aa r X i v : . [ m a t h . F A ] S e p ANALYTIC LIFTS OF OPERATOR CONCAVEFUNCTIONS
MIKL ´OS P ´ALFIA
Dedicated to the memory of my grandfather B´ela P´alfia (1928-2020).
Abstract.
The motivation behind this paper is threefold. Firstly, tostudy, characterize and realize operator concavity along with its appli-cations to operator monotonicity of free functions on operator domainsthat are not assumed to be matrix convex. Secondly, to use the obtainedSchur complement based representation formulas to analytically extendoperator means of probability measures and to emphasize their studythrough random variables. Thirdly, to obtain these results in a decentgenerality. That is, for domains in arbitrary tensor product spaces ofthe form
A ⊗ B ( E ), where A is a Banach space and B ( E ) denotes thebounded linear operators over a Hilbert space E . Our arguments alsoapply when A is merely a locally convex space. Introduction
The main object of our investigations is a free function F : D ( E )
7→ B ( E )with self-adjoint domain and range satisfying(1) F (( I A ⊗ W ∗ ) X ( I A ⊗ W )) ≥ W ∗ F ( X ) W for each isometry W : E K for Hilbert spaces E, K and X ∈ D ( K )such that also W ∗ XW ∈ D ( E ) ⊆ A ⊗ B ( E ), where I A denotes the identitymap of a Banach space A . In particular if ( D ( E )) is matrix convex, then W ∗ XW ∈ D ( E ) always when X ∈ D ( K ), however this is not required inthis paper. It is known by [20], that (1) characterizes operator concave freefunctions determined by(2) F ((1 − λ ) X + λY ) ≥ (1 − λ ) F ( X ) + λF ( Y )for λ ∈ [0 , X, Y ∈ D ( E ) on a matrix convex domain ( D ( E )), so onemight think of an F above in (1) as a partially defined operator concavefunction. A conceptually similar problem in the particular case of powermeans of positive numbers is treated successfully in [14] and then in [13] bylifting the real function into a fully non-commutative through characterizingnonlinear operator equations. The other paper known to the author that Date : September 29, 2020.2010
Mathematics Subject Classification.
Primary 46L52, 47A56 Secondary 47A64.
Key words and phrases. free function, operator concave function, operator monotonefunction, operator mean. does not assume matrix convexity of the domain is the foundational mate-rial [1], in which local monotonicity is characterized for a real multivariablefunction through construction of analytic extensions to upper half-planes.However [1] does not succeed in lifting up the real function into a full non-commutative free function on an enclosing matrix convex domain, nor canit show that it preserves the partial order. In this paper this problem ofnon-commutative lifts is eliminated by transforming the problem in section2 into a more suitable form handled in Theorem 3.11 for real multivariablefunctions that preserves the partial order between commuting tuples of ma-trices. Theorem 3.11 constructs full non-commutative analytic lifts of thesereal multivariable functions to matrix convex hulls of the original domain.This is based on the more general Corollary 3.7 characterizing functionssatisfying (1).Usually matrix convexity of the domain ( D ( E )) is essential for the ma-chinery of various further existing results characterizing operator monotoneor operator concave (2) functions, like [10, 20] and [22] when A = C k for apositive integer k , and in [8, 23, 24] when A is an operator system. Exclud-ing [20], the other existing results in the field are restricted to the case whendim( E ) < ∞ and F is continuous with respect to finitely open topologiesused to study holomorphic functions in general, see the monograph [29].This essentially renders investigations in [8, 23, 24] in the norm topologyrestricted to matrix convex matricial domains of M n ( A ) ≃ M n ( C ) ⊗ A foran operator space A . In this paper A can be any Banach space and E canbe infinite dimensional over the same ground field. Actually all results ofsection 3 can be worked out in exactly the same way for a locally convexvector space A as remarked there. However we will not need that here, forour applications in the last section, the Banach space case suffices.The above mentioned restrictions of the state of the art of free func-tion theory become apparent if we consider the recent developments in thetheory of operator means of probability measures of positive operators in[11, 12, 15, 21]. There, one studies functions F : P ∞ ( P ( E )) P ( E ) on thecone of probability measures P ∞ ( P ( E )) over the positive invertible opera-tors P ( E ), which preserve the stochastic order [11] of probability measures.We show that one can lift such a function F into an operator monotone freefunction ˆ F : L ∞ ([0 , , λ ) + ⊗ P ( E ) P ( E ) of P ( E )-valued random variables,thus satisfies (1). Then we apply our results to this setting, to analyticallycontinue an operator mean in several variables to the probability measuresetting, thus obtaining ˆ F : L ∞ ([0 , , λ ) + ⊗ P ( E ) P ( E ). This providesa realization for a class of operator means whose study were initiated in[21] and put into a framework in [12]. The main results in this topic are insection 4, specifically Theorem 4.7, Corollary 4.8 and Definition 4.2The results of the paper are self-contained in the sense, that we essentiallyuse only standard operator theoretic results available for example in [28] tostudy free functions. Detailed structure theory of free functions like themonograph [29] is not required. In section 4 we also make use of the theory NALYTIC LIFTS OF OPERATOR CONCAVE FUNCTIONS 3 of the stochastic order of probability measures and some related resultswhich are from probability theory.In the following, we explicitly introduce the basic definitions of the objectsto be studied in this paper. All vector spaces are over the ground fields R or C respectively. Let A be a vector space and let I A : A 7→ A denote theidentity map.
Definition 1.1 (Free set and matrix convex set) . A collection ( D ( E )) ofsets of operators D ( E ) ⊆ A ⊗ B ( E ) for each Hilbert space E over the groundfield R or C is a called a free set whenever for all Hilbert spaces E, K wehave the following:1) ( I A ⊗ U ∗ ) D ( E )( I A ⊗ U ) ⊆ D ( K ) for all unitary U : K E .2) D ( E ) ⊕ D ( K ) ⊆ D ( E ⊕ K ).If additionally (2) holds for any linear isometry U : K E , then ( D ( E ))is a matrix convex set .Sometimes the collection ( D ( E )) will be restricted to the case dim( E ) < ∞ . In that case, for all other involved Hilbert spaces K we assume dim( K ) < ∞ as well.We remark that if a given free set ( D ( E )) is matrix convex, then accordingto [9] each D ( E ) is convex in the usual sense. Definition 1.2 (Free function) . Let L be a fixed Hilbert space. A collectionof functions F : D ( E )
7→ B ( L ⊗ E ) indexed by E for a free set D ( E ) ⊆A⊗B ( E ) defined for all Hilbert spaces E, K is called a free function wheneverfor all A ∈ D ( E ) and B ∈ D ( K ), we have1) unitary invariance, that is F (( I A ⊗ U ∗ ) A ( I A ⊗ U )) = ( I L ⊗ U ∗ ) F ( A )( I L ⊗ U )holds for all unitaries U : E K ;2) direct sum invariance, that is F ( A ⊕ B ) = F ( A ) ⊕ F ( B ) . In the paper we assume that L = C , since given a free function F : D ( E )
7→ B ( L ⊗ E ), one can study its slices l ( F ) : D ( E )
7→ B ( E ) instead,where l ∈ B ( L ) ∗ + is a state of B ( L ), since l ( F ) is then also a free function inthe same class as F itself regarding operator concavity or monotonicity, soessentially the same techniques apply to them.2. Lifted hypographs are matrix convex
We use P ( E ) to denote the cone of invertible positive and S ( E ) ⊃ P ( E )the self-adjoint bounded linear operators over the Hilbert space E , so that P ( C ) denotes the positive reals. Definition 2.1 (cf. [1]) . A real function f : P ( C ) k P ( C ) is said to be globally operator monotone , if for any X ≤ Y ∈ CP ( E ) k , dim( E ) < + ∞ wehave f ( X ) ≤ f ( Y ), where CP ( E ) k denotes the set of pairwise commuting MIKL ´OS P ´ALFIA k -tuples of invertible positive bounded linear operators on E , and f ( X ) := U ∗ f (Λ) U where X = U ∗ Λ U denotes the joint spectral decomposition of thepairwise commuting tuple X and f (Λ) := L ki =1 f ( { Λ } ii , . . . , { Λ k } ii ). Proposition 2.1.
