is a ∗ -linear mapping, we conclude that e X is aself-adjoint subset of M ( X ) in the sense of e X ∗ = e X , and e X ∩ H ⊥ = ∅ . Thus e S ( c ) is a w ∗ -closed,convex, ∗ -subset of M ( X ) disjoint with H ⊥ . Corollary 5.1.
The measured state space e S ( c ) of the unital cone c is the disjoint union of all δ s + H ⊥ , that is, e S ( c ) = W (cid:8) δ s + H ⊥ : s ∈ S ( c ) (cid:9) . In particular, P ( X ) = e X ∩ M ( X ) + . Proof. If δ s − δ t ∈ H ⊥ for some s, t ∈ X then s = t as we have just confirmed above. Therefore theunion ∪ (cid:8) δ s + H ⊥ : s ∈ S ( c ) (cid:9) is a disjoint union which is e S ( c ). Moreover, P ( X ) ⊆ e X ∩ M ( X ) + thanks to Lemma 5.1. Conversely, if µ = δ s + ν ∈ M ( X ) + with ν ∈ H ⊥ , then R dµ = h e ( · ) , µ i = h e ( · ) , δ s i + h e ( · ) , ν i = e ( s ) = 1, for e ( · ) ∈ H and h e ( · ) , ν i = 0. Whence µ ∈ P ( X ). (cid:3) Finally, notice that Re µ ∈ e X whenever µ ∈ e X . Indeed, by Corollary 5.1, we have µ = δ s + ν forsome ν ∈ H ⊥ . Then Re µ = δ s + Re ν . But H ⊥ is a ∗ -subspace of M ( X ), therefore Re ν ∈ H ⊥ ,and Re µ ∈ e X . Note also that Im µ = Im ν being an element of H ⊥ stays out of e X . Similarly,in the general case the positive part µ + of a hermitian µ ∈ e X may stay out of e X . The set of allunital measures on X is denoted by U ( X ), that is, U ( X ) = P ( X ) ∩ (cid:0) δ e + H ⊥ (cid:1) is a convex subsetof P ( X ). Notice that U ( X ) = M ( X ) + ∩ (cid:0) δ e + H ⊥ (cid:1) = M ( X ) + ∩ (cid:0) δ e + H ⊥ ∩ M ( X ) h (cid:1) thanksto Corollary 5.1.5.2. The unital measures on X . Fix µ ∈ U ( X ). Put u = e ( · ), which is a unit of L ( X, µ )and it represents µ in M ( X ). Consider the related Hilbert space L ( X, µ ) (which is a subspaceof L ( X, µ ) out of compactness of X ) with its norm k·k , the unital cone L ( X, µ ) + with the unit u , and the unital ∗ -linear mapping ι : C ( X ) → L ( X, µ ). The latter in turn defines the followingbounded, unital ∗ -linear mapping ικ : H → L ( X, µ ) of Hilbert spaces. If η ∈ L ( X, µ ) then ι ∗ ( η ) ∈ C ( X ) ∗ = M ( X ), ηµ ∈ M ( X ) and h h, ι ∗ ( η ) i = h ι ( h ) , η i = ( ι ( h ) , η ) = Z h ( t ) η ∗ ( t ) dµ = h h, η ∗ µ i for all h ∈ C ( X ), where the inner product is taken in L ( X, µ ). Thus ι ∗ : L ( X, µ ) ∗ → M ( X ) isreduced to the canonical identification ι ∗ ( η ) = η ∗ µ . In particular, ( ικ ) ∗ ( η ) = κ ∗ ι ∗ ( η ) = κ ∗ ( η ∗ µ )for all η ∈ L ( X, µ ), which justifies to use a brief notation ι : H → L ( X, µ ) instead of ικ .In this case, ι ( e ) = u , and its dual is reduced to κ ∗ | L ( X, µ ) for the exact quotient mapping κ ∗ : M ( X ) → H considered above in Subsection 5.1.Now let ι ( H ) − be the closure of the subspace ι ( H ) in the Hilbert space L ( X, µ ), and let P ∈ B ( L ( X, µ )) be the orthogonal projection onto ι ( H ) − . Since L ( X, µ ) = ι ( H ) − ⊕ ι ( H ) ⊥ and ι is a ∗ -linear mapping, it follows that both ι ( H ) − and ι ( H ) ⊥ are ∗ -subspaces, and u ∈ ι ( H ) ⊆ im ( P ). In particular, P is a unital ∗ -linear mapping. If η ∈ ι ( H ) ⊥ for η ∈ L ( X, µ ),then ηµ ∈ M ( X ) and h ζ ( · ) , ηµ i = ( ζ ( · ) , η ) = ( ι ( ζ ) , η ) = 0 for all ζ ∈ H , which means that ηµ ∈ H ⊥ . Hence L ( X, µ ) ∩ H ⊥ = ι ( H ) ⊥ (up to the canonical identification). The orthogonalto u subspace of im ( P ) is denoted by im ( P ) u whereas im ( P ) uh denotes the closed real subspaceim ( P ) u ∩ L ( X, µ ) h . Lemma 5.2.
