Jensen's inequality for separately convex noncommutative functions
aa r X i v : . [ m a t h . OA ] F e b JENSEN’S INEQUALITY FOR SEPARATELY CONVEXNONCOMMUTATIVE FUNCTIONS
ADAM HUMENIUK
Abstract.
Classically, Jensen’s Inequality asserts that if X is a compact con-vex set, and f : K → R is a convex function, then for any probability measure µ on K , that f (bar( µ )) ≤ R f dµ , where bar( µ ) is the barycenter of µ . Recently,Davidson and Kennedy proved a noncommutative (“nc”) version of Jensen’sinequality that applies to nc convex functions, which take matrix values, withprobability measures replaced by ucp maps. In the classical case, if f is only a separately convex function, then f still satisfies the Jensen inequality for anyprobability measure which is a product measure. We prove a noncommutativeJensen inequality for functions which are separately nc convex in each variable.The inequality holds for a large class of ucp maps which satisfy a noncommu-tative analogue of Fubini’s theorem. This class of ucp maps includes any freeproduct of ucp maps built from Boca’s theorem, or any ucp map which is con-ditionally free in the free-probabilistic sense of M lotkowski. As an applicationto free probability, we obtain some operator inequalities for conditionally freeucp maps applied to free semicircular families. Introduction
Noncommutative convexity is now an exciting and devloping toolbox for use inoperator algebras and functional analysis. Wittstock [17] introduced the central no-tion of a matrix convex set. The main idea is that matrix convex sets are graded bymatrix levels, and include points at each level. Here, the classical notion of “convexcombination” P i t i x i is replaced with a “matrix convex combination” P i α ∗ i x i α i ,where the “points” x i are matrices of possibly different sizes, and α i are rectangularmatrices satisfying P i α ∗ i α i = I . Matrix convexity is a more natural notion for thestudy of operator algebras, where the study of structure at all matrix levels viacompletely positive or completely bounded maps is a central part of the theory. Infact, Webster and Winkler [16] showed that the category of compact matrix convexsets is contravariantly equivalent to the category of operator systems, so matrixconvex sets faithfully encode the information of any operator system.The theory of matrix convex sets contains many noncommutative analogues ofclassical facts in convexity and Choquet theory. For instance, Effros and Winkler[7] gave noncommutative analogues of the Hahn-Banach Separation Theorem andBipolar Theorem. A persistent difficulty in matrix convexity was the search forthe right notion of extreme point and a working Krein-Milman theorem. The bestknown version of a Krein-Milman type theorem in matrix convexity was givenby Webster and Winkler in [16]. Recently, Davidson and Kennedy [6] improved Date : February 8, 2021.2010
Mathematics Subject Classification.
Key words and phrases. noncommutative functions, noncommutative convexity, Jensen in-equality, free products, completely positive maps.
Webster and Winkler’s result by working in a framework of noncommutative–or“nc”, convex sets, obtaining a Krein-Milman theorem and even a noncommutativeChoquet-Bishop-De Leeuw Integral Representation Theorem.The key idea in Davidson and Kennedy’s framework is that one needs to includeinfinite matrix levels, and we refer to such convex sets as “noncommutative” or“nc” convex sets as opposed to “matrix” convex sets. Noncommutative convexsets are determined by their finite levels (see [6, Proposition 2.2.10]), so in a sensethe two theories are the same. However, nc extreme points may occur at infinitematrix levels, and there are simple examples where the only extreme points occur asinfinite matrices. Using the dual equivalence to the category of operator systems,the extreme points in the state space of an operator system S are exactly the(restrictions of) boundary representations of S , which completely norm the C*-envelope of S , in complete analogue to the Choquet boundary for a classical functionsystem.On matrix or nc convex sets, classical functions are more naturally replaced bynoncommutative functions. Usually, one requires an nc function to be graded alongmatrix levels, to preserve direct sums, and respect similarities either by arbitraryinvertible matrices, or just unitary equivalences. The theory of similarity invariantfunctions parallels complex analysis, because similarity invariant nc functions turnout to be automatically analytic. See [12] for a detailed treatment. Studying thenotion of convex nc functions requires selfadjoint-valued functions, so that there isan ordering on the codomain. Because similarities don’t preserve selfadjointness,we instead only require our nc functions in this context to be unitarily invariant.If X is a (classical) compact convex set, any convex function f : X → R satisfies Jensen’s inequality f (bar( µ )) ≤ Z X f dµ for any probability measure µ ∈ Prob( X ). The barycenter bar( µ ) is the uniquepoint in X that satisfies ϕ (bar( µ )) = R ϕ dµ for every affine function ϕ : X → C .In fact, Jensen’s inequality characterizes convexity, because we can take µ to be aconvex combination of point masses.In [6, Section 7], Davidson and Kennedy show that a noncommutative convexfunction f : K = [ n K ( n ) → M = [ n M n ( C )satisfies the Jensen inequality f (bar( µ )) ≤ µ ( f )whenever µ is a ucp map C ( K ) → M k defined on the C*-algebra C ( K ) of continuousnc functions on K . Here, the barycenter bar( µ ) of µ is the unique point in the k thmatrix level K ( k ) of K that satisfies a (bar( µ )) = µ ( a ) for all nc affine functions a on K .This noncommutative Jensen Inequality sheds some light on classical operatorconvexity. A function f : I → R defined on some interval I ⊆ R is operator convexif its associated functional calculus defines a convex function, i.e. if f ((1 − t ) x + ty ) ≤ (1 − t ) f ( x ) + tf ( y ) ENSEN’S INEQUALITY FOR SEP. NC CONVEX FUNCTIONS 3 for all t ∈ [0 ,
1] and all selfadjoint matrices x, y with spectrum in I . Hansen andPedersen [9] demonstrated that operator convexity is equivalent to a noncommu-tative Jensen inequality. Hansen and Pedersen’s characterization can be obtainedas a special case of Davidson and Kennedy’s nc Jensen Inequality, by restricting tothe case where K = MIN( I ) = { x ∈ M sa n | σ ( x ) ⊆ I } is the unique minimal compact nc convex set with first level MIN( I )(1) = I , see[4, Section 4]. Operator convexity for multivariate functions is more delicate. Forinstance, Hansen [10] established a multivariate nc Jensen inequality for an operatorconvex function f of two variables, but this inequality is more technical and can’t besimply obtained from Davidson and Kennedy’s inequality by working in MIN( I × J ),because on this domain an nc function may not be determined entirely by its firstlevel.1.1. Main Results.
Our main result is a fully noncommutative analogue of thefollowing classical fact (see Proposition 4.2). Let X , . . . , X d be compact convexsets, and let f : X × · · · × X d → R be a separately convex function, meaning f isconvex as a function in any one variable as long as all other variables are fixed. Thisis a much weaker assumption than convexity of f . Then the function f satisfies theJensen inequality f (bar( µ )) ≤ Z X ×···× X d f dµ for any product measure of the form µ = µ × · · · × µ d , where µ i ∈ Prob( X i ). Infact, this characterizes separate convexity of f .We are interested in convex nc functions of multiple variables. Therefore, inSection 3 we first study nc convex sets of the form K × · · · × K d , where K i arecompact nc convex sets, and the product is taken separately at each matrix level.By the categorical duality between compact nc convex sets, each compact nc convexset K i corresponds to the operator system A ( K i ) of continuous nc affine functionson K i . Because K × · · · × K d is the categorical product, it follows from theequivalence of categories that (Proposition 3.1) A ( K × · · · × K d ) ∼ = A ( K ) ⊕ · · · ⊕ A ( K d )is the categorical coproduct of the associated operator systems. This is the “unitaldirect sum” S ⊕ T = S ⊕ T C ((1 S , − (0 , T ))constructed by Fritz in [8].Davidson and Kennedy [6, Section 4.4] showed that if K is a compact nc convexset, the operator system A ( K ) of continuous affine functions generates the C*-algebra C ( K ) of (point-ultrastrong- ∗ ) continuous nc functions on K , and that C ( K ) = C ∗ max ( A ( K ))is in fact the maximal C*-algebra. By comparing the right universal properties, itfollows (Corollary 3.3) that C ( K × · · · × K d ) ∼ = C ( K ) ∗ · · · ∗ C ( K d )is a free product of the associated maximal C*-algebras, with amalgamation over C . This is evidently a noncommutative analogue of the classical result that C ( X ×· · · × X d ) ∼ = C ( X ) ⊗ · · · ⊗ C ( X d ) for compact Hausdorff spaces X , . . . , X d . ADAM HUMENIUK
So, in analogy to the classical case, we should expect that a selfadjoint nc function f : K × · · · × K d → M sa which is separately nc convex (Definition 4.3) shouldsatisfy an nc Jensen inequality for any ucp map µ : C ( K × · · · × K d ) ∼ = C ( K ) ∗ · · · ∗ C ( K d ) → M k which is a “free product” of ucp maps µ i : C ( K i ) → M k . The central difficultyis that the notion of “free product” for ucp maps is not uniquely defined. If A i , i ∈ I , are unital C*-algebras, then Boca’s theorem [2], or its generalized version in[5], gives a standard recipe for how to glue a collection of ucp maps µ i : A i → M k to a ucp map µ : ∗ i ∈ I A i → M k with µ | A i = µ i . However, such a map is not unique, and many such gluings mightexist. Nonetheless, we show f satisfies an nc Jensen inequality for any ucp mapbuilt from Boca’s theorem. We call any ucp map glued together as in the proof ofBoca’s theorem or the more general construction in [5, Theorem 3.1] a free productucp map (see Definition 4.13), and get the following result. Theorem 1.1.