Let f : P ( C ) k P ( C ) be a globally operator monotonefunction. Then for any isometry W : E K between finite dimensionalHilbert spaces E, K and any X ∈ CP ( K ) k such that W ∗ XW ∈ CP ( E ) k wehave (3) W ∗ f ( X ) W ≤ f ( W ∗ XW ) . In particular f is concave and continuous as a real function.Proof. Let U := (cid:20) W ( I − W W ∗ ) / ( I − W ∗ W ) / − W ∗ (cid:21) = (cid:20) W ( I − W W ∗ ) / − W ∗ (cid:21) denote a unitary dilation of the isometry W , i.e. U ∗ U = U U ∗ = I on E ⊕ K .Now choose arbitrary A ∈ CP ( E ) k and let C := ( I − W W ∗ ) / X ( I − W W ∗ ) / + W AW ∗ . Then we have U ∗ (cid:20) X A (cid:21) U = (cid:20) W ∗ XW W ∗ X ( I − W W ∗ ) / ( I − W W ∗ ) / XW C (cid:21) . Set D := − W ∗ X ( I − W W ∗ ) / and notice that for any given ǫ > (cid:20) W ∗ XW + ǫI
00 2 zI (cid:21) − U ∗ (cid:20) X A (cid:21) U ≥ (cid:20) ǫI DD ∗ zI (cid:21) if zI ≥ C = ( I − W W ∗ ) / X ( I − W W ∗ ) / + W AW ∗ for z ∈ P ( C ) k . Thelast k -tuple of block matrices above is positive semi-definite if z i I ≥ ǫ D i D ∗ i for all 1 ≤ i ≤ k . So, for sufficiently large positive k -tuple z we have U ∗ (cid:20) X A (cid:21) U ≤ (cid:20) W ∗ XW + ǫI
00 2 zI (cid:21) . For such z >
0, by the global operator monotonicity of f we get f (cid:18) U ∗ (cid:20) X A (cid:21) U (cid:19) ≤ (cid:20) f ( W ∗ XW + ǫI ) 00 f (2 z ) I (cid:21) . We also have that f (cid:18) U ∗ (cid:20) X A (cid:21) U (cid:19) = U ∗ (cid:20) f ( X ) 00 f ( A ) (cid:21) U = W ∗ f ( X ) W W ∗ f ( X )( I − W W ∗ ) / ( I − W W ∗ ) / f ( X ) W ( I − W W ∗ ) / f ( X )( I − W W ∗ ) / ++ W f ( A ) W ∗ , hence we obtain that(4) W ∗ f ( X ) W ≤ f ( W ∗ XW + ǫI ) . NALYTIC LIFTS OF OPERATOR CONCAVE FUNCTIONS 5
Now since f is monotone, f ( X + ǫI ) for ǫ > f ( X ), thus the right limit f + ( X ) := inf ǫ> f ( X + ǫI ) = lim ǫ → f ( X + ǫI )exists for all X ∈ CP ( K ) k and f + is a multivariable real function. Hencefor any ǫ >
0, using (4) with W = [ λ / I, (1 − λ ) / I ] where λ ∈ [0 ,
1] and X = (cid:20) a b (cid:21) with a, b ∈ CP ( C ) k , we obtain λf + ( a )+(1 − λ ) f + ( b ) ≤ λf ( a + ǫI )+(1 − λ ) f ( b + ǫI ) ≤ f ( λa +(1 − λ ) b +2 ǫI ) . Taking the limit ǫ →
0+ we obtain that λf + ( a ) + (1 − λ ) f + ( b ) ≤ f + ( λa + (1 − λ ) b ) , i.e. the real function f + is concave, thus continuous. Also f ( X ) ≤ f + ( X ) ≤ f ( X + ǫI )for all ǫ >
0. Since f is monotone increasing, we have f + ( X − ǫI ) ≤ f ( X ) ≤ f + ( X ) , and since f + is continuous we get that f = f + by taking the limit ǫ → ǫ →
0+ in (4) proving (3). (cid:3)
Corollary 2.2.
Under the assumptions of Proposition 2.1, (3) remains truewith contractions W : E K , that is k W k ≤ .Proof. If k W k ≤
1, then ( I − W ∗ W ) / is not necessarily 0. However theblock operator matrix U is still unitary and we can choose A = 0 in theproof with f (0) := 0 since f | P ( C ) k >
0, and the same block operator matrixargumentation goes through, leading to (3). (cid:3)
Definition 2.2 (Matrix convex hull) . Given a disjoint union of sets ( C ( E ))for each Hilbert space dim( E ) < + ∞ , its matrix convex hull, denoted as(co mat C ( E )) for each Hilbert space dim( E ) < + ∞ , is defined as the smallestmatrix convex set containing ( C ( E )). By Proposition 2.6 in [9], it is knownthat if ( C ( E )) is closed under direct sums, thenco mat C ( E ) := { V ∗ XV : X ∈ C ( K ) , dim( K ) < + ∞ , V : E K an isometry } . Notice that in general convex combinations are themselves matrix convexcombinations, since (1 − λ ) A + λB = V ∗ ( A ⊕ B ) V , where V = (cid:20) (1 − λ ) λ (cid:21) is an isometry for λ ∈ [0 , Lemma 2.3.
We have co mat CP k ( E ) = P k ( E ) for each dim( E ) < ∞ . MIKL ´OS P ´ALFIA
Proof.