The unital ∗ -linear mapping ι : H → L ( X, µ ) is a Hilbert-Schmidt operator with k ι k ≤ √ and ι (ball H e ) ⊆ ball im ( P ) u , whose dual ι ∗ : L ( X, µ ) → H is a unital ∗ -linearmapping given by the following H -valued integral ι ∗ ( η ) = Z η ( t ) tdµ, η ∈ L ( X, µ ) . In particular, ι (ball H eh ) ⊆ ball im ( P ) uh , im ( P ) uh = ι ( H eh ) − , and R tdµ = e . EPARABLE MORPHISMS OF OPERATOR HILBERT SYSTEMS 41
Proof.
The fact that ι is a Hilbert-Schmidt operator follows from Remark 2.2. In the present case,if F is a (hermitian) Hilbert basis for H containing e , then ι = P f ι ( f ) ⊙ f with k ι k = X f k ι ( f ) k = X f = e Z | ( f, t ) | dµ + Z | ( e, t ) | dµ = Z X f = e | ( f, t ) | dµ + 1 ≤ Z k t k dµ + 1 ≤ , which means that k ι k ≤ √
2. Note that ι ∗ ( u ) = κ ∗ ( u ∗ µ ) = κ ∗ ( µ ) = e , that is, ι ∗ is a unitalmapping. Further, h ζ , ι ∗ ( η ) i = h ζ , κ ∗ ( η ∗ µ ) i = h ζ ( · ) , η ∗ µ i = ( ζ ( · ) , η ) = Z ( ζ , t ) η ∗ ( t ) dµ = Z ( ζ , η ( t ) t ) dµ = (cid:18) ζ , Z η ( t ) tdµ (cid:19) for all ζ ∈ H . It follows that ι ∗ ( η ) = R η ( t ) tdµ for all η ∈ L ( X, µ ). In particular, ι ∗ ( η ∗ ) = Z η ∗ ( t ) tdµ = (cid:18)Z η ( t ) tdµ (cid:19) ∗ = ι ∗ ( η ) ∗ for all η ∈ L ( X, µ ), which means that ι ∗ is a unital ∗ -linear mapping, and R tdµ = ι ∗ ( u ) = e .Now prove that ι (ball H e ) ⊆ ball im ( P ) u . Take ζ ∈ H e and a (hermitian) Hilbert basis F for H containing e . Since µ is unital, we conclude that( ι ( ζ ) , u ) = Z ζ ( t ) dµ = h ζ ( · ) , µ i = h ζ ( · ) , δ e i = ( ζ , e ) = 0 , that is, ι ( ζ ) ⊥ u . Thus ι ( H e ) ⊆ im ( P ) u and ι ( F \ { e } ) ⊆ im ( P ) uh . If ζ ∈ ball H e then ζ = P f = e ( ζ , f ) f and k ι ( ζ ) k ≤ X f = e | ( ζ , f ) | k ι ( f ) k ≤ X f = e | ( ζ , f ) | ! / X f = e k ι ( f ) k ! / ≤ X f = e | ( ζ , f ) | ! / (cid:18)Z k t k dµ (cid:19) / ≤ k ζ k ≤ , that is, ι ( ζ ) ∈ ball im ( P ) u . Since ι is ∗ -linear, we also deduce that ι (ball H eh ) ⊆ ball im ( P ) uh . Inparticular, ι ( H eh ) − ⊆ im ( P ) uh .Finally, prove that im ( P ) uh = ι ( H eh ) − . Take θ ∈ im ( P ) uh . Since ι ( H ) is dense in im ( P )and ι is a ∗ -linear mapping, it follows that θ = lim n ζ n ( · ) for a certain sequence ( ζ n ) n ⊆ H h .But ζ n ( · ) = ζ n, ( · ) + r n u with ζ n, ∈ H eh , r n ∈ R , and lim n r n = lim n ( ζ n, ( · ) , u ) + r n ( u, u ) =lim n ( ζ n ( · ) , u ) = ( θ, u ) = 0. Thereby θ = lim n ζ n, ( · ) ∈ ι ( H eh ) − . (cid:3) Remark 5.1.
Notice also that k ι ∗ ( η ) k ≤ R | η | k t k dµ ≤ √ k η k ≤ √ k η k for all η ∈ L ( X, µ ) . Put c µ = { ζ ∈ H h : k ζ ( · ) k ≤ ( ζ , e ) } , which is a cone in H . Lemma 5.3.
The cone c µ in H is a unital cone containing c , ι ( c µ ) ⊆ L ( X, µ ) + and im ( P ) ∩ L ( X, µ ) + = ι ( c µ ) − . In particular, ι : H → L ( X, µ ) is a unital positive mapping in the sense of ι ( e ) = u and ι ( c ) ⊆ L ( X, µ ) + , and P is a unital positive projection (a conditional expectation). Proof.