Let K , . . . , K d be compact nc convex sets, and suppose f : K × · · · × K d → M sa is a continuous separately nc convex nc function. Suppose µ : C ( K × · · · × K d ) ∼ = C ( K ) ∗ · · · ∗ C ( K d ) → M k is a free product ucp map of any ucp maps µ i : C ( K i ) → M k , i = 1 , . . . , d . Then f satisfies the Jensen inequality f ( bar ( µ )) ≤ µ ( f ) . In fact, free product ucp maps are not the most general class of ucp maps on C ( K ) ∗ · · · ∗ C ( K d ) for which we get a Jensen inequality. Our strongest version ofTheorem 1.1 is Theorem 4.10, which shows that f satisfies the Jensen inequalityfor any ucp map which satisfies a certain dilation-theoretic analogue of Fubini’stheorem. We call such ucp maps “Fubini type” (Definition 4.8), and they forma larger family than just maps coming from Boca’s theorem. This class is largeenough that the Jensen inequality characterizes separate nc convexity of f .1.2. Connection to Free Probability.
In [3], Bo˙zejko, Leinert, and Speicherintroduced the notion of a conditionally free or c-free product of states on a freeproduct A ∗ · · · ∗ A d of C*-algebras. M lotowski [13] generalized this definition toinclude a conditionally free product of ucp maps as follows. Suppose we have anindex set I , unital C*-algebras A i , i ∈ I , and prescribed ucp maps µ i : A i → M k , and states ϕ i : A i → C , for i ∈ I . The ( ϕ i ) i ∈ I -conditionally free product of the ucp maps µ i is a ucp map µ : ∗ i ∈ I A i → M k which satisfies µ | A i = µ i for each i ∈ I , and whenever a · · · a m ∈ ∗ i ∈ I A i is areduced word (meaning a ℓ ∈ A j ℓ with j = j = · · · 6 = j m ) that satisfies ϕ j ℓ ( a ℓ ) = 0 ENSEN’S INEQUALITY FOR SEP. NC CONVEX FUNCTIONS 5 for each ℓ = 1 , . . . , m , then one has the independence rule µ ( a · · · a m ) = 0 . If µ is a conditionally free product for any tuple of states ( ϕ i ) i ∈ I , we will simplysay µ is a conditionally free or c-free ucp map. We can decompose each A i as adirect sum A i = ker ϕ i ⊕ C A i , the value of µ on any reduced word is recursivelydetermined by µ i and ϕ i for i ∈ I . Thus the ( ϕ i ) i ∈ I -c-free product is uniquelydetermined.Existence and complete positivity of the ( ϕ i ) i ∈ I -c-free product µ follows fromBoca’s theorem. Indeed, examining [2, Theorem 3.1] or [5, Theorem 3.4] in the caseof amalgamation over C shows that the constructed ucp map on the free product isthe unique c-free product ucp map on the unital free product. Since c-free productscan be built from Boca’s theorem, they are product ucp maps in our definition andso Theorem 1.1 in this context gives Corollary 1.2.
Suppose K , . . . , K d are compact nc convex sets and let A i = C ( K i ) , i = 1 , . . . , d . Then for any continuous separately nc convex function f : K × · · · × K d → M sa , and any conditional free ucp map µ : A ∗ · · · ∗ A d → M k , the Jensen inequality f ( bar ( µ )) ≤ µ ( f ) holds. Note that Corollary 1.2 applies exactly to those unital C*-algebras A i which areof the form A i = C ∗ max ( S i ) for any operator systems S i . In this case, we may assume K i = S ( S i ) is the nc state space S n UCP( S i , M n ). For example, the result appliesto commutative C*-algebras of the form C ( X i ), where X i ⊆ R m are simplices.As an application of Corollary 1.2, we obtain some operator inequalities forconditionally free ucp maps on free semicircular families. For instance, let a and b be free semicircular elements in a C*-probability space ( A, ϕ ), where ϕ is faithfuland tracial. Let S = span { A , a } and T = span { A , b } be the operator systemsthey generate Then because the spectra σ ( a ) and σ ( b ) are closed intervals, thecontinuous functional calculus implies that C ∗ ( a ) ∼ = C ∗ max ( S ) and C ∗ ( b ) ∼ = C ∗ max ( T ) . Therefore Corollary 1.2 applies to C ∗ ( a, b ) ∼ = C ∗ ( a ) ∗ C ∗ ( b ) . With this identification, elements such as ab + ba or ab a correspond to separatelync convex functions. Consequently, if µ : C ∗ ( a ) ∗ C ∗ ( b ) → B ( H ) is a c-free ucp map,or a ucp map built from Boca’s theorem, in Example 5.2 we obtain the operatorinequalities µ ( a ) µ ( b ) + µ ( b ) µ ( a ) ≤ µ ( ab + ba ) and µ ( a ) µ ( b ) µ ( a ) ≤ µ ( ab a ) . ADAM HUMENIUK
More generally, if a , . . . , a k is any free semicircular family in ( A, ϕ ), and µ is ac-free ucp map, we show (Corollary 5.3) that µ ( a ) · · · µ ( a k ) + µ ( a k ) · · · µ ( a ) ≤ µ ( a · · · a k + a k · · · a ) , and µ ( a ) · · · µ ( a k − ) µ ( a k ) µ ( a k − ) · · · µ ( a ) ≤ µ ( a · · · a k − a k a k − · · · a ) . The appearance of conditional freeness in Corollary 1.2 suggests that some ana-logue of free independence for ucp maps may play a role in our main Theorem4.10.
Question 1.3.
Is the class of ucp maps C ( K ) ∗· · ·∗ C ( K d ) → M k for which a non-commutative Jensen inequality for separately nc convex functions holds describedby some free independence condition? In the language of Section 4.2, are ucp mapsof Fubini type, or free product ucp maps, characterized by some generalized freeindependence condition? Question 1.3 has a positive answer for states, in which case k = 1 and M k = C .Indeed it is straightforward to check that if ϕ : ∗ i ∈ I A i → C is a state which is a free product ucp map (Definition 4.13), then the C*-subalgebras A i are freely independent, and this occurs if and only if ϕ is the unique free productof the states ϕ i := ϕ | A i built from Boca’s theorem or by [1, Proposition 1.1].2. Background
Noncommutative Convexity.