Let A ∈ P k ( E ) with dim( E ) = n , so that A i ≥ ǫI E for a small enough ǫ >
0. In spectral decomposition form, we also have A = n X i =1 a i u i u ∗ i , . . . , n X i =1 a ki u ki u k ∗ i ! where u il are eigenvectors and a il are the corresponding eigenvalues of A i . Bylooking at the above form, it is clear that each A i is written as a finite convexcombination of rank one matrices of the form cuu ∗ + dI E where c, d ∈ P ( C )and u ∈ E , k u k = 1. We can write cuu ∗ = ( e ⊗ u ∗ ) ∗ ce e ∗ ( e ⊗ u ∗ ), anisometric inclusion of ce e ∗ ∈ P ( C ), so by extending e ⊗ u ∗ into a unitary U , we get that cuu ∗ + dI E = U ∗ (( c + d ) ⊕ ( ⊕ nj =2 d )) U , itself a matrix convexcombination. Thus it follows that A is actually a finite convex combinationof elements where only one coordinate of the k -tuple is not necessarily equalto zI E for some z ∈ P ( C ), and such elements are also finite matrix convexcombinations of elements of P k ( C ). (cid:3) Now consider the hypographhypo( f ) := (hypo( f )( K )) := ( { ( Y, X ) ∈ S ( K ) × CP ( K ) k : Y ≤ f ( X ) } )of a real function f : P ( C ) k P ( C ) for dim( K ) < + ∞ . We should thinkabout the real function f and its hypo( f ) as a partially defined free functionand its partially defined hypograph. Theorem 2.4.
Let f : P ( C ) k P ( C ) be a real function. Then f is globallyoperator monotone if and only if for each ( Y, X ) ∈ co mat (hypo( f ))( E ) with dim( E ) < + ∞ and X ∈ CP ( E ) k we have that Y ≤ f ( X ) .Proof. Suppose first that f is globally operator monotone. Let ( Y, X ) ∈ co mat (hypo( f )( E )) with dim( E ) < + ∞ and X ∈ CP ( E ) k . Then by thedefinition of the matrix convex hull there exists an isometry W : E K between the finite dimensional Hilbert spaces E, K and a ( y, x ) ∈ S ( K ) × CP ( K ) k with y ≤ f ( x ) such that Y = W ∗ yW and X = W ∗ xW . Then itfollows that Y ≤ W ∗ f ( x ) W , so by Proposition 2.1 we get that W ∗ f ( x ) W ≤ f ( W ∗ xW ) = f ( X ).To see the converse implication, consider the function h v ( X ) := sup { v ∗ Y v : (
Y, X ) ∈ co mat (hypo( f ))( E ) } for v ∈ E and X ∈ P ( E ) k . Since co mat (hypo( f )) is matrix convex, wehave that co mat (hypo( f ))( E ) is a convex set, it follows that h v is a boundedfrom below concave, thus by Proposition 3.5.4 in [19], norm-continuous realvalued function. Moreover by the assumption if X ∈ CP ( E ) k then for each( Y, X ) ∈ co mat (hypo( f ))( E ) we have that Y ≤ f ( X ), thus we must have h v ( X ) = v ∗ f ( X ) v . It is also clear by the definition of co mat (hypo( f ))( E )that h v ≥ P ( E ) k by Lemma 2.3. Now NALYTIC LIFTS OF OPERATOR CONCAVE FUNCTIONS 7 assume that
A, B ∈ CP ( E ) k and A < B . Let t ∈ (0 , tB = tA + (1 − t ) (cid:20) t − t ( B − A ) (cid:21) where t − t ( B − A ) ∈ P ( E ) k . Thus the concavity and positivity of h v yields h v ( tB ) ≥ th v ( A ) + (1 − t ) h v (cid:18) t − t ( B − A ) (cid:19) ≥ th v ( A ) , so letting t → − in the above implies h v ( B ) ≥ h v ( A ). Then again usingthe continuity of h v we obtain h v ( B ) ≥ h v ( A ) as well, when we have B ≥ A .From this, since v ∈ E was arbitrary, we obtain f ( B ) ≥ f ( A ) as desired. (cid:3) Free analytic lifts through models
Let A denote a Banach space in this section. All tensor products in thesubsequent sections are understood to be projective, see chapter IV.2. in[28] for more information, however this particular choice of cross-norm doesnot make an essential difference in the calculations. For a vector space V the map I V is understood to be the identity homomorphism. The first resultcharacterizes concavity through isometric conjugations. Such maps are alsocalled Jensen-type maps if they satisfy a similar reversed inequality [18].The characterizing inequality (5) will play a key role in this section. Proposition 3.1.
Let ( D ( E )) with D ( E ) ⊆ A ⊗ B ( E ) denote a self-adjointmatrix convex set and let F : D ( E )
7→ B ( E ) be a free function. Then F isoperator concave if and only if for each isometry W : E K and X ∈ D ( K ) we have (5) F (( I A ⊗ W ∗ ) X ( I A ⊗ W )) ≥ W ∗ F ( X ) W. Proof. ( ⇒ ) : Let U := (cid:20) W ( I − W W ∗ ) / ( I − W ∗ W ) / − W ∗ (cid:21) = (cid:20) W ( I − W W ∗ ) / − W ∗ (cid:21) denote the unitary dilation of the isometry W , i.e. U ∗ U = U U ∗ = I on E ⊕ K . Now choose arbitrary A ∈ D ( E ) and let C := ( I A ⊗ ( I − W W ∗ ) / ) X ( I A ⊗ ( I − W W ∗ ) / ) + ( I A ⊗ W ) A ( I A ⊗ W ∗ ) . Then we have( I A ⊗ U ∗ ) (cid:20) X A (cid:21) ( I A ⊗ U )= (cid:20) ( I A ⊗ W ∗ ) X ( I A ⊗ W ) ( I A ⊗ W ∗ ) X ( I A ⊗ ( I − W W ∗ ) / )( I A ⊗ ( I − W W ∗ ) / ) X ( I A ⊗ W ) C (cid:21) . MIKL ´OS P ´ALFIA
Also notice that12 ( I A ⊗ U ∗ ) (cid:20) X A (cid:21) ( I A ⊗ U ) + 12 (cid:20) I − I (cid:21) ( I A ⊗ U ∗ ) (cid:20) X A (cid:21) × ( I A ⊗ U ) (cid:20) I − I (cid:21) = (cid:20) ( I A ⊗ W ∗ ) X ( I A ⊗ W ) 00 C (cid:21) . Then we have (cid:20) F (( I A ⊗ W ∗ ) X ( I A ⊗ W )) 00 F ( C ) (cid:21) = F (cid:18)(cid:20) ( I A ⊗ W ∗ ) X ( I A ⊗ W ) 00 C (cid:21)(cid:19) = F (cid:18)
12 ( I A ⊗ U ∗ ) (cid:20) X A (cid:21) ( I A ⊗ U )+ 12 (cid:20) I − I (cid:21) ( I A ⊗ U ∗ ) (cid:20) X A (cid:21) ( I A ⊗ U ) (cid:20) I − I (cid:21)(cid:19) ≥ F (cid:18) ( I A ⊗ U ∗ ) (cid:20) X A (cid:21) ( I A ⊗ U ) (cid:19) + 12 F (cid:18)(cid:20) I − I (cid:21) ( I A ⊗ U ∗ ) (cid:20) X A (cid:21) ( I A ⊗ U ) (cid:20) I − I (cid:21)(cid:19) = 12 ( I A ⊗ U ∗ ) (cid:20) F ( X ) 00 F ( A ) (cid:21) ( I A ⊗ U )+ 12 (cid:20) I − I (cid:21) ( I A ⊗ U ∗ ) (cid:20) F ( X ) 00 F ( A ) (cid:21) ( I A ⊗ U ) (cid:20) I − I (cid:21) = ( I A ⊗ W ∗ ) F ( X )( I A ⊗ W ) 00 ( I A ⊗ ( I − W W ∗ ) / ) F ( X ) × ( I A ⊗ ( I − W W ∗ ) / )+( I A ⊗ W ) F ( A )( I A ⊗ W ∗ ) . Thus (5) follows.( ⇐ ) : For t ∈ [0 ,
1] let W = (cid:20) (1 − t ) / I E t / I E (cid:21) so that W ∗ W = I E , anisometry. Let X, Y ∈ D ( E ). Then by (5) we have F ((1 − t ) X + tY ) = F (cid:18) ( I A ⊗ W ∗ ) (cid:20) X Y (cid:21) ( I A ⊗ W ) (cid:19) ≥ ( I A ⊗ W ∗ ) F (cid:18)(cid:20) X Y (cid:21)(cid:19) ( I A ⊗ W )= ( I A ⊗ W ∗ ) (cid:20) F ( X ) 00 F ( Y ) (cid:21) ( I A ⊗ W )= (1 − t ) F ( X ) + tF ( Y ) . (cid:3) NALYTIC LIFTS OF OPERATOR CONCAVE FUNCTIONS 9
Corollary 3.2.