Take ζ = ζ + re ∈ c with ζ ∈ H eh and k ζ k ≤ r . Note that ι ( ζ ) = ι ( ζ )+ ru , ι ( ζ ) = ζ ( · ) ∈ im ( P ) uh and k ζ ( · ) k ≤ k ζ k ≤ r by virtue of Lemma 5.2, that is, ζ ∈ c µ . Thus c ⊆ c µ and c µ is aunital cone in H . Prove that ι ( c µ ) ⊆ L ( X, µ ) + . Take ζ = ζ + re ∈ c µ . By Lemma 5.2, ι ( ζ ) ⊥ u and k ζ ( · ) k ≤ ( ζ , e ) = h ζ ( · ) , δ e i = h ζ ( · ) , µ i = ( ζ ( · ) , u ), which means that ι ( ζ ) ∈ L ( X, µ ) + .Thus ι ( c ) ⊆ ι ( c µ ) ⊆ L ( X, µ ) + and ι is a unital positive mapping. If η = η + ru ∈ L ( X, µ ) + then P ( u ) = u , P ( η ) = P ( η ) + ru ∈ L ( X, µ ) h , ( P ( η ) , u ) = ( η , u ) = 0 in L ( X, µ ), and k P ( η ) k ≤ k η k ≤ r , which means that P is a unital positive projection.Finally prove that im ( P ) ∩ L ( X, µ ) + = ι ( c µ ) − . We saw above ι ( c µ ) ⊆ L ( X, µ ) + ∩ im ( P ),which results in ι ( c µ ) − ⊆ L ( X, µ ) + ∩ im ( P ). Take η = η + ru ∈ im ( P ) ∩ L ( X, µ ) + with k η k ≤ r . Prove that η ∈ ι ( c µ ) − . Since η = lim n (1 − n − ) η + ru , we can assume that k η k < r .By Lemma 5.2, η ∈ im ( P ) uh = ι ( H eh ) − , therefore η = lim n ζ ,n ( · ) + ru for some ( ζ ,n ) n ⊆ H eh .But lim n k ζ ,n ( · ) k = k η k < r , therefore we can assume that k ζ ,n ( · ) k < r for all n . Thus ζ n = ζ ,n + re ∈ c µ and η = lim n ζ n ( · ) = lim n ι ( ζ n ) ∈ ι ( c µ ) − . (cid:3) The description of the state space of the cone L ( X, µ ) + in terms of the measured state space e S ( c ) considered above in Subsection 5.1 is provided in the following assertion. Theorem 5.1. If µ ∈ U ( X ) then S (cid:0) L ( X, µ ) + (cid:1) = e X ∩ √ L ( X, µ ) h . In particular, ι ∗ (cid:0) L ( X, µ ) + (cid:1) ⊆ c , which means that ι ∗ : L ( X, µ ) → H is a unital positive mapping.Proof. By Lemma 5.3, ι : H → L ( X, µ ) is a unital positive mapping. This fact in turn impliesthat ι ∗ (cid:0) S (cid:0) L ( X, µ ) + (cid:1)(cid:1) ⊆ X . Let us show the details of this inclusion. Take a state η = η + u ∈ S (cid:0) L ( X, µ ) + (cid:1) of the unital cone L ( X, µ ) + , where η ∈ ball L ( X, µ ) uh . Notice that k η k = (cid:0) k η k + k u k (cid:1) / ≤ √
2. Further, ι ∗ ( η ) = κ ∗ ( ηµ ) = (( η + u ) µ ) | H , which means that h ζ , ι ∗ ( η ) i = h ζ ( · ) , ( η + u ) µ i = h ζ ( · ) , η µ i + h ζ ( · ) , µ i = h ζ ( · ) , η µ i + ( ζ , e )for all ζ ∈ H . But η µ | H is a hermitian linear functional on H such that k η µ | H k ≤ k η µ k = k η k = Z | η | dµ ≤ (cid:18)Z dµ (cid:19) / k η k ≤ . It follows that the functional η µ | H is given by a hermitian vector s ∈ H h such that h ζ ( · ) , η µ i =( ζ , s ). But ( e, s ) = h e ( · ) , η µ i = R η dµ = ( u, η ) = 0, that is, s ∈ ball H eh . Consequently, s η = s + e ∈ X and h ζ , ι ∗ ( η ) i = h ζ , s η i for all ζ ∈ H , which means that s η = ι ∗ ( η ) ∈ X . Inparticular, S (cid:0) L ( X, µ ) + (cid:1) ⊆ e X ∩ √ L ( X, µ ) h .Conversely, suppose ι ∗ ( η ) = s ∈ X for some η ∈ L ( X, µ ) h with k η k ≤ √
2. Prove that η ∈ S (cid:0) L ( X, µ ) + (cid:1) . Taking into account that ker ( ι ∗ ) = ι ( H ) ⊥ , we can also assume that η ∈ ι ( H ) − h .Then η = lim n ι ( ζ ,n ) + r n u for some ζ ,n ∈ H eh and r n ∈ R . By Lemma 5.2, { ι ( ζ ,n ) } ⊆ ι ( H eh ) ⊆ im ( P ) uh , and lim n r n = lim n ( ι ( ζ ,n ) + r n u, u ) = ( η, u ) = Z ηdµ = h e ( · ) , ηµ i = h ι ( e ) , ηµ i = h e, κ ∗ ( ηµ ) i = h e, ι ∗ ( η ) i = h e, s i = ( e, s ) = 1 . It follows that η = η + u with η = lim n ι ( ζ ,n ) ∈ ι ( H eh ) − = im ( P ) uh (see Lemma 5.2), and k η k + 1 = k η k ≤
2, which means that η ∈ ball L ( X, µ ) uh . Whence η = η + u ∈ S (cid:0) L ( X, µ ) + (cid:1) and s = s η . Hence S (cid:0) L ( X, µ ) + (cid:1) = e X ∩ √ L ( X, µ ) h . EPARABLE MORPHISMS OF OPERATOR HILBERT SYSTEMS 43
Finally, prove that ι ∗ : L ( X, µ ) → H (or ι ∗ : L ( X, µ ) → H ) is positive either. Since ι ∗ isunital and S (cid:0) L ( X, µ ) + (cid:1) ⊆ e X = ( ι ∗ ) − ( X ), it follows that ι ∗ (cid:16) L ( X, µ ) + (cid:17) = ι ∗ (cid:0) R + S (cid:0) L ( X, µ ) + (cid:1)(cid:1) ⊆ R + ι ∗ (cid:0) S (cid:0) L ( X, µ ) + (cid:1)(cid:1) ⊆ R + X = c or equivalently we have ι ∗ (cid:0) L ( X, µ ) + (cid:1) ⊆ c , which means that ι ∗ is positive. (cid:3) Remark 5.2.