Throughout, we work in the framework of ncconvexity developed by Davidson and Kennedy in [6]. Because their results are stillfairly novel, we devote a larger-than-normal portion of this section to an expositionof the main results of their paper that play a role here.Given an operator system E , we let M ( E ) = a n ≤ κ M n ( E ) , where κ is any fixed sufficiently large cardinal greater than the density characterof E . In the special case where E is separable, usually κ = ℵ . When E = C , wewrite M := M ( E )for simplicity. The key difference from the existing theory of matrix convex sets isthat we allow for infinite matrix levels, i.e. we consider all cardinals n ≤ α , withthe convention that M n ( E ) ∼ = M n ⊗ min E . Note that when E = C , by conventionwe have M n := M n ( C ) = B ( H n ) for any Hilbert space H n of dimension n .If E is an operator system, we call a subset K ⊆ M ( E )an nc convex set if it is closed under direct sums and compression by isometries.Equivalently, given x i ∈ K ( n i ) for some index set i ∈ I , and matrices α i ∈ M n i ,n satisfying X i ∈ I α ∗ i α i = I n , ENSEN’S INEQUALITY FOR SEP. NC CONVEX FUNCTIONS 7 where the series converges weak- ∗ , the “nc convex combination” X i ∈ I α ∗ i x i α i is also in K . The n th matrix level of K is K ( n ) := K ∩ M n ( E ) , and if K is nonempty and nc convex, then each K ( n ) is nonempty. If E = ( E ∗ ) ∗ isa dual operator system, then we may identify M n ( E ) = M n (CB( E ∗ , C )) ∼ = CB( E ∗ , M n ) ⊆ B ( E ∗ , M n ) , which has a standard weak- ∗ topology. Hence M n ( E ) has an induced weak- ∗ topol-ogy, with agrees with the topology of pointwise-weak- ∗ convergence on B ( E ∗ , M n )on bounded subsets. When E = C , this is just the usual weak- ∗ topology on eachlevel M ( n ) = M n = B ( H n ) of M . If E is a dual operator space, we say that K isa compact nc convex set if it is nc convex and each level K ( n ) ⊆ M n ( E ) is compactin the weak- ∗ topology.Given nc convex sets K and L , a function f : K → L is called an nc function if f is graded (i.e. f ( K ( n )) ⊆ L ( n )), preserves unitary equivalence, and directsums. If K and L are compact nc convex sets, we say f is continuous if it isweak- ∗ continuous on each level of K . Moreover f is nc affine if it also preservescompressions, so if x ∈ K ( n ), and α ∈ M n,k is an isometry, then f ( α ∗ xα ) = α ∗ f ( x ) α. Compact nc convex sets are a noncommutative analogue of classical compactconvex sets (in locally convex spaces). In the classical case, given a compact convexset X , the space A ( X ) of continuous affine function X → C forms a function system.Kadison’s Representation Theorem [11] shows that the functor A : x A ( X ) isan equivalence of categories between the category of compact convex sets and thecategory of function systems. The essential inverse functor is F
7→ S ( F ), where S ( F ) is the state space of F , and so the map C → S ( A ( C )) which embeds C aspoint evaluations is a natural isomorphism.In the noncommutative setting, operator systems are the correct analogue offunction systems. Given a compact nc convex sets K and L , we form the space A ( K, L ) = { a : K → L | a nc affine } of nc affine functions with values in L = S n L n . When L = M , we set A ( K ) := A ( K, M ). The space A ( K ) is an operator system, with ∗ -structure a ∗ ( x ) := a ( x ) ∗ . The matrix order unit is the “constant function”1 A ( K ) ( x ) = 1 M n , x ∈ K ( n ) . The matrix order structure on M n ( A ( K )) ∼ = A ( K, M n M ) is pointwise. The functor K A ( K ) implements an equivalence of categories between the category of com-pact nc convex sets, with continuous nc affine maps as morphisms, and the categoryof operator systems, with ucp maps as morphisms [6, Theorem 3.2.5]. The essentialinverse takes an operator system S to the nc state space S ( S ) ⊆ M ( S ∗ ) ADAM HUMENIUK whose n th level is S ( S )( n ) = { µ : S → M n | µ ucp } . In particular S ( A ( K )) ∼ = K naturally, and the isomorphism means that every ucpmap A ( K ) → M n is of the form f f ( x ), for some x ∈ K ( n ).Given an operator system S , the maximal C*-algebra C ∗ max ( S ) satisfies the fol-lowing universal property. We have an embedding S ⊆ C ∗ max ( S ) = C ∗ ( S ) , and for any unital complete order embedding ι : S → B ( H ), there is a unique ∗ -homomorphism π : C ∗ max ( S ) → B ( H ) with π | S = ι . In the classical setting, fora compact convex set X , one has C ∗ max ( A ( X )) ∼ = C ( X ), via the usual inclusion A ( X ) ⊆ C ( X ). Let K be a compact nc convex set. Let B ( K ) denote the C*-algebra of bounded nc functions K → M , where the C*-operations are pointwise.Then A ( K ) ⊆ B ( K ), and we set C ( K ) := C ∗ ( A ( K )) ⊆ B ( K ) . Davidson and Kennedy demonstrated a noncommutative analogue of the classicalresult C ∗ max ( A ( X )) ∼ = C ( X ) in [6, Theorem 4.4.3]. The C*-algebra C ( A ( K )) isboth • the C*-algebra of all bounded nc functions K → M which are continuouslevelwise in the point-ultrastrong- ∗ topology on each K ( n ) ⊆ B ( E ∗ , M n ),and • the maximal C*-algebra C ∗ max ( A ( K )), with the usual inclusion A ( K ) ⊆ C ( K ).By the universal property and category duality, any ∗ -homomorphism π : C ( K ) → M n is of the form π = δ x : f f ( x ), for some x ∈ K ( n ). Then, Stinespring’stheorem implies that any ucp map µ : C ( K ) → M n has the form µ = α ∗ δ x α : f α ∗ f ( x ) α, for some x ∈ K k and isometry α ∈ M k,n .On a classical compact convex set X , a function f : X → R is convex if for all x , . . . , x n ∈ X and t , . . . , t n ∈ [0 ,
1] with P nk =1 t k = 1, we have f (cid:16) n X k =1 t k x k (cid:17) ≤ n X k =1 t k f ( x k ) . If f is continuous, then f is convex if and only if f satisfies Jensen’s inequality f (bar( µ )) ≤ Z f dµ for all Radon probability measures µ ∈ Prob( X ). Here, the barycenter bar( µ ) is theunique point in X such that a (bar( µ )) = R a dµ , which exists by Kadison duality.In the noncommutative case, a selfadjoint nc function f : K → M sa on a com-pact nc convex set K is nc convex if whenever x i ∈ K ( n i ) and α i ∈ M n i ,n with P i α ∗ i α i = I n , we have f X i ∈ I α ∗ i x i α i ! ≤ X i ∈ I α ∗ i f ( x i ) α i . ENSEN’S INEQUALITY FOR SEP. NC CONVEX FUNCTIONS 9 f is a continuous nc function, it automatically preserves direct sums, and so it isequivalent to simply require(1) f ( α ∗ xα ) ≤ α ∗ f ( x ) α whenever x ∈ K ( k ) and α ∈ M k,n is an isometry. Consequently, a continuousbounded nc function f ∈ C ( K ) sa is nc convex if and only if it satisfies the ncJensen inequality (2) f (bar( µ )) ≤ µ ( f )for all ucp maps µ : C ( K ) → M n . Here the barycenter