Under the assumptions of Proposition 3.1 if also ∈ D ( C ) and F (0) ≥ , then the equivalence in Proposition 3.1 remains true withcontractions W : E K in (5) , that is k W k ≤ .Proof. Only the ( ⇒ ) implication in Proposition 3.1 requires further consid-eration, since if k W k ≤ I − W ∗ W ) / is not necessarily 0.However the block operator matrix U is still unitary and since 0 ∈ D ( C ), wecan choose A = 0 in the proof of ( ⇒ ) and the same block operator matrixargumentation goes through leading to (5). (cid:3) The following lemma has been proved and used by a number of authorsbefore. For its proof we refer to [7, 20].
Lemma 3.3 (Lemma 3.6. [20]) . Suppose F is a convex set of weak- ∗ con-tinuous affine linear mappings f : B +1 ( E ) ∗ R with respect to a duality. Iffor each f ∈ F there exists a T ∈ B +1 ( E ) ∗ such that f ( T ) ≥ , then thereexists a T ∈ B +1 ( E ) ∗ such that f ( T ) ≥ for every f ∈ F . Lemma 3.4.
Let D = ( D ( E )) be a matrix convex set, where D ( E ) ⊆A ⊗ B ( E ) and ∈ D ( C ) . Let a linear functional Λ :
A ⊗ B ( N ) R begiven for a fixed N . If Λ( X ) ≤ for each X ∈ D ( N ) , then there exists a T ∈ B +1 ( N ) ∗ such that for each Hilbert space E , and each Y ∈ D ( E ) andeach contraction V : N E we have Λ(( I A ⊗ V ∗ ) Y ( I A ⊗ V )) ≤ T ( V ∗ V ) . Proof.
For a Hilbert space K , a point Y ∈ D ( K ) and a V : N K contraction, define f Y,V : B +1 ( N ) ∗ R by f Y,V ( T ) := T ( V ∗ V ) − Λ(( I A ⊗ V ∗ ) Y ( I A ⊗ V )) . We claim that the collection F := { f Y,V : Y, V } is a convex set. Let λ i ≥ ≤ i ≤ n for a fixed integer n and let P ni =1 λ i = 1. Also let ( Y i , V i )be given where Y i ∈ D ( K i ) for a Hilbert space K i and V i : N K i bea contraction for each 1 ≤ i ≤ n . Let Z := ⊕ ni =1 Y i and let F denote thecolumn operator matrix with entries √ λ i V i . Then Z ∈ D ( ⊕ K i ) and F ∗ F = n X i =1 λ i V ∗ i V i ≤ n X i =1 λ i I = I. By definition n X i =1 λ i ( I A ⊗ V ∗ i ) Y i ( I A ⊗ V i ) = ( I A ⊗ F ∗ ) Z ( I A ⊗ F )and n X i =1 λ i T ( V ∗ i V i ) = T ( F ∗ F ) for T ∈ B +1 ( N ) ∗ . Hence n X i =1 λ i f Y i ,V i ( T ) = f Z,F ( T ) . If V has operator norm 1, by Proposition II.6.3.3. in [4] we can choose anorming state γ ∈ B +1 ( N ) ∗ so that1 = k V k = γ ( V ∗ V ) . Then for T = γ it follows that f Y,V ( T ) = T ( V ∗ V ) − Λ(( I A ⊗ V ∗ ) Y ( I A ⊗ V )) = 1 − Λ(( I A ⊗ V ∗ ) Y ( I A ⊗ V )) . Since ( I A ⊗ V ∗ ) Y ( I A ⊗ V ) ∈ D ( N ), the right hand side above is nonnegative.If the operator V does not have norm one, we can rescale it to have norm 1and follow the same argument to show that f Y,V ( T ) ≥
0. So, for each f Y,V there exists a T ∈ B +1 ( N ) ∗ such that f Y,V ( T ) ≥
0, moreover each f Y,V isweak- ∗ continuous. Thus, by Lemma 3.3 there exists a T ∈ B +1 ( N ) ∗ suchthat f Y,V ( T ) ≥ Y and V . (cid:3) Similarly to the finite dimensional case of Definition 2.2, given a disjointunion of sets ( C ( E ) ⊆ A ⊗ B ( E )) for each Hilbert space E closed underdirect sums, its matrix convex hull is given asco mat C ( E ) := [ K a Hilbert space { V ∗ XV : X ∈ C ( K ) , V : E K an isometry } . If 0 ∈ C ( C ) then we also haveco mat C ( E ) = [ K a Hilbert space { V ∗ XV : X ∈ C ( K ) , V : E K, k V k ≤ } . Given a collection of sets ( D ( E ) ⊆ A ⊗ B ( E )) closed under direct sumsand a collection of functions F : D ( E )
7→ B ( E ) preserving direct sums, weconsider its hypographhypo( F ) := (hypo( F )( E )) := ( { ( Y, X ) ∈ B ( E ) × D ( E ) : Y ≤ f ( X ) } ) . Proposition 3.5.