As follows from the proof of Theorem 5.1, if ι ∗ ( η ) = s ∈ X for some η ∈ L ( X, µ ) h ,then η = η + u with η ∈ im ( P ) uh . In particular, ι ∗ ( η ) = s for s = s + e with s ∈ ball H eh . Recall that a point s ∈ X is called a µ -mass if µ ( s ) > Corollary 5.2.
Let µ ∈ U ( X ) and let s be a µ -mass in X with µ ( s ) ≥ / . Then s is given bya certain state η of L ( X, µ ) + , that is, s = ι ∗ ( η ) for η ∈ S (cid:0) L ( X, µ ) + (cid:1) .Proof. By Lemma 2.3, δ s = s ′ µ for s ′ = µ ( s ) − [ s ] ∈ L ( X, µ ) h . Then s = κ ∗ ( s ′ µ ), which meansthat s ′ µ ∈ e X . But k s ′ k = µ ( s ) − / ≤ √
2, that is, s ′ µ ∈ e X ∩ √ L ( X, µ ) h . By Theorem 5.1, s ′ µ ∈ S (cid:0) L ( X, µ ) + (cid:1) , and the result follows. (cid:3) In the case of any µ -mass s in X we have s ′ = µ ( s ) − [ s ] ∈ L ( X, µ ) h , ( s ′ , u ) = R s ′ dµ = 1, and s ′ = P s ′ +(1 − P ) s ′ = s ′ + u +(1 − P ) s ′ with s ′ ∈ im ( P ) uh . It follows that s = κ ∗ ( s ′ µ ) = ι ∗ ( s ′ ) = ι ∗ ( s ′ ) + ι ∗ ( u ) = ι ∗ ( s ′ ) + e (see Remark 5.2), which in turn implies that ι ∗ ( s ′ ) = s ∈ ball H eh . Corollary 5.3.
Let µ ∈ U ( X ) and let s be a µ -mass in X . Then µ ( s ) / k s k ≤ k s ( · ) k ≤ k s k ≤ k s ′ k ≤ (cid:0) µ ( s ) − − (cid:1) / . In particular, µ ( s ) ≤ (cid:0) k s k (cid:1) − , and µ ( s ) ≤ / whenever k s k = 1 .Proof. As in the proof of Corollary 5.2, we have µ ( s ) − = k s ′ k + 1 + k (1 − P ) s ′ k ≥ k s ′ k + 1,that is, k s ′ k ≤ (cid:0) µ ( s ) − − (cid:1) / . Further, notice that s ′ ∈ im ( P ) uh , s = ι ∗ ( s ′ ) ∈ H eh , and ι (ball H eh ) ⊆ ball im ( P ) uh thanks to Lemma 5.2. It follows that k s k = sup | (ball H eh , ι ∗ ( s ′ )) | = sup | ( ι (ball H eh ) , s ′ ) | ≤ sup | (ball im ( P ) uh , s ′ ) | = k s ′ k , that is, k s k ≤ k s ′ k . Using again Lemma 5.2, we deduce that µ ( s ) / k s k = (cid:0) µ ( s ) ( s , s ) (cid:1) / ≤ (cid:18)Z | ( s , t ) | dµ (cid:19) / = k s ( · ) k ≤ k s k . Hence µ ( s ) / k s k ≤ k s ( · ) k ≤ k s k ≤ k s ′ k ≤ (cid:0) µ ( s ) − − (cid:1) / . In particular, k s k + 1 ≤ µ ( s ) − or µ ( s ) ≤ (cid:0) k s k (cid:1) − . (cid:3) Corollary 5.4.
For every λ ∈ e X there corresponds µ ∈ U ( X ) such that λ ∈ √ L ( X, µ ) h modulo H ⊥ .Proof. The assertion is trivial for dim ( H ) ≤
1. Suppose that dim ( H ) ≥
2. Take λ ∈ e X with s = κ ∗ ( λ ) ∈ X . Notice that s = s + e for s ∈ ball H eh . Put s − = − s + e , which is another pointof X . Then µ = 2 − δ s +2 − δ s − is a unital measure on X (see Example 5.1). Moreover, µ ( s ) = 1 / s = ι ∗ ( η ) for a certain η ∈ S (cid:0) L ( X, µ ) + (cid:1) . Then ηµ ∈ e X ∩ √ L ( X, µ ) h thanks to Theorem 5.1, and κ ∗ ( ηµ ) = s . Whence κ ∗ ( λ − ηµ ) = 0 or λ − ηµ ∈ H ⊥ . (cid:3) µ -concentration sets. As above for every s ∈ X we use the notation s = s + e with s ∈ ball H eh . Let S ⊆ X be a subset. By probabilistic mass on S we mean a summable function m : S → R + such that P s ∈ S m ( s ) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X s ∈ S m ( s ) s (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ − X s ∈ S m ( s ) . In this case, we say that S is a concentration set with a mass m . Note that P s ∈ S m ( s ) s convergesabsolutely, for P s ∈ S k m ( s ) s k ≤ P s ∈ S m ( s ) ≤
1. The function m = 0 is a mass on each subset S automatically. Basically, we deal with a nontrivial mass m on S , in this case, we say that m isa positive mass on S . Lemma 5.4.