of a ucp map µ : C ( K ) → M n is the unique point bar( µ ) ∈ K ( n ) such that µ ( a ) = a (bar( µ )) for all a ∈ A ( K ) ⊆ C ( K ). The nc Jensen inequality above follows directly from (1) together with theobservation that any ucp map µ : C ( K ) → M n must be of the form µ = α ∗ δ x α for a point x ∈ K and isometry α , and in this case bar( µ ) = α ∗ xα ∈ K ( n ).In fact [6, Theorem 7.6.1] applies even to matrix-valued bounded nc functions f : K → M k ( M ) sa which are lower semicontinuous in the sense that their ncepigraph epi( f ) = [ n ≤ κ { ( x, y ) ∈ K ( n ) × M k ( M n ) | y ≥ f ( x ) } is levelwise weak- ∗ closed, see [6, Theorem 7.6.1].2.2. Minimal nc convex sets. If X ⊆ C d is a compact convex set, there is aminimal compact nc convex set K = MIN( X ) ⊆ M d with K (1) = X [4, Definition4.1]. A tuple x = ( x , . . . , x d ) ∈ M d ( n ) = M dn lies in MIN( I )( n ) if and only if thereis a normal tuple n = ( n , . . . , n d ) ∈ M d which dilates x and has joint spectrum σ ( n ) ⊆ X. In particular, if d = 1 and X = [ a, b ] is an interval, thenMIN([ a, b ]) = { x ∈ M sa | σ ( x ) ⊆ [ a, b ] } . For general X ⊆ C d , if a ∈ A ( K ) with first level f := a | K (1) = a | X , because a preserves unitaries and direct sums, an application of the spectral theorem showsthat a ( n , . . . , n d ) = f ( n , . . . , n d ) , in the sense of the functional calculus, for every normal tuple n = ( n , . . . , n d ) withjoint spectrum σ ( n ) ⊆ X . Because a preserves compressions, and every x ∈ MIN( I )is a compression of a normal tuple, the nc affine function a is determined by itsrestriction to the first level. Consequently A (MIN( X )) ∼ = A ( X ) via the isomorphismthat restricts to the first level.2.3. Dilations and Notation. If E is an operator space, x ∈ M n ( E ), and y ∈ M k ( E ), we say that y dilates x if there is an isometry α ∈ M k,n such that α ∗ yα = x .In this case we write, x ≺ α y , or just x ≺ y if the associated isometry is clear.If S is an operator system and µ : S → B ( H ), ν : S → B ( K ) are ucp maps,then we say ν dilates µ if α ∗ να = µ , for some isometry α : H → K . Usuallyup to a unitary we assume α is just an inclusion map H ⊆ K . This is really thesame perspective as above, where E = S ∗ is the dual operator space and there is astandard identification M n ( E ) ∼ = CB( S, M n ) , where for infinite cardinals n we take M n = B ( H n ) for an n -dimensional Hilbertspace H n , and identify H ∼ = H dim H and K ∼ = H dim K up to some fixed hidden unitary. Since our nc functions are always unitarily equi-variant, these hidden unitaries are harmless, so we may freely switch between work-ing with M n and B ( H ) as long as dim H = n .A dilation x ≺ y is trivial if y ∼ = x ⊕ z with respect to the range of α , orequivalently αα ∗ y = yαα ∗ . For ucp maps µ : S → B ( H ) and ν : S → B ( K ), with H ⊇ K and µ ≺ ν , the dilation µ ≺ ν is trivial if and only if H is invariant/reducingfor ν ( S ).2.4. Free Products. If A i , i ∈ I are unital C*-algebras, we denote their unitalfree product C*-algebra by ∗ i ∈ I A i , or, if I = { , . . . , d } , by A ∗· · ·∗ A d . Here, we amalgamate only over the subalgebras C ∼ = C A i ⊆ A i . For convenience, we freely identify each C*-algebra A j as a literalsubalgebra A j ⊆ ∗ i ∈ I A i of the free product.3. Products of NC Convex Sets
Suppose K ⊆ M ( E ) and K ⊆ M ( E ) are compact nc convex sets, where E i = ( E i, ∗ ) ∗ are dual operator systems. As in [6], compactness is meant levelwisein the weak- ∗ topology. The Cartesian product K × K := a n ( K )( n ) × ( K )( n ) ⊆ M ( E × E )is also an nc convex set. By convention E × E is the usual ℓ ∞ -product of operatorspaces. We have the standard operator space duality [15, Section 2.6] E × E = (( E ) ∗ × ( E ) ∗ ) ∗ and the corresponding weak- ∗ topology agrees with the product topology on K × K . Hence K × K is a compact nc convex set when given the product topology.It is straightforward to verify that E × E is the categorical product of E and E in the category of compact nc convex sets with continuous nc affine maps asmorphisms.Davidson and Kennedy [6, Theorem 3.2.5] showed that the functor K A ( K )implements an equivalence of categories between this category of compact nc convexsets and the category of operator systems with ucp maps as morphisms. Fritz [8,Proposition 3.3] showed that the categorical coproduct in the category of operatorsystems S, T is the unital direct sum S ⊕ T := S × T C ((1 S , − (0 , T )) . Here, we naturally identify M n ( S ⊕ T ) ∼ = M n ( S ) × M n ( T ) M n ( C ((1 S , − (0 , T ))) . Write the coset of a pair ( s, t ) ∈ M n ( S ) × M n ( T ) in M n ( S ⊕ T ) as s ⊕ t . Thematrix order structure is determined by declaring s ⊕ t ≥ ⇐⇒ s − λ ≥ t + λ ≥ λ ∈ M n ( C ) . ENSEN’S INEQUALITY FOR SEP. NC CONVEX FUNCTIONS 11
Proposition 3.1.
Let K ⊆ M ( E ) and K ⊆ M ( E ) be compact nc convex sets.Then there is a natural complete order isomorphism A ( K × K ) ∼ = A ( K ) ⊕ A ( K ) . Here a ⊕ b ∈ A ( K ) ⊕ A ( K ) corresponds to the continuous nc affine function ( a ⊕ b )( x, y ) = a ( x ) + b ( y ) , ( x, y ) ∈ K × K . Proof.
By [6, Theorem 3.2.5], the functor K A ( K ) is a contravariant equivalenceof categories. Let π i : K × K → K i be the usual projection. Since the diagram K K × K π rrrrr π & & ▲▲▲▲▲ K is a categorical product in the category of compact nc convex sets, the diagram A ( K ) ǫ ( ( PPPPP A ( K × K ) A ( K ) ǫ ♥♥♥♥♥ is a coproduct of operator systems, where ǫ i ( a ) = a ◦ π i . Since the coproduct A ( K ) ⊕ A ( K ) is unique up to isomorphism, the induced map ǫ ⊕ ǫ : A ( K ) ⊕ A ( K ) → A ( K × K )given by ( ǫ ⊕ ǫ )( a, b ) = ǫ ( a ) + ǫ ( b ) is an isomorphism. (cid:3) For a compact nc convex set K , recall that C ( K ) = C ∗ max ( A ( K )) is the maximalC*-algebra generated by the operator system A ( K ). Since A ( K × K ) = A ( K ) ⊕ A ( K ) is a coproduct of operator systems, it is natural to expect that C ( K × K ) ∼ = C ∗ max ( A ( K ) ⊕ A ( K ))is itself a coproduct in the category of unital C*-algebras, with unital ∗ -homomorphismsas morphisms. Indeed this is the case. Here, the coproduct of unital C*-algebras A and B is the unital free product A ∗ B with amalgamation over C ∼ = C A ∼ = C B . Proposition 3.2.
Let S and T be operator systems. Then C ∗ max ( S ⊕ T ) ∼ = C ∗ max ( S ) ∗ C ∗ max ( T ) naturally. The ∗ -isomorphism C ∗ max ( S ) ∗ C ∗ max ( T ) → C ∗ max ( S ⊕ T ) is induced bythe ∗ -monomorphisms ι S : C ∗ max ( S ) → C ∗ max ( S ⊕ T ) , ι T : C ∗ max ( T ) → C ∗ max ( S ⊕ T ) , which are themselves induced by the natural complete order embeddings ι S : S → S ⊕ T and ι T : T → S ⊕ T . Proof.