Let a collection of self-adjoint sets ( D ( E ) ⊆ A ⊗ B ( E )) closed under direct sums and a collection of functions F : D ( E )
7→ B ( E ) preserving direct sums be given. Then for each isometry W : E K and X ∈ D ( K ) such that ( I A ⊗ W ∗ ) X ( I A ⊗ W ) ∈ D ( E ) we have that (6) F (( I A ⊗ W ∗ ) X ( I A ⊗ W )) ≥ W ∗ F ( X ) W, if and only if for each ( Y, X ) ∈ co mat (hypo( F ))( E ) with X ∈ D ( E ) we havethat Y ≤ F ( X ) .Moreover if ∈ D ( C ) and F (0) ≥ then the statement holds with con-tractions W : E K in (6) .Proof. A straightforward argumentation similar to the proof of Theorem 2.4based on the expression of the matrix convex hull. (cid:3)
NALYTIC LIFTS OF OPERATOR CONCAVE FUNCTIONS 11
Remark 3.1.
For unitary W we have that W − is also an isometry, thususing (6) twice, we can see that in this special case (6) holds with equality.In [18] it was shown that if a map satisfies (6) for all contractions W andconvexity (thus called a map of Jensen-type, see Definition 1.1. [18]), then itpreserves direct sums since the proof of Lemma 3.1. (i) in [18] goes through.Their proof carries over to our case with minor modifications if the domain( D ( E )) is matrix convex, thus in this case the direct sum invariance of F inProposition 3.5 can be dropped. Proposition 3.6.
Let ( D ( E )) ∋ and F be as in Proposition 3.5 with F | D > . Assume that co mat ( D )( E ) has nonempty interior for each E .Let N be a Hilbert space. Then for each interior point A ∈ D ( N ) andeach unit vector v ∈ N there exists a completely bounded affine linear map L F,A,v : ( B ( E ) , A ⊗ B ( E ))
7→ B ( N ) ∗ ⊗ B ( E ) given as L F,A,v ( Y, X ) := T ( F, A, v ) ⊗ I E − vv ∗ ⊗ Y + Λ F,A,v ( X ) , where ≤ T ( F, A, v ) ∈ B ( N ) ∗ and Λ F,A,v : A 7→ B ( N ) ∗ is a self-adjointcompletely bounded linear map, such that (a) T ( F, A, v )( I N ) = v ∗ F ( A ) v − Λ F,A,v ( A ) and there exists ǫ > suchthat (1+ ǫ ) A ∈ co mat ( D )( N ) and − Λ F,A,v ( A ) ≤ v ∗ F ( A ) v − v ∗ F ((1+ ǫ ) A ) vǫ ; (b) For all ( Y, X ) ∈ hypo( F ) we have L F,A,v ( Y, X ) ≥ ; (c) γ ∗ L F,A,v ( F ( A ) , A ) γ = 0 where γ = I N ; (d) For every X in the interior of co mat ( D )( E ) there exists an ǫ > such that h V, L
F,A,v (0 , X ) V i ≥ ǫT ( F, A, v )( V ∗ V ) .Proof. Define the real valued function h v : co mat ( D )( N ) R as h v ( X ) :=sup { v ∗ Y v : (
Y, X ) ∈ co mat (hypo( F )( N )) } . Since co mat (hypo( F )) is matrixconvex, we have that co mat (hypo( F ))( E ) is a convex set, also F | D ≥
0, soit follows that h v is a bounded from below concave function on the realBanach-space of the self-adjoint part of A ⊗ B ( N ), thus norm-continuous byProposition 3.5.4 in [19]. Moreover by Proposition 3.5 if X ∈ D ( N ) then foreach ( Y, X ) ∈ co mat (hypo( F ))( N ) we have that Y ≤ F ( X ), thus we musthave h v ( X ) = v ∗ F ( X ) v for all X ∈ D ( N ).It follows from the supporting hyperplane version of the Hahn-Banach the-orem, more precisely Theorem 7.12 and 7.16 [2], that the norm-continuousconvex function g ( X ) := − h v ( X ) has a subgradient at each interior point ofits domain, thus at A . That is, there exists a self-adjoint continuous linearfunctional λ ∈ ( A ⊗ B ( N )) ∗ such that(7) h v ( X ) − h v ( A ) ≤ λ ( X − A )for X ∈ co mat ( D )( N ) and for X = A we have equality. Now let ( Y, X ) ∈ co mat (hypo( F ))( N ). Then it follows from (7) and the definition of h v that(8) v ∗ Y v − λ ( X ) ≤ h v ( A ) − λ ( A ) . Notice that by the assumption we have 0 ∈ D ( C ) and F >
0, thus 0 ∈ co mat (hypo( F ))( E ) for any Hilbert space E and h v ( A ) − λ ( A ) >
0. Thus the linear functional Λ(
Y, X ) := h v ( A ) − λ ( A ) ( v ∗ Y v − λ ( X )) satisfies Λ( Y, X ) ≤ Y, X ) ∈ co mat (hypo( F ))( N ), also co mat (hypo( F )) is a matrix convex set.Thus by Lemma 3.4 there exists a 0 ≤ T ∈ B +1 ( N ) ∗ such that T ( I N ) = 1and for any contraction V : N E we have(9) 0 ≤ ( h v ( A ) − λ ( A )) T ( V ∗ V ) − v ∗ V ∗ Y V v + λ (( I A ⊗ V ∗ ) X ( I A ⊗ V ))for any ( Y, X ) ∈ hypo( F )( E ), moreover by (7) choosing V = I N and Y = F ( A ) , X = A we get(10) 0 = ( h v ( A ) − λ ( A )) T ( I N ) − v ∗ F ( A ) v + λ (( I A ⊗ I N ) A ( I A ⊗ I N )) . Now define T ( F, A, v ) := ( h v ( A ) − λ ( A )) T. Next, by assumption co mat ( D )( E ) contains an open neighborhood of 0 in A ⊗ B ( E ), thus there exists a ˆ ρ > B (0 , ˆ ρ ) ⊆ co mat ( D )( E ). Thusby (9) and norm continuity we have that0 ≤ T ( F, A, v )( V ∗ V ) + λ (( I A ⊗ V ∗ ) X ( I A ⊗ V ))for X ∈ B (0 , ˆ ρ ) where B (0 , ˆ ρ ) denotes the norm closure of B (0 , ˆ ρ ). Moreover X ∈ B (0 , ˆ ρ ) if and only if − X ∈ B (0 , ˆ ρ ), so we also have0 ≤ T ( F, A, v )( V ∗ V ) − λ (( I A ⊗ V ∗ ) X ( I A ⊗ V )) . Then from the above it follows that(11) − T ( F, A, v )( V ∗ V ) ≤ λ (( I A ⊗ V ∗ ) X ( I A ⊗ V )) ≤ T ( F, A, v )( V ∗ V )for X ∈ B (0 , ˆ ρ ). This together with Theorem IV.2.3. [28] ensure that thetranspose map Λ F,A,v : A 7→ B ( N ) ∗ of λ ∈ ( A ⊗ B ( N )) ∗ is completelybounded and self-adjoint since λ is a self-adjoint linear functional.Consider the Hilbert space B ( N, E ) T ( F,A,v ) that we obtain by completingthe quotient space B ( N, E ) / { V ∈ B ( N, E ) : T ( F, A, v )( V ∗ V ) = 0 } equippedwith the positive definite Hermitian form(12) h W, V i T ( F,A,v ) := T ( F, A, v )( W ∗ V )for W, V ∈ B ( N, E ). Then the right hand side of (9) determines a quadraticform in V ∈ B ( N, E ), which gives rise to the densely defined symmetriclinear operator h W, L