A subset S ⊆ X is a concentration set with a positive mass m iff there is a unitalmeasure µ on X such that µ : S → R + is a nozero function. In this case, µ defines a mass on S and µ ≥ m .Proof. First assume that there is µ ∈ P ( X ) such that µ : S → R + , s µ ( s ) is a nontrivial func-tion. Then { µ ( s ) δ s : s ∈ S } is a summable family of measures on X and µ = P s ∈ S µ ( s ) δ s + ν forsome ν ∈ M ( X ) + [1, Ch. 5, 5.10, Proposition 15]. Certainly, P s ∈ S µ ( s ) = sup { µ ( F ) : F ⊆ S } ≤ µ ( X ) = 1, where F is running over all finite subsets of S . But( ζ , e ) = h ζ , e i = h ζ ( · ) , µ i = * ζ ( · ) , X s ∈ S µ ( s ) δ s + + h ζ ( · ) , ν i = X s ∈ S µ ( s ) ζ ( s ) + h ζ ( · ) , ν i = X s ∈ S µ ( s ) ( ζ , s ) + X s ∈ S µ ( s ) ( ζ , e ) + h ζ ( · ) , ν i , which in turn implies that h ζ , η i = ( ζ , η ) = h ζ ( · ) , ν i for η = − P s ∈ S µ ( s ) s + (cid:0) − P s ∈ S µ ( s ) (cid:1) e ∈ H . It follows that ι ∗ ( ν ) = η . But ν ≥
0, therefore η ∈ c or η ∈ c . Since − P s ∈ S µ ( s ) s ∈ H eh , wededuce that (cid:13)(cid:13)P s ∈ S µ ( s ) s (cid:13)(cid:13) ≤ − P s ∈ S µ ( s ). The latter means that µ is a mass on S .Conversely, suppose that m is a nonzero mass on S . Then η = − X s ∈ S m ( s ) s + − X s ∈ S m ( s ) ! e ∈ c and η defines a positive functional, which in turn admits an extension up to a positive measure ν on X . Put µ = P s ∈ S m ( s ) δ s + ν ∈ M ( X ) + . Then h ζ ( · ) , µ i = X s ∈ S m ( s ) ( ζ , s ) + ( ζ , η ) = ( ζ , e ) = h ζ ( · ) , e i for all ζ ∈ H . In particular, µ ( X ) = h e ( · ) , µ i = ( e, e ) = 1, which means that µ is a unital measureon X . Finally, µ ( s ) = m ( s ) + ν ( s ) ≥ m ( s ) for all s ∈ S (see [1, Ch. 5, 3.5, Corollary 1]). (cid:3) Notice that if m = 0 the assertion of Lemma 5.4 follows with any unital measure µ on X .5.4. The exact and finite H -measures on X . A unital measure µ on the state space X issaid to be a finite H -measure if dim ι ( H ) < ∞ . Notice that the latter is equivalent to the factthat ι ( H ) u = im ( P ) u and dim ι ( H ) u < ∞ . For example, if µ is a unital atomic measure withits finite support then it is a finite H -measure on X . By Lemma 5.2, ι (ball H eh ) − ⊆ ball im ( P ) uh for every unital measure µ on X . If the inclusion ball im ( P ) uh ⊆ ι (ball H eh ) − (or the equality ι (ball H eh ) − = ball im ( P ) uh ) holds for µ , we say that µ is an exact H - measure on X . EPARABLE MORPHISMS OF OPERATOR HILBERT SYSTEMS 45
Proposition 5.1.