It suffices to show that C ∗ max ( S ) ι S ) ) ❘❘❘❘❘❘ C ∗ max ( S ⊕ T ) C ∗ max ( T ) ι T ❧❧❧❧❧❧ is a coproduct in the category of unital C*-algebras, and the natural isomorphism C ∗ max ( S ⊕ T ) ∼ = C ∗ max ( S ) ∗ C ∗ max ( T ) follows by uniqueness of coproducts up toisomorphism.Suppose A = B ( H ) is a C*-algebra and we have ∗ -homomorphisms π S : C ∗ max ( S ) → A and π T : C ∗ max ( T ) → A . Set ϕ S = π S | S and ϕ T = π T | T , which are ucp maps S → A and T → A , respectively. By the universal property, these induce a ucpmap ϕ : S ⊕ T → A with ϕι S = ϕ S and ϕι T = ϕ T . The ucp map ϕ induces a ∗ -homomorphism π : C ∗ max ( S ⊕ T ) → B ( H )with π | S ⊕ T = ϕ . Because S ⊕ T generates C ∗ max ( S ⊕ T ), and ϕ ( S ⊕ T ) ⊆ A ,we in fact have π ( C ∗ max ( S ⊕ T )) = C ∗ ( ϕ ( S ⊕ T )) ⊆ A. Since ι S ( C ∗ max ( S )) = C ∗ ( ι S ( S )) is generated by ι S ( S ) = S ⊕
0, and πι S | S = ϕι S = ϕ S = π S | S , we have πι S = π S . Identically, we find πι T = π T . Since the ∗ -homomorphism π is determined by its action on the generating set S ⊕ T = ι S ( S ) + ι T ( T ), it follows that π is unique. (cid:3) Corollary 3.3.
Let K ⊆ M ( E ) and K ⊆ M ( E ) be compact nc convex sets.Then C ( K × K ) ∼ = C ( K ) ∗ C ( K ) naturally via the isomorphism ǫ ∗ ǫ : C ( K ) ∗ C ( K ) , where ǫ ( f )( x, y ) = f ( x ) and ǫ ( g )( x, y ) = g ( y ) for f ∈ C ( K ) and g ∈ C ( K ) .Proof. This follows from combining Proposition 3.1 with Proposition 3.2. Note thateach inclusion A ( K i ) → A ( K ) ⊕ A ( K ) ∼ = A ( K × K )is exactly the map ǫ i , which clearly extends to a ∗ -homomorphism C ( K i ) → C ( K × K ). Thus when identifying C ∗ max ( A ( K i )) ∼ = C ( K i ), in the notation of Proposition3.2 we must have ι A ( K i ) = ǫ i , so the isomorphism is implemented by ι A ( K ) ∗ ι A ( K ) = ǫ ∗ ǫ . (cid:3) Remark 3.4.
Propositions 3.1 and 3.2, and Corollary 3.3 all extend immediatelyto finitely many variables. Since categorical products and coproducts such as × , ⊕ , and ∗ are all associative up to natural isomorphism, a straightforward inductionshows that for any d ∈ N , we have natural isomorphisms A ( K × · · · × K d ) , ∼ = A ( K ) ⊕ · · · ⊕ A ( K d ) C ∗ max ( S ⊕ · · · ⊕ S d ) ∼ = C ∗ max ( S ) ∗ · · · ∗ C ∗ max ( S d ) , and C ( K × · · · × K d ) ∼ = C ( K ) ∗ · · · ∗ C ( K d ) , ENSEN’S INEQUALITY FOR SEP. NC CONVEX FUNCTIONS 13 whenever K , . . . , K d are compact nc convex sets and S , . . . , S d are operator sys-tems. 4. Jensen’s Inequality for Separately NC Convex Functions
The commutative case.
Let K , . . . , K d be (classical) compact convex sets.A function f : K ×· · ·× K d → R is separately convex if it is convex in each variableseparately. That is, if f ((1 − t ) x + ty ) ≤ (1 − t ) f ( x ) + tf ( y )whenever the points x = ( x , . . . , x d ) and y = ( y , . . . , y d ) in K × · · · × K d differin at most one coordinate. Example 4.1.
The function f : R → R given by f ( x, y ) = xy is separately convexbut not convex. Indeed, it’s affine in each variable, yet for example we find f (1 / , − /
2) = − / > − / f (1 , − f (0 , . Separately convex functions satisfy Jensen’s inequality for product measures.
Proposition 4.2.
Let K , . . . , K d be compact convex sets, and let f : K × · · · × K d → R be a continuous function. Then f is separately convex if and only if f ( bar ( µ )) ≤ Z K ×···× K d f dµ for all product measures µ = µ ×· · ·× µ d , where µ k ∈ Prob ( K k ) for each ≤ k ≤ d .Proof. For simplicity we will prove only the case d = 2. Suppose f satisfies f (bar( µ )) ≤ µ ( f ) for any product of probability measures. Suppose x, y ∈ K , t ∈ [0 , z ∈ K . Define the product measure µ := ((1 − t ) δ x + tδ y ) × δ z Then f ((1 − t ) x + ty, z ) = f (bar( µ )) ≤ µ ( f )= (1 − t ) f ( x, z ) + tf ( y, z ) . Therefore the function f is convex in its first argument, and a symmetrical argumentworks in the second argument.Conversely, suppose f is separately convex, and let µ = µ × µ be a product ofprobability measures. By Fubini’s theorem, we find µ ( f ) = Z K Z K f ( x, y ) dµ ( y ) dµ ( x ) ≥ Z K f ( x, bar( µ )) dµ ( x ) ≥ f (bar( µ ) , bar( µ )) = f (bar( µ )) . Here, in the first inequality, we had f ( x, bar( µ )) ≤ Z K f ( x, y ) dµ ( y )by the one-variable version of Jensen’s inequality applied to the convex function y f ( x, y ), and similarly in the second. (cid:3) Taking a more algebraic perspective, we have the standard identification C ( K × · · · × K d ) ∼ = C ( K ) ⊗ · · · ⊗ C ( K d )as C*-algebras. Product measures, as states on C ( K × · · · × K d ), are exactly thosestates of the form µ ⊗ · · · ⊗ µ d , where each µ i ∈ S ( C ( K i )) is a state.4.2. Noncommutative analogue.
In the setting of noncommutative convexity,we’ve seen in Corollary 3.3 that C ( K × · · · × K d ) ∼ = C ( K ) ∗ · · · ∗ C ( K d )for compact nc convex sets K , . . . , K d . The expected noncommutative analogue ofProposition 4.2 should be that a “separately nc convex” nc function f : K × · · · × K d → M sa satisfies a Jensen-type inequality for any nc state/ucp map µ : C ( K × · · · × K d ) ∼ = C ( K ) ∗ · · · ∗ C ( K d ) → M m that arises as a “free product” of ucp maps µ i : C ( K i ) → M m . Defining a noncom-mutative version of separate convexity is straightforward, but the main difficultyis identifying which ucp maps should “count” as free products. Boca’s theorem [2]and its more general version given by Davidson and Kakariadis [5] provide a stan-dard recipe for taking free products for ucp maps. We will show ucp maps builtfrom Boca’s theorem satisfy a Jensen inequality for separately nc convex functions,but this is not the biggest class that works. Definition 4.3.
Suppose K i , i ∈ I , are compact nc convex sets, for i ∈ I . Let f : K = Q i ∈ I K i → M sa be an nc function. We say f is separately nc convex ifwhenever x, y ∈ K , and x ≺ α y is a dilation in K such that at most one of thedilations x i ≺ α y i , i ∈ I, is not trivial, we have f ( x ) ≤ α ∗ f ( y ) α. The following observation justifies the terminology “separately nc convex”.
Proposition 4.4.