F,A,v ( Y, X ) V i T ( F,A,v ) := T ( F, A, v )( W ∗ V ) − v ∗ W ∗ Y V v + λ (( I A ⊗ W ∗ ) X ( I A ⊗ V ))for V, W ∈ B ( N, E ) , Y ∈ B ( E ) and X ∈ A ⊗ B ( E ). Then (b) and (c) of theassertion follows from (9) and (10) respectively and they also yield the firstequality in (a).Furthermore inequality (11) ensures that L F,A,v is a completely boundedaffine linear map that admits the continuous linear extension L F,A,v ( Y, X )which then is a bounded self-adjoint operator acting on B ( N, E ) T ( F,A,v ) .To see the remaining parts of (a), first we realize that by assumption A is in the interior of co mat ( D )( N ), thus there exists an ǫ > NALYTIC LIFTS OF OPERATOR CONCAVE FUNCTIONS 13 X := (1 + ǫ ) A is in co mat ( D )( N ) as well. Then choosing Y = F ((1 + ǫ ) A ) , X = (1 + ǫ ) A , from (7) we calculate0 ≤ h v ((1 + ǫ ) A ) ≤ h v ( A ) − λ ( A ) + (1 + ǫ ) λ ( A ) ǫh v ( A ) − ǫλ ( A ) ≤ (1 + ǫ ) h v ( A ) − h v ((1 + ǫ ) A ) h v ( A ) − λ ( A ) ≤ h v ( A ) + h v ( A ) − h v ((1 + ǫ ) A ) ǫ , thus the last inequality in (a) follows, since T ( F, A, v )( I N ) = h v ( A ) − λ ( A ) . Now to see (d), by assumption V ∈ B ( N, E ) T ( F,A,v ) and X is in theinterior of co mat ( D )( E ), thus there exists an ǫ > B ( X, ǫ ) ⊆ co mat ( D )( E ). Then there exists an r >
1, such that rX ∈ B ( X, ǫ ). Let c := λ (( I A ⊗ V ∗ ) X ( I A ⊗ V )). Then by (9) we have that0 ≤ h V, ( L F,A,v (0 , rX ) V i = T ( F, A, v )( V ∗ V ) + rc, thus 0 < (cid:18) − r (cid:19) T ( F, A, v )( V ∗ V ) ≤ T ( F, A, v )( V ∗ V ) + c = T ( F, A, v )( V ∗ V ) + λ (( I A ⊗ V ∗ ) X ( I A ⊗ V ))= h V, L
F,A,v (0 , X ) V i (13)proving (d). (cid:3) Remark 3.2.
In order to allow locally convex vector spaces A in Proposi-tion 3.6, one needs to establish the continuity of h v . This can be done usingProposition 4.4. in [6] which generalizes Proposition 3.5.4 in [19] to locallyconvex vector spaces under the same assumptions.Let ( D ( E )) ∋ E anda dense set E ∈ { x ∈ E : k x k = 1 } define the auxiliary vector space H E, := M ( X,v ) ∈ ( D ( E ) ,E ) E and its completion H E with respect to the usual inherited direct sum innerproduct. We denote by I ( X,v ) ∈ B ( H E , E ) the isometry that equals to I E − vv ∗ on the ( X, v ) slot and 0 elsewhere.
Corollary 3.7.
Let ( D ( E )) ∋ and F be as in Proposition 3.5 with F | D > . Fix a Hilbert space E and an η > . Assume that co mat ( D )( E ) hasnonempty interior for E . Then there exists a vector e ∈ H E with k e k = 1 ,a completely bounded affine map L F : A ⊗ B ( E )
7→ B ( H E ) ∗ ⊗ B ( E ) given as L F ( X ) := T F ⊗ I E + Λ F ( X ) , where ≤ T F ∈ B ( H E ) ∗ and Λ F : A 7→ B ( H E ) ∗ is a self-adjoint completelybounded linear map that is completely absolutely continuous with respect to T F , such that (a) For all X ∈ co mat ( D )( E ) we have L F ( X ) ≥ ; (b) For all (1 + η ) X ∈ D ( E ) in the interior of co mat ( D )( E ) and v ∈ E we have (14) (cid:10) W, L F ( X )( I ( X,v ) + ve ∗ ) (cid:11) T F = e ∗ W ∗ F ( X ) v for all W ∈ B ( H E , E ) T F with the notation of (12) , and there existsan ǫ > such that h W, L F ( X ) W i T F ≥ ǫT F ( W ∗ W ) .Proof. Rewriting (14), it essentially becomes(15) T ( W ∗ ( I ( X,v ) + ve ∗ )) + λ (( I ⊗ W ∗ ) X ( I ⊗ ( I ( X,v ) + ve ∗ ))) = e ∗ W ∗ F ( X ) v for 0 ≤ T ∈ B ( H E ) ∗ , λ ∈ ( A⊗B ( H E )) ∗ and an e ∈ H E . If we equip B ( H E ) ∗ ,( A ⊗ B ( H E )) ∗ with their respective weak- ∗ topologies, then the set of alluniformly norm bounded ( T, λ ) that satisfies (15) for all v, W, X and a fixed e ∈ H E is closed in the product topology. The free function F preserves di-rect sums, so for any finite set of points { ( X , v ) , . . . , ( X n , v n ) } with v i ∈ E and (1 + η ) X i ∈ D ( E ) in the interior of co mat ( D )( E ), by applying Propo-sition 3.6 with e = ⊕ ni =1 v i to the single point data set ( ⊕ ni =1 X i , ⊕ ni =1 v i ) itfollows that the set of all such ( T, λ ) is also nonempty. Moreover by (a) inProposition 3.6 we can assume that the norms of such
T, λ are uniformlybounded since (1+ η ) X is also in the interior of co mat ( D )( E ). After a changeof basis in H E we conclude that we can also choose e ∈ H E arbitrarily in(15). These norm bounded closed sets of ( T, λ ) are compact it their re-spective weak- ∗ topologies by Banach-Alaoglu, thus their product is alsocompact. Furthermore these closed compact sets of ( T, λ ) form a collectionindexed by (
X, v ) and thus can be partially ordered by inclusion within opensets of (
X, v ) and then this collection has the finite intersection property.Thus, for a fixed e ∈ H E , by compactness there exists a ( T F , λ F ) for which(15) holds for all (1 + η ) X ∈ D ( E ) in the interior of co mat ( D )( E ), v ∈ E and W ∈ B ( H E , E ) T F . Then L F is determined by the transpose of λ F , andthe positivity condition h W, L F ( X ) W i T F ≥ ǫT F ( W ∗ W ) follows from (13).Thus (a) and (b) are proved. (cid:3) Theorem 3.8 (Theorem 3 cf. [3]) . Let Z be a positive semi-definite lin-ear operator on a Hilbert space and S a subspace. Let the matrix of Z be partitioned as Z = (cid:20) Z Z Z Z (cid:21) with Z : S S , Z : S S ⊥ .Then ran( Z ) ⊂ ran( Z ) / and there exists a bounded linear operator C : S S ⊥ such that Z = ( Z ) / C and Z = (cid:20) Z − C ∗ C
00 0 (cid:21) + (cid:20) C ∗ Z ) / (cid:21) (cid:20) C ( Z ) / (cid:21) . NALYTIC LIFTS OF OPERATOR CONCAVE FUNCTIONS 15
The bounded positive semi-definite operator S S ( Z ) = Z − C ∗ C is calledthe shorted operator or Schur complement of Z . It satisfies S S ( Z ) ≤ Z andit is maximal among all self-adjoint operators X : S S such that X ≤ Z . Theorem 3.9.