An exact H -measure µ on X is a finite H -measure automatically. In this case,we have ball im ( P ) h = ι (ball H h ) − , im ( P ) ∩ L ( X, µ ) + = ι ( c ) − and k ι ∗ ( η ) k = k η k for every η ∈ im ( P ) uh . In particular, k s k = k s ′ k whenever s = s + e is a µ -mass in X .Proof. Suppose that ball im ( P ) uh ⊆ ι (ball H eh ) − . Then ι : H eh → im ( P ) uh turns out to be anopen mapping thanks to the Open Mapping Theorem. But it is a compact operator by virtue ofLemma 5.2. It follows that dim im ( P ) uh < ∞ , which in turn implies that dim im ( P ) < ∞ . Thusim ( P ) = ι ( H ) − = ι ( H ), which means that µ is a finite H -measure.Now prove that ball im ( P ) h ⊆ ι (ball H h ) − whenever µ is an exact H -measure. Take θ ∈ im ( P ) h with k θ k <
1. Then θ = ζ ( · ) for some ζ ∈ H h . But ζ = ζ + re , ζ ∈ H eh . Then θ = ζ ( · ) + ru , ζ ( · ) ∈ ι ( H eh ) ⊆ im ( P ) uh (see Lemma 5.2) and k ζ ( · ) k + r = k θ k <
1, that is, ζ ( · ) ∈√ − r ball im ( P ) uh = √ − r ι (ball H eh ) − . In particular, ζ ( · ) = lim n ζ ,n ( · ) for some sequence( ζ ,n ) n ⊆ √ − r ball H eh . It follows that θ = lim n ζ n ( · ) with ζ n ( · ) = ζ ,n ( · ) + ru ∈ ι (ball H h ),that is, θ ∈ ι (ball H h ) − .Further prove that im ( P ) ∩ L ( X, µ ) + = ι ( c ) − . Based on Lemma 5.3, it suffices to provethat ζ ( · ) ∈ ι ( c ) − for every ζ ∈ c µ . Take ζ = ζ + re ∈ c µ with ζ ∈ H eh , k ζ ( · ) k ≤ r .Since r − ζ ( · ) ∈ ball ι ( H eh ) ⊆ ι (ball H eh ) − , it follows that r − ζ ( · ) = lim n ζ ,n ( · ) for a sequence( ζ ,n ) n ⊆ ball H eh . Then ζ n = rζ ,n + re ∈ c and ζ ( · ) = lim n ζ n ( · ) ∈ ι ( c ) − in L ( X, µ ).Finally, take η ∈ im ( P ) uh . Taking into account that ball im ( P ) uh = ι (ball H eh ) − and ι ∗ ( η ) ∈ H eh , we deduce that k η k = sup | ( ι (ball H eh ) , η ) | = sup | (ball H eh , ι ∗ ( η )) | = k ι ∗ ( η ) k , that is, k ι ∗ ( η ) k = k η k . If s = s + e is a µ -mass in X then s ′ ∈ im ( P ) uh and ι ∗ ( s ′ ) = s .Whence k s k = k s ′ k . (cid:3) Now take s ∈ X such that s = 0. As in the proof of Corollary 5.4, we use the notation s − = − s + e for the symmetric opposite of s in X . Notice that the unital measure µ s = 2 − δ s + 2 − δ s − on X is a finite H -measure, and the mapping U s : L ( X, µ s ) → C , U s η = (cid:0) η ( s ) / √ , η ( s − ) / √ (cid:1) implements a unitary equivalence of the Hilbert spaces. Indeed, k U s η k = (cid:0) | η ( s ) | / | η ( s − ) | / (cid:1) / = (cid:18)Z | η | dµ s (cid:19) / = k η k for all η ∈ L ( X, µ s ). Moreover, U s u = (cid:0) / √ , / √ (cid:1) = e s and L ( X, µ s ) u = C U ∗ s f s for f s = (cid:0) / √ , − / √ (cid:1) ∈ C . Thus ( C , e s ) = ℓ (2) is a unital Hilbert space equipped with the unitalcone ℓ (2) + = { λf s + re s : λ, r ∈ R , | λ | ≤ r } . In particular, ι : H → L ( X, µ s ) is reduced to the following mapping ι s : H → ℓ (2), ι s ( ζ ) = (cid:0) ( ζ , s ) / √ , ( ζ , s − ) / √ (cid:1) . If ζ ∈ H eh then ι s ( ζ ) = (cid:0) ( ζ , s ) / √ , − ( ζ , s ) / √ (cid:1) = ( ζ , s ) f s . Inparticular, ι s ( s ) = f s , which in turn implies that P is the identity projection and ι s (ball H eh ) = { ( ζ , s ) f s : k ζ k ≤ } = { ( rs , s ) f s : | r | ≤ } = { rf s : | r | ≤ } = ball (cid:0) C (cid:1) e s h , which means that µ s is an exact H -measure on X . Notice that s is a µ -mass with µ ( s ) = 1 / s ′ = 2 [ s ] and U s s ′ = (cid:0) / √ , (cid:1) = e s + f s , which in turn implies that ι s ( s ) = f s = s ′ (seeProposition 5.1). If ζ = ζ + re ∈ c then ι ( ζ ) = ( ζ , s ) f s + re s and | ( ζ , s ) | ≤ k ζ k ≤ r , that is, ι s ( ζ ) ∈ ℓ (2) + . Conversely, if η = λf s + re s ∈ ℓ (2) + , λ, r ∈ R , | λ | ≤ r then ζ = λs + re ∈ c and ι s ( ζ ) = ( λs , s ) f s + re s = ζ , that is, ι s ( c ) = ℓ (2) + . The factorization problem.