Let f : K = Q i ∈ I K i → M sa be an nc function. Then f isseparately nc convex if and only the restriction f | K n is a separately convex functionfor each matrix level K ( n ) of K .Proof. The proof is essentially the same as for [6, Proposition 7.2.3], so we onlyprovide a sketch in the two-variable case I = { , } .Suppose f : K × K → M sa is separately nc convex. Given x, y ∈ K ( n ), λ ∈ [0 , z ∈ K ( n ), the dilation((1 − λ ) x + λy, z ) = (cid:0) √ − λ √ λ (cid:1) (cid:18)(cid:18) x y (cid:19) , (cid:18) z z (cid:19)(cid:19) (cid:18) √ − λ √ λ (cid:19) is trivial in every entry. So, separate nc convexity gives f ((1 − λ ) x + λy, z ) ≤ (cid:0) √ − λ √ λ (cid:1) (cid:18) f ( x, z ) 00 f ( y, z ) (cid:19) (cid:18) √ − λ √ λ (cid:19) = (1 − λ ) f ( x, z ) + λf ( y, z ) . Thus f is convex in its first argument, and a symmetrical argument works for thesecond argument ENSEN’S INEQUALITY FOR SEP. NC CONVEX FUNCTIONS 15
Conversely, suppose f is separately convex at each level. Take any dilation ofthe form ( x, z ) ≺ α ( y, w )in K × K where w ∼ = z ⊕ z ′ with respect to ran( α ). Up to a unitary, which thenc function f respects, we may write y = (cid:18) x ∗∗ ∗ (cid:19) , w = (cid:18) z z ′ (cid:19) , f ( y, w ) = (cid:18) a ∗∗ ∗ (cid:19) , and α = (cid:18) I (cid:19) as operator matrices. Define a selfadjoint unitary U := αα ∗ − ( I − αα ∗ ) = (cid:18) I − I (cid:19) . Then by assumption f (cid:18) y + U yU , w (cid:19) = (cid:18) f ( x, w ) 00 f ( ∗ , w ) (cid:19) ≤ f ( y, w ) + U f ( y, w ) U (cid:18) a ∗ (cid:19) . Cutting down to the (1,1) corner shows f ( x, z ) ≤ b = α ∗ f ( y, w ) α. Hence f respects dilations which are trivial in the second argument, and a symmet-ric argument works when the dilation is trivial in its first argument. This showsthat f is separately nc convex. (cid:3) Example 4.5.
Let K , K ⊆ M sa be compact nc convex sets. Proposition 4.4implies that the nc function f : K × K → M sa defined by f ( x, y ) = xy + yx K × K ) is the non-convex function f ( x, y ) = xy from Example4.1. Example 4.6.
More generally, Proposition 4.4 shows that any function of the form a · · · a k + a ∗ k · · · a ∗ ∈ C ( K × · · · × K d )for nc affine functions a i ∈ A ( K j i ) in separate variables, i.e. with i = i ′ imply-ing j i = j i ′ , is separately nc convex. Moreover, sums of such functions are alsoseparately nc convex. Example 4.7. If I is a finite set and J ⊆ I , we identify C Y j ∈ J K j ∼ = ∗ j ∈ J C ( K j ) ⊆ ∗ i ∈ I C ( K i ) ∼ = C Y i ∈ I K i ! as a C*-subalgebra. If f ∈ C ( Q j ∈ J K j ) + is a positive separately nc convex ncfunction, i ∈ I \ J , and a ∈ A ( K i ) ⊆ C ( K i ) ⊆ C ( Q i K i ) is an nc affine function,then the function F := a ∗ f a ∈ C ( Q i K i ) is separately nc convex.Indeed, suppose x = ( x i ) i ∈ I ≺ α y = ( y i ) i ∈ I is a dilation that is nontrivial in atmost one coordinate. Indeed, if π J : Q i ∈ I K i → Q j ∈ J K J is the natural projection,then F ( x ) = α ∗ a ( y i ) ∗ f ( α ∗ π J ( y ) α ) α ∗ a ( y i ) α ≤ α ∗ a ( y i ) ∗ ( αα ∗ ) f ( π J ( y ))( αα ∗ ) a ( y i ) α. If the dilation x i ≺ α y i is trivial, then a ( x i ) commutes with αα ∗ and the right handside becomes α ∗ a ( y i ) ∗ f ( π J ( y )) a ( y i ) α = α ∗ F ( y ) α. Otherwise, x j ≺ α y j for each j ∈ J , so f ( π J ( y )) commutes with αα ∗ . Since f ( π J ( y )) ≥
0, the right hand side is α ∗ a ( y i ) ∗ f ( π J ( y )) / aa ∗ f ( π J ( y )) / a ( y i ) α ≤ α ∗ a ( y i ) ∗ f ( π J ( y )) a ( y i ) α = α ∗ F ( y ) α. In either case, F ( x ) ≤ α ∗ F ( y ) α .For example, if a ∈ A ( K i ) and b ∈ A ( K j ) are nc affine functions in separatevariables i = j , then F = a ∗ b ∗ ba is separately nc convex. An easy induction shows that if a ∈ A ( K i ) , . . . , a k ∈ A ( K i k ) with i , . . . , i k distinct indices, then the function F = a ∗ · · · a ∗ k a k · · · a is separately nc convex.The following definition is designed to exactly capture the largest class of ucpmaps for which our approach can prove a Jensen-type inequality that characterizesseparately nc convex functions. The name and involved chain of dilations are meantas a noncommutative analogue of the role that Fubini’s theorem plays in the proofof Proposition 4.2. Definition 4.8.
Let A i , i ∈ I , be unital C*-algebras. A ucp map µ : ∗ i ∈ I A i → B ( H ) is of Fubini type if there exists an ordinal α and a chain of dilations { µ λ : ∗ i ∈ I A i → B ( H ) | λ ≤ α } such that(i) µ = µ and H = H ,(ii) µ α is a ∗ -homomorphism,(iii) λ ≤ ρ ≤ α implies that H λ ⊆ H ρ and µ λ ≺ µ ρ ,(iii) each dilation µ λ ≺ µ λ +1 is nontrivial in at most one of the algebras A i , i ∈ I ,(iv) if β ≤ α is a limit ordinal, then H β = S λ ≤ β H λ .In most examples, when I is finite we will take α to be a finite ordinal with | α | = d := | I | . Usually, we can arrange that µ = µ ≺ µ ≺ · · · ≺ µ d = π where π is a ∗ -homomorphism and each dilation µ k − ≺ µ k is nontrivial only in the k th coordinate. Example 4.9.
Suppose µ : ∗ i ∈ I A i → B ( H ) is a ucp map such that all but oneof the ucp maps µ | A i is a ∗ -homomorphism. Then µ is of Fubini type. Let π : ∗ i ∈ I A i → B ( K ) be the minimal Stinespring dilation of µ , with K ⊇ H . Then since µ = P H π | H is a ∗ -homomorphism on all but one algebra A i , it follows that H isreducing for π ( A i ) for all but possibly one i ∈ I . Hence the dilation µ =: µ ≺ µ := π is trivial in all but one algebra A i . ENSEN’S INEQUALITY FOR SEP. NC CONVEX FUNCTIONS 17
In the following theorem, we freely identify C ( Q i K i ) with ∗ i C ( K i ). Theorem 4.10.
Let K , . . . , K d be compact nc convex sets, and let f : K × · · · × K d → M sa be a bounded and (weak- ∗ to weak- ∗ or ultrastrong- ∗ to ultrastrong- ∗ ) upper semi-continuous nc function. If f is separately nc convex, and µ : ∗ i ∈ I C ( K i ) → M m is a ucp map of Fubini type, then the Jensen inequality f ( bar ( µ )) ≤ µ ( f ) holds.Proof. Conversely, suppose f is continuous and separately nc convex. Let µ : C ( Y i ∈ I K i ) ∼ = ∗ i ∈ I C ( K i ) → M m = B ( H )be a ucp map of Fubini type, with associated dilation chain { µ λ | λ ≤ α } , where µ = µ and π := µ α is a ∗ -homomorphism. Set x λ := bar( µ λ ) for all λ ≤ α .Because π is a ∗ -homomorphism, we have π = δ x α . Since the barycenter map iscontinuous and nc affine, then λ ≤ ρ ≤ α implies x λ ≺ x ρ , and whenever λ + 1 ≤ α ,the dilation x λ ≺ α λ +1 x λ +1 in K ×· · ·× K d is trivial in all but at most one variable.That is, the chain of dilations { x λ | λ ≤ α } shows that x ∈ K ∼ = S ( A ( K )) is itselfof Fubini type when viewed as a ucp map on A ( K ).We will show that f ( x ) ≤ P H f ( x λ ) | H for any λ by transfinite induction on λ . This is a tautology when λ = 0. Supposefor λ ≤ α that f ( x ) ≤ P H f ( x λ ) | H . Because the dilation x λ ≺ x λ +1 is trivial inall but one variable, and f is separately nc convex, we have f ( x λ ) ≤ P H λ f ( x λ +1 ) | H λ . Compressing to H and using the inductive hypothesis yields f ( x ) ≤ P H f ( x λ ) | H ≤ P H f ( x λ +1 ) | H . Finally, suppose β ≤ α is a limit ordinal, and f ( x ) ≤ P H f ( x λ ) | H for all λ < β .Fix any constant vector c ∈ ( K × · · · × K d ) . Define a net z λ = x λ ⊕ c d λ with respect to the decomposition H β = H λ ⊕ ( H β ⊖ H λ ), where d λ = dim( H β ⊖ H λ ).Then z λ converges to x β ultrastrong- ∗ . For ρ ≤ λ < β we have P H ρ f ( z λ ) | H ρ = P H ρ f ( x λ ) | H ρ as f respects direct sums. Compressing to H gives P H f ( z λ ) | H = P H f ( x λ ) | H ≥ f ( x )by inductive hypothesis. Since z λ → x and f is upper semicontinuous, we have P H f ( x β ) | H ≥ P H (lim sup λ<β f ( z λ )) | H ≥ lim sup λ<β P H f ( z λ ) | H ≥ f ( x ) . This completes the induction, and by taking λ = α we conclude f (bar( µ )) = f ( x ) ≤ P H f ( x α ) | H = P H π ( f ) | H = µ ( f ) . (cid:3) . Remark 4.11.