Let ( D ( E )) ∋ and F be as in Proposition 3.5 with F | D > .Fix a Hilbert space E . Assume that co mat ( D )( E ) has nonempty interior for E . Then for each X ∈ D ( E ) in the interior of co mat ( D )( E ) we have (16) F ( X ) = ( e ⊗ I E ) S e ∗ ⊗ E ( L F ( X ))( e ∗ ⊗ I E ) where L F and e are as in Corollary 3.7 for an arbitrary, but sufficientlysmall fixed η > . Moreover the right hand side of (16) is well defined foreach interior point X ∈ co mat ( D )( E ) .Proof. By (b) in Corollary 3.7 the self-adjoint completely bounded linearmap L F ( X ) = T F ⊗ I E + Λ F ( X ) is strictly positive definite for all X in theinterior of co mat ( D )( E ) and is completely absolutely continuous with respectto 0 ≤ T F ∈ B ( H E ) ∗ . Thus by Theorem 3.8 its Schur complement pivotingon the subspace e ∗ ⊗ E of B ( H E , E ) T F exists. By the strict positivity of L F ( X ), the Z block of L F ( X ) in Theorem 3.8 has closed range and isinvertible. So, in (14) block Gaussian elimination applies and thus F ( X ) v = ( e ⊗ I E ) S e ∗ ⊗ E ( L F ( X ))( e ∗ ⊗ v )for each v ∈ E and (1+ η ) X ∈ D ( E ) in the interior of co mat ( D )( E ). Taylor’spower series formula ensures the uniqueness of analytic continuations to thewhole interior of co mat ( D )( E ) thus the assertion holds for η = 0 as well.This implies (16). (cid:3) The converse of the above also holds:
Theorem 3.10.
Let H a Hilbert space and e ∈ H with k e k = 1 be fixed. Leta completely bounded affine map L : A ⊗ B ( E )
7→ B ( H ) ∗ ⊗ B ( E ) be given as L ( X ) := T ⊗ I E + Λ( X ) , where ≤ T ∈ B ( H ) ∗ and Λ :
A 7→ B ( H ) ∗ is a self-adjoint completelybounded linear map that is completely absolutely continuous with respect to T . Then the function (17) F ( X ) = ( e ⊗ I E ) S e ∗ ⊗ E ( L ( X ))( e ∗ ⊗ I E ) is well defined and analytic for each X ∈ { Y ∈ A⊗B ( E ) : L ( ℜ ( Y )) > } andsatisfies the assumptions of Proposition 3.5 with D ( E ) := { Y ∈ A ⊗ B ( E ) : L ( Y ) ≥ } , in particular (6) holds.Proof. By the positivity assumption L ( ℜ ( X )) > L ( X ) is the maximal in the positive definite order on thesubspace e ⊗ E among those which are dominated by L ( X ) on the subspace e ⊗ E over its matrix convex domain ( D ( E )). From this maximality property concavity of F ( X ) readily follows. Thus by Proposition 3.1 F ( X ) satisfies(6). (cid:3) Now as a combination of Theorem 3.9 and Theorem 2.4 we obtain thefollowing result establishing the analytic lifts for globally monotone functionson P ( C ) k . Then from this, further considerations prove the same for anyother rectangular domain in R k . We use the notations Π( E ) := { X ∈ B ( E ) : ℑ ( X ) > } and Π( E ) := { X ∈ B ( E ) : ℑ ( X ) ≥ } , also Π( E ) ∗ := { X ∈B ( E ) : ℑ ( X ) < } and Π( E ) ∗ := { X ∈ B ( E ) : ℑ ( X ) ≤ } . Theorem 3.11.
Let f : P ( C ) k P ( C ) be a real function. Then the follow-ing are equivalent: (a) f is globally operator monotone; (b) f has a free analytic extension f : P ( E ) k P ( E ) that is operatormonotone; (c) There exists a Hilbert space K , a vector e ∈ K , ≤ B i ∈ B ( K ) , ≤ i ≤ k with B ≥ P ki =1 B i such that for all X ∈ CP ( E ) k we have (18) f ( X ) = ( e ⊗ I E ) S e ∗ ⊗ E ( L f ( X ))( e ∗ ⊗ I E ) where (19) L f ( X ) := B ⊗ I E + k X i =1 B i ⊗ ( X i − I E ) , (d) f has a free analytic continuation to Π( E ) k and to (Π( E ) ∗ ) k across P ( E ) k , mapping Π( E ) k to Π( E ) and (Π( E ) ∗ ) k to Π( E ) ∗ .Proof. First we prove that (a) implies (c). By Theorem 3.9 the represen-tation formula follows for the translated function g ( x ) := f ( x + 1) withdomain ( − , ∞ ) k whose matrix convex hull is D ( E ) := { X ∈ B ( E ) : X = X ∗ , X i ≥ − I E } k which contains an open neighborhood of 0 for the oper-ator system A = C k . Since the domain contains arbitrarily large positiveoperators and by (a) of Corollary 3.7 we have L f ( X ) ≥
0, it follows thatΛ f ( X ) is completely positive, thus of the the form as in (19) with B i ≥ B ≥ P ki =1 B i as well, since L f ( tI ) ≥ t >
0. In a similar waywe show that (b) implies (c).Next we claim that (c) implies (a). Indeed, L f ( X ) is order preserving andthe maximality characterization of the Schur complement in Theorem 3.8ensures that the right hand side of (18) is an operator monotone function.In a similar way we also prove that (c) implies (b).That (c) implies (d) essentially follows from B i ≥ ℜ ( X i ) > ℑ ( L f ( X )) for arbi-trary ℑ ( X ) ∈ Π( E ) k and E implies the strict lower boundedness, thus theexistence of the inverse operator in the formula of the Schur complement,thus the Schur complement itself as an analytic function on Π k .Lastly, that (d) implies (b) can be found in the main theorem of [20]. (cid:3) NALYTIC LIFTS OF OPERATOR CONCAVE FUNCTIONS 17
At this point, much like as in [20], one can use M¨obius transformationsto transform the domain P ( C ) k into any open rectangle in R k to prove thatglobal monotonicity implies free analytic continuation for the function tothe whole matrix convex hull of its domain, thus arriving at its operatormonotone non-commutative lift. Remark 3.3.