As above consider the canonical ∗ -representation H → C ( X ), ζ ζ ( · ) from Subsection 5.1, where X = S ( c e ), and fix µ ∈ U ( X ). By Lemma 5.3, ι : H → L ( X, µ ) is a unital positive mapping, that is, ι ( e ) = u and ι ( c e ) ⊆ L ( X, µ ) + . Based onProposition 4.1, we conclude that ι = ι σ for a certain unital L ( X, µ )-support σ = { σ χ : χ ∈ B } ⊆ H h , where B is a (hermitian) Hilbert basis for L ( X, µ ) containing u , σ u = e , σ χ ⊥ e for all χ = u ,and P χ = u ( ζ , σ χ ) ≤ k ζ k for all ζ ∈ H eh (see Remark 4.1). Thus { σ χ : χ = u } ⊆ H eh , and( ζ , σ χ ) = ( ζ ( · ) , χ ) = Z ζ ( t ) χ ( t ) dµ = ( ζ ( · ) , P χ )for all ζ ∈ H and χ = u . Thus we can assume that B is a basis for im P , and ( ζ , σ χ ) = ( ζ ( · ) , χ ) forall ζ ∈ H and χ = u . Taking into account that ι ∈ B ( H, L ( X, µ )) (see Lemma 5.2), we concludethat σ is of type 2, that is, P χ k σ χ k = P χ k ι ∗ χ k ≤ k ι ∗ k ≤ √ k σ χ k = k ι ∗ χ k ≤ k χ k ≤ χ ∈ B \ { u } , that is, { σ χ : χ = u } ⊆ ball H eh . Put s χ = σ χ + e , χ = u and s u = e . Then S = { s χ : χ ∈ B } is a subset of X containing e , and ( ζ , s χ ) = ( ζ , σ χ ) + ( ζ , e ) = ( ζ ( · ) , χ ) + ( ζ ( · ) , u ) =( ζ ( · ) , χ + u ) for all ζ ∈ H . Thus B + u = { χ + u } ⊆ e X ∩ √ L ( X, µ ) h = S (cid:0) L ( X, µ ) + (cid:1) and ι ∗ ( B + u ) = S thanks to Theorem 5.1, and P s = e h ζ ( · ) , δ s i ≤ k ζ k for all ζ ∈ H eh .5.6. L -factorization. Now let (
K, u ) be a unital Hilbert space, X = S ( c u ) with the canonical ∗ -representation K → C ( X ), η η ( · ), and fix µ ∈ U ( X ). As above there is a unital L ( X, µ )-support σ = { σ χ : χ ∈ B } ⊆ K h such that ι = ι σ : K → L ( X, µ ) is a unital positive mapping.Put K µ = im ( ι ∗ ) h , which is a real subspace in K h , and σ ⊆ K µ . Since ι ∗ = ι ∗ P , we concludethat K µ = ι ∗ (im ( P ) h ) and ι ∗ : im ( P ) h → K µ is injective. In particular, for every η ∈ K µ therecorresponds a unique η ′ ∈ im ( P ) h such that η = ι ∗ ( η ′ ). Put k η k µ = k η ′ k , which defines a normon K µ such that k η k ≤ k ι ∗ k k η k µ for all η ∈ K µ . If µ is a finite K -measure on X (see Subsection5.1), then B is a finite hermitian basis for im ( P ) and the L ( X, µ )-support σ is finite, which inturn implies that K µ is a finite dimensional real subspace of K h . Lemma 5.5.
Take η ∈ K h . Then η ∈ K µ iff η = P χ α χ σ χ is a sum of an absolutely convergentseries in K for a certain α = ( α χ ) χ ∈ ℓ ( B ) h . In this case, η ′ = P χ α χ χ and k η k µ = k α k .Proof. Note that η ∈ K µ iff η = ι ∗ ( η ′ ) for some η ′ ∈ im ( P ) h . But η ′ = P χ α χ χ has a uniqueexpansion through the basis B such that k η k µ = k η ′ k = k α k , where α = ( α χ ) χ ∈ ℓ ( B ) h . Itfollows that η = P χ α χ σ χ and X χ k α χ σ χ k = X χ | α χ | k σ χ k ≤ X χ | α χ | ! / X χ k σ χ k ! / = k α k X χ k ι ∗ χ k ! / ≤ k α k k ι k < ∞ , which means that η is a sum of an absolutely convergent series in K . (cid:3) Let T : ( K, u ) → ( H, e ) be a unital positive mapping given by a unital H -support k ⊆ K h . Wesay that T is L ( X, µ ) -factorable mapping if T = Sι for a certain S ∈ B ( L ( X, µ ) , H ). Takinginto account ι ∈ B ( K, L ( X, µ )), we conclude that T = Sι ∈ B ( K, H ). Based on Remark 4.2,we obtain that k is of type 2 automatically. Lemma 5.6. If T : ( K, u ) → ( H, e ) is a unital positive mapping given by a unital H -support k ⊆ K h and T = Sι for a positve mapping S : L ( X, µ ) → H , then k ⊆ K µ with sup k k k µ < ∞ . EPARABLE MORPHISMS OF OPERATOR HILBERT SYSTEMS 47
Proof.
Since T = SP ι and P is a positive projection (see Lemma 5.3), we can assume that S = S m for a certain H -support m = { m f : f ∈ F } in im ( P ), where m f ⊥ u , f = e and m e = m ue + u , m ue ∈ ball im ( P ) uh (see Proposition 4.1). Then k f = ι ∗ S ∗ f = ι ∗ ( m f ) = ι ∗ X χ ( m f , χ ) χ ! = X χ ( m f , χ ) ι ∗ ( χ ) = X χ ( m f , χ ) σ χ . By Lemma 5.5, k ′ f = m f and k k f k µ = P χ ( m f , χ ) = k m f k < ∞ . If f = e then k f = P χ = u ( m f , χ ) σ χ ∈ K µ ∩ K uh , whereas k e = P χ = u ( m ue , χ ) σ χ + u . Hence k ⊆ K µ and sup k k k µ =sup k m k < ∞ . (cid:3) Notice that T is L ( X, µ )-factorable iff k T η k ≤ C k η ( · ) k , η ∈ K for some positive constant C .In particular, T transforms k·k -bounded sequences from K to bounded sequences in H . Lemma 5.7.