Theorem 4.10, applies to weak- ∗ or ultrastrong- ∗ continuous ncfunctions, including all functions in C ( K × · · · × K d ) sa . The same proof alsoshows that a non-continuous separately nc convex function still satisfies a Jenseninequality f (bar( µ )) ≤ α ∗ π ( f ) α whenever µ is a Fubini type ucp map with anassociated dilation chain of finite length. Remark 4.12.
In fact, the Jensen inequality in Theorem 4.10 completely charac-terizes separate nc convexity of f . Suppose f satisfies the claimed Jensen inequalityfor ucp maps of Fubini type. Let x ≺ α y be a dilation in Q i ∈ I K i which is trivial in all but at most one coordinate. Let µ = α ∗ δ y α . Because the dilation µ ≺ δ y is trivial in all but possibly one algebra C ( K i ), the ucp map µ is of Fubini type. Therefore f ( x ) = f (bar( µ )) ≤ µ ( f ) = α ∗ f ( y ) α, so f is separately nc convex.Theorem 4.10 applies to the following wide class of ucp maps which we mightconsider “free products” of ucp maps. This definition is just a reorganizing of theconstruction given by Davidson and Kakariadis [5]. Given an index set I , let S I denote the set of finite words in I without repeated letters. We include the emptyword ∅ in S I . Definition 4.13.
Let A i , i ∈ I , be unital C*-algebras with unital free product A = ∗ i ∈ I A i . Let µ : A → B ( H ) be a ucp map with minimal Stinespring dilation π : A → B ( K ), where K ⊇ H . Define subspaces H w for w ∈ S I by setting K ∅ = H, and inductively K iw := π ( A i ) K w ⊖ K w whenever w = w · · · w m ∈ S I with w = i ∈ I . We call µ a free product ucp map (of the ucp maps µ i = µ | A i , i ∈ I ) if the spaces K w are pairwise orthogonal for w ∈ S I .Minimality of the Stinespring dilation in Definition 4.13 implies that K = X w ∈ S I K w , so the definition of “free product” is just that this sum is direct. Remark 4.14.
By definition, any ucp map µ : A → B ( H ) which is built from ucpmaps µ i : A i → B ( H ) via the proof of [5, Theorem 3.1] is a free product ucp map.Examining the proof of [5, Theorem 3.4] also shows that any ucp map built fromBoca’s theorem, which produces a unique product map µ given the additional dataof prescribed states ϕ i : A i → C , is also a free product ucp map. Proposition 4.15.
Let A i , i ∈ I be unital C*-algebras with unital free product A = ∗ i ∈ I A . Then any free product ucp map µ : A → B ( H ) is a ucp map of Fubinitype. ENSEN’S INEQUALITY FOR SEP. NC CONVEX FUNCTIONS 19
Proof.
Suppose µ is a free product ucp map. Let π : A → B ( K ) be its minimalStinespring dilation and define the spaces K w , w ∈ S I , exactly as in Definition4.13, so that K = M w ∈ S I K w as an orthogonal direct sum. Up to a fixed bijection we may assume I is just someordinal α . For any ordinal λ ≤ α , let H λ = M w ∈ S λ K w . Then H = H , H α = K , and the sequence ( H λ ) λ ≤ α is increasing with H β = [ λ<β H λ for any limit ordinal β .Set µ λ := P H λ π | H λ . It suffices to show that for any λ with λ + 1 ≤ α thatthe dilation µ λ ≺ µ λ +1 is trivial in all variables except λ + 1. Since the sum K = L w K w is direct, we find H λ +1 ⊖ H λ = M { K w | w = w · · · w m ∈ S λ +1 with some w j = λ + 1 } . Given i ∈ I and w = w · · · w m ∈ S I , by definition π ( A i ) maps K w into ( K w ⊕ K iw i = w ,K w ⊕ K w ··· w m i = w . Knowing this, it follows that for any i ∈ I with i = λ +1 that π ( A i ) maps H λ +1 ⊖ H λ into M { K w | w = w · · · w m ∈ S ( λ +1) ∪{ i } with some w j = λ + 1 }⊆ M { K w | w = w · · · w m ∈ S I with some w j = i } = ( H λ ) ⊥ . Compressing to H λ +1 shows that H λ +1 ⊖ H λ is invariant for µ λ +1 ( A i ) = P H λ +1 π ( A i ) | H λ +1 .So, the dilation µ λ ≺ µ λ +1 is trivial in any variable i = λ + 1. The chain of dilations { µ λ | λ ≤ α } shows that µ is of Fubini type. (cid:3) Example 4.16.
Fix some large M ≥
2, and let I = [ − M, M ] ⊆ R be an interval,so we have a compact nc convex setMIN( I ) = { x ∈ M sa | σ ( x ) ⊆ I } . Set A = A = C (MIN( I )). By Corollary 3.3 we may identify A ∗ A ∼ = C (MIN( I ) × MIN( I )) . Let y = ( y , y ) := , ∈ (MIN( I ) × MIN( I ))(3) ⊆ M ( C ) . Setting K := C , the evaluation map π := δ y : A ∗ A ∼ = C (MIN( I ) × MIN( I )) → M = B ( K ) is a ∗ -homomorphism. Define subspaces H = H := C ⊕ ⊕ H := C ⊕ C ⊕ . Compress to get ucp maps µ := µ := P H π | H ,µ := P H π | H . Because H is reducing for µ ( A ), and H is reducing for π ( A ), the chain ofdilations µ ≺ µ ≺ π demonstrates that µ is a ucp map of Fubini type.However, µ is not a free product ucp map. For I = { , } , define the subspaces K w ⊆ K for w ∈ S I as in Definition 4.13. Because C (MIN( I )) ∼ = C ( I )is generated by the polynomials, it is straightforward to see that π ( A ) H = C ∗ ( y ) H = C ⊕ C ⊕ H . Hence K = π ( A ) H ⊖ H = 0 ⊕ C ⊕ . Then, we have π ( A ) K = C ∗ ( y ) K = C = K. For instance, if { e , e , e } is the standard basis for C , then e = ( y − y ) e and e = ( y − I ) e both lie in C ∗ ( y ) K . Thus K = π ( A ) K ⊖ K = C ⊕ ⊕ C , which is not orthogonal to H ∅ = H = C ⊕ ⊕
0. Therefore K = H + K + K , but this sum is not direct, and so µ isn’t a free product ucp map.5. Connection to Free Probability
Given C*-algebras A i , i ∈ I , and prescribed states ϕ i : A i → C , a ucp map µ : ∗ i ∈ I A i → B ( H )is conditionally free (with respect to the family ( ϕ i ) i ∈ I ) if for every reduced word a · · · a m ∈ ∗ i ∈ I A i (reduced meaning that a k ∈ A j ( k ) with j (1) = · · · 6 = j ( m )) that satisfies ϕ j ( k ) ( a k ) = 0 for all k = 1 , . . . , m, the multiplication rule µ ( a · · · a m ) = µ ( a ) · · · µ ( a m )holds [3, 13]. Boca’s theorem ([2] or see [5, Theorem 3.4]) shows that if µ i : A i → B ( H ) are any ucp maps and ϕ i ∈ S ( A i ) are prescribed states, there exists a uniqueucp map µ : ∗ i ∈ I A i → B ( H )which is conditionally free with respect to ( ϕ i ) i ∈ I . ENSEN’S INEQUALITY FOR SEP. NC CONVEX FUNCTIONS 21 If µ is any ucp map which is conditionally free with respect to some familyof states ( ϕ i : A i → C ) i ∈ I , then µ is a free product ucp map in the sense ofDefinition 4.13. This follows from the uniqueness in Boca’s theorem. Any suchconditionally free map agrees with the one constructed by Boca’s theorem, and soRemark 4.14 applies. Or, examining the proof of [5, Theorem 3.2] in the case ofamalgamation over C shows how to build the Stinespring dilation. The minimalStinespring dilation π : ∗ i ∈ I A i → B ( K )lives on a direct sum K = M w ∈ S I K w , where S I is the set of words with letters in I without repeated letters. The dilationis constructed recursively so that whenever i ∈ I and w = w · · · w m ∈ S I with i = w , the sum H w ⊕ H iw is reducing for π ( A i ), and the compression P H w π | H w to H w satisfies P H w π | H w = ϕ i ⊗ id H w . Applying Theorem 4.10 in this context yields the following.
Corollary 5.1.
Suppose A , . . . , A d are unital C*-algebras such that A i = C ∗ max ( S i ) for some operator systems S = A ( K ) , . . . , S d = A ( K d ) , where K i ∼ = S ( S i ) arecompact nc convex sets. If µ : A ∗ · · · ∗ A d → B ( H ) is a conditionally free ucp map, and f ∈ ( A ∗ · · · ∗ A d ) sa ∼ = C ( K × · · · × K d ) sa is a continuous separately nc convex function, then the Jensen inequality f ( bar ( µ )) ≤ µ ( f ) holds. As in Remark 4.6, Corollary 5.1 applies e.g. to any function f which is a (limitof) sum(s) of the form a · · · a k + a ∗ k · · · a ∗ , where each a i ∈ A ( K j i ) ∼ = S i is continuous and nc affine, and no two a i ’s dependon the same variable, i.e. i = i ′ implies j i = j i ′ .In the commutative case, if K i ⊆ R m i are (classical) Choquet simplices, then C (MIN( K i )) ∼ = C ( K i )is a commutative C*-algebra, where the isomorphism is implemented by restrictionto the first level. In particular, the theorem applies to commutative C*-algebras ofthe form A i = C ( I i ), where I i ⊆ R are intervals. Any selfadjoint element a i withspectrum σ ( a i ) = I i generates such a C*-algebra. What follows is an applicationof Corollary 5.1 in the special case where a i are free semicircular elements. Example 5.2.
Suppose (
A, ϕ ) is a C*-probability space, where A is a unital C*-algebra, and ϕ is a faithful tracial state on A . Suppose a, b ∈ A sa are free semi-circular elements. This means that a and b have the semicircular ∗ -distributionwith some radii r > s >
0, respectively, and that the C*-algebras C ∗ ( A )and C ∗ ( b ) are freely independent with respect to ϕ . Then σ ( a ) = I := [ − r, r ] and σ ( b ) = J = [ − s, s ]. By [14, Theorem 7.9], faithfulness and traciality of ϕ impliesthat C ∗ ( a, b ) ∼ = C ∗ ( a ) ∗ C ∗ ( b )via the natural map.Consider the operator systems S = span { A , a } and T = span { A , b } generatedby a and by b . Because σ ( a ) = I and σ ( b ) = J are closed intervals, the functionalcalculus gives standard isomorphisms C ∗ ( a ) ∼ = C ( I ) ∼ = C (MIN( I )) and C ∗ ( b ) ∼ = C ( J ) ∼ = C (MIN( J )) . We then identify C ∗ ( a, b ) ∼ = C (MIN( I )) ∗ C (MIN( J )) ∼ = C (MIN( I ) × MIN( J )) . Suppose µ : C ∗ ( a ) ∗ C ∗ ( b ) ∼ = C (MIN( I ) × MIN( J )) → B ( H )is a conditionally free ucp map, or a ucp map built from Boca’s theorem. Thebarycenter bar( µ ) ∈ MIN( I ) × MIN( J ) corresponds to the point evaluation σ = δ bar( µ ) : C ∗ ( a ) ∗ C ∗ ( b ) → B ( H )that is the unique ∗ -homomorphism determined by σ ( a ) = µ ( a ) and σ ( b ) = µ ( b ).Note that such a ∗ -homomorphism exists because σ ( µ ( a )) ⊆ I and σ ( µ ( b )) ⊆ J .Theorem 4.10 implies that for every element x ∈ C ∗ ( a ) ∗ C ∗ ( b ) ∼ = C (MIN( I ) × MIN( J )) which corresponds to a separately nc convex function on MIN( I ) × MIN( J ),the operator inequality σ ( x ) ≤ µ ( x ) holds. By Examples 4.6 and 4.7, elements ofthe form x = ab + ba , or x = ab a correspond to separately convex functions.Therefore if a and b are free semicircular elements and µ is a conditionally free ucpmap, we get the inequalities µ ( a ) µ ( b ) + µ ( b ) µ ( a ) ≤ µ ( ab + ba ) , (3) µ ( a ) µ ( b ) µ ( a ) ≤ µ ( ab a ) . (4)In this same context, the “one-variable” nc Jensen inequality of Davidson andKennedy [6, Theorem 7.6.1] implies that if y ∈ S + T = span { A , a, b } , we have µ ( y ) ∗ µ ( y ) ≤ µ ( y ∗ y )because the element x = y ∗ y corresponds to a (jointly) nc convex function. Inthis case, such an inequality trivially reduces to the usual Schwarz inequality forucp maps. In contrast, the inequalities (3) and (4) do not reduce to some trivialapplication of the Schwarz inequality because they do not hold for general ucp maps µ . For instance, one could take A = M , a = (cid:18) (cid:19) , b = (cid:18) (cid:19) , and let µ : M → C be the normalized trace. (In this example, ( A, ϕ ) := ( M , tr / a and b are not freely independentand not semicircular.)The reasoning of Example 5.2 generalizes readily to free semicircular families ofarbitrary size as follows. ENSEN’S INEQUALITY FOR SEP. NC CONVEX FUNCTIONS 23
Corollary 5.3.
Let ( A, ϕ ) be a C*-probability space with faithful tracial state ϕ ∈S ( A ) . Suppose a , . . . , a d ∈ A is a free family of semicircular elements. If µ : C ∗ ( a , . . . , a d ) ∼ = C ∗ ( a ) ∗ · · · ∗ C ∗ ( a d ) → B ( H ) is a conditionally free ucp map, or a ucp map built from Boca’s theorem, then forevery list of distinct indices i , . . . , i k ∈ { , . . . , d } , the operator inequalities µ ( a i ) · · · µ ( a i k ) + µ ( a i k ) · · · µ ( a i ) ≤ µ ( a i · · · a i k + a i k · · · a i ) and µ ( a i ) · · · µ ( a i k − ) µ ( a i k ) µ ( a i k − ) · · · µ ( a i ) ≤ µ ( a i · · · a i k · · · a i ) hold. Note that in Corollary 5.3, the ucp map µ need not be conditionally free withrespect to the states ϕ i := ϕ | C ∗ ( a i ) . Conditional freeness with respect to any familyof states is enough. Funding
This work was supported by an NSERC Alexander Graham Bell Canada Grad-uate Scholarship-Doctoral (CGS-D) [grant number 401230185].
Acknowledgements
The author would like to thank Kenneth Davidson and Matthew Kennedy formany helpful discussions. The author would also like to thank Paul Skoufranis forhelpful discussions regarding the connections of this work to free probability.
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