Let T : ( K, u ) → ( H, e ) be a unital positive mapping given by a unital H -support k ⊆ K h , which transforms k·k -bounded sequences from K to bounded sequences in H for some µ ∈ U ( S ( c u )) . If k ⊆ K µ with sup k k k µ < ∞ , then T = Sι for a certain unital ∗ -linear mapping S : L ( X, µ ) → H .Proof. As above we put X = S ( c u ). Since k ⊆ K µ , it follows that k f = P χ α f,χ σ χ with α f =( α f,χ ) χ ∈ ℓ ( B ) h , k k f k µ = k α f k thanks to Lemma 5.5. Note that α f,u = 0, f = e , and α e,u = 1. Put m f = P χ α f,χ χ ∈ im ( P ) h , f ∈ F , and m = { m f : f ∈ F } . Then m f = k ′ f and k k f k µ = k m f k for all f (see Lemma 5.5). Notice that { m f : f = e } ⊆ im ( P ) uh and m e = m ue + u with m ue = P χ = u α e,χ χ ∈ im ( P ) uh . Moreover, sup k m k = sup (cid:8) k α f k : f ∈ F (cid:9) = sup k k k µ < ∞ ,that is, m is a bounded family in L ( X, µ ) h . Take θ ∈ im ( P ). Then θ = lim n η n ( · ) for acertain sequence ( η n ) n ⊆ K . Thus ( η n ) n is a k·k -bounded sequence in K . By assumption,( T η n ) n is a bounded sequence in H . Put S ( θ ) = P f ( θ, m f ) f . Using again Lemma 5.5 and thelowersemicontinuity argument, we deduce that k S ( θ ) k = X f | ( θ, m f ) | = X f lim n | ( η n ( · ) , m f ) | ≤ lim inf n X f | ( η n ( · ) , m f ) | = lim inf n X f | ( η n , k f ) | = lim inf n k T η n k < ∞ . By the Uniform Boundedness Principle, we obtain that S ∈ B ( L ( X, µ ) , H ) with S = SP .Moreover, S ( ι ( η )) = P f ( η ( · ) , m f ) f = P f ( η, k f ) f = T ( η ), η ∈ K thanks to Lemma 5.5. (cid:3) Notice that the unital ∗ -linear mapping S from Lemma 5.7 may not be positive being just ι ( c u ) − -positive. But that is the case of an exact K -measure µ . Proposition 5.2.
Let T : ( K, u ) → ( H, e ) be a unital positive mapping given by a unital H -support k ⊆ K h , and let µ be an exact K -measure on the state space X = S ( c u ) . Then T isfactorized as T = Sι throughout the canonical mapping ι : K → L ( X, µ ) and a unital positivemapping S : L ( X, µ ) → H iff k ⊆ K µ . In this case, T is of finite rank automatically.Proof. If T admits a positive L ( X, µ )-factorization then k ⊆ K µ thanks to Lemma 5.6. Con-versely, assume that the latter inclusion holds. Using Lemma 5.5 and Proposition 5.1, we deducethat k k f k µ = (cid:13)(cid:13) k ′ f (cid:13)(cid:13) = k k f k ≤ sup k k k < ∞ for all f = e . In particular, sup k k k µ < ∞ . As in theproof of Lemma 5.7, we have the bounded family m = { m f : f ∈ F } = k ′ in im ( P ). Take η ∈ c u . Using Lemma 5.5, we derive that(5.2) X f = e ( η ( · ) , m f ) = X f = e ( η, k f ) ≤ ( η, k e ) = ( η ( · ) , m e ) in L ( X, µ ). Now take θ ∈ L ( X, µ ) + . Then P ( θ ) ∈ im ( P ) ∩ L ( X, µ ) + = ι ( c u ) − thanks toProposition 5.1, that is, P ( θ ) = lim n η n ( · ) in L ( X, µ ) for a certain sequence ( η n ) n ⊆ c u . Using(5.2) and the lowersemicontinuity argument, we deduce that X f = e ( θ, m f ) = X f = e ( P ( θ ) , m f ) = X f = e lim n ( η n ( · ) , m f ) ≤ lim inf n X f = e ( η n ( · ) , m f ) ≤ lim inf n ( η n ( · ) , m e ) = ( θ, m e ) , which means that m is a unital H -support in ( L ( X, µ ) , u ). By Proposition 4.1, S : L ( X, µ ) → H , S ( θ ) = P f ( θ, m f ) f is a unital positive mapping that responds to m , and Sι = T . (cid:3) In particular, the assertion of Proposition 5.2 is applicable to a unital atomic measure µ on X with its finite support . References [1] Bourbaki N., Elements of Mathematics, Integration, Ch. I-V, Nauka, Moscow (1967)[2] Choi M.D., Effros E.G., Injectivity and operator spaces, J. Functional Anal., 24 (1977) 156-209.[3] Chen Z., Huan Z.,
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Middle East Technical University NCC, Guzelyurt, KKTC, Mersin 10, Turkey
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