In the above corollaries we assumed A = C k , however onecan work out similar results in the same way for any operator system A .4. Representation of operator means of probability measures
Let P ( P ( E )) denote the set of fully supported Borel probability measureson the complete metric space ( P ( E ) , d ∞ ) where E is a Hilbert space and d ∞ ( A, B ) = k log( A − / BA − / ) k denotes the Thompson metric [12, 21].Let P ∞ ( P ( E )) ⊂ P ( P ( E )) denote the subset of probability measures withbounded support. For a µ ∈ P ( P ( E )) the support supp( µ ) is a separableclosed subset of P ( E ) and it has full measure µ (supp( µ )) = 1. Also notethat the relative operator norm topology on P ( E ) agrees with the metrictopology of d ∞ , for this and further references see [21]. Proposition 4.1.
The collection of sets ( P ∞ ( P ( E ))) indexed by E is a self-adjoint matrix convex set. In particular P ∞ ( P ( E )) embeds into L ∞ ([0 , , λ ) + ⊗ P ( E ) , the strictly positive cone of the projective tensor product L ∞ ([0 , , λ ) ⊗B ( E ) .Proof. Let µ ∈ P ∞ ( P ( E )). Then supp( µ ) is separable and closed, thus µ is concentrated on the complete Polish space (supp( µ ) , d ∞ ). Thus bythe Skorokhod representation Theorem 8.5.4. in [5], there exists a Borelmap ξ µ : [0 , supp( µ ) ⊆ P ( E ) such that µ = ( ξ µ ) ∗ λ . Since supp( µ )is a bounded subset of P ( E ), we have that ess sup k ξ µ k < ∞ . Thus ξ µ ∈ L ∞ ([0 , , λ, P ( E )) ⊆ L ∞ ([0 , , λ, B ( E )) where L ∞ ([0 , , λ, B ( E )) = { f : [0 ,
7→ B ( E ) , f is strongly measurable , ess sup( f ) < ∞} . Next, we claim that L ∞ ([0 , , λ, B ( E )) ≃ L ∞ ([0 , , λ ) ⊗ B ( E ) as von Neu-mann algebras, where the latter is the projective tensor product. This fol-lows from the same argument leading to Proposition 12.5. in [25] showingthat C ( X ) ⊗ A ≃ C ( X, A ) for any C ∗ -cross norm with the isomorphism φ : P ni =1 f i ⊗ a i −→ P ni =1 f i ( t ) a i for f i ∈ C ( X ) and a i ∈ A , where A is a C ∗ -algebra and X is a compact Hausdorff space. Now it is straightforwardto see that the Borel map constructed above ξ µ is a strictly positive elementof the positive cone of L ∞ ([0 , , λ ) ⊗ B ( E ) which is L ∞ ([0 , , λ ) + ⊗ P ( E )by Lemma 2.2. in [27], where L ∞ ([0 , , λ ) + = { f ≥ f ∈ L ∞ ([0 , , λ ) } .Now the positive cone ( L ∞ ([0 , , λ ) + ⊗ P ( E )) is closed under direct sumsand isometric conjugations, thus ( L ∞ ([0 , , λ ) + ⊗ P ( E )) is an (open) matrixconvex set. Moreover for any ξ ∈ L ∞ ([0 , , λ ) + ⊗ P ( E ) the pushforward( ξ ) ∗ λ ∈ P ∞ ( P ( E )), so ( P ∞ ( P ( E ))) is matrix convex as well. (cid:3) Remark 4.1.
Notice that L ∞ ([0 , , λ ) is an injective von Neumann algebra,or in other words L ∞ ([0 , , λ ) is nuclear as a C ∗ -algebra. Nuclearity ensuresthat all C ∗ -cross norms on ( L ∞ ([0 , , λ ) + ⊗ P ( E )) are equivalent. So inparticular the projective and injective C ∗ -cross norms are the same. Formore details see for example Chapter IV in [28]. Remark 4.2.
In Proposition 4.1 the embedding of ( P ∞ ( P ( E ))) into thecone L ∞ ([0 , , λ ) + ⊗ P ( E ) does not appear to be injective. Elements of L ∞ ([0 , , λ ) + ⊗ P ( E ) can be classified into equivalence classes by almostsure identification with elements of ( P ∞ ( P ( E ))).A set U ⊆ P ( E ) is upper if X ≤ Y ∈ P ( E ) and X ∈ U imply that Y ∈ U . Definition 4.1 (Stochastic order, cf. [11]) . For µ, ν ∈ P ( P ( E )) the stochas-tic partial order µ ≤ ν is defined by requiring µ ( U ) ≤ ν ( U ) for all closedupper sets U ⊆ P ( E ).The following result is essentially due to Strassen for Polish spaces witha closed partial order. It can be found in a suitable form as Theorem 1 in[16]. Theorem 4.2.
Let µ, ν ∈ P ∞ ( P ( E )) . Then the following are equivalent: (i) µ ≤ ν ; (ii) there exists ξ µ : [0 , supp( µ ) and ξ ν : [0 , supp( ν ) such that µ = ( ξ µ ) ∗ λ and ν = ( ξ ν ) ∗ λ with ξ µ ( t ) ≤ ξ ν ( t ) almost surely for all t ∈ [0 , . One might wonder for µ ∈ P ∞ ( P ( E )), ν ∈ P ∞ ( P ( K )) what is the correctway to define µ ⊕ ν ? In [12] the authors define it as the pushforward ( g ) ∗ ( µ × ν ) of the direct sum g ( A, B ) := A ⊕ B , and they show that their operatormeans preserve this direct sum. However in free function theory if we have n -tuples of operators X := ( X , . . . , X n ) ∈ B ( E ) n and Y := ( Y , . . . , Y n ) ∈B ( K ) n , their direct sum is element-wise, that is X ⊕ Y = ( X ⊕ Y , . . . , X n ⊕ Y n ). Free functions, including as well all operator means [20], preserve thisdirect sum. Notice that if we regard X, Y as discrete probability measures,that is X = P ni =1 1 n δ X i , Y = P ni =1 1 n δ Y i , then both definitions of X ⊕ Y aremeasures in P ( B ( E ⊕ K )) with marginals µ and ν , where the σ -algebra isinduced by the norm. Also operator means are permutation invariant, thatis the ordering of coordinates in ( X , . . . , X n ) does not matter [12, 20]. Thisand Remark 4.2 seem to suggest that we should allow some non-uniquenesswhen considering direct sums of measures. This leads to the following. Definition 4.2 (Direct sums of probability measures) . For µ ∈ P ∞ ( P ( E )), ν ∈ P ∞ ( P ( K )), let Γ( µ, ν ) ⊆ P ∞ ( P ( E ⊕ K )) denote the set of couplings of µ, ν , that is γ ∈ Γ( µ, ν ) if γ ( A × P ( K )) = µ ( A ) and γ ( P ( E ) × B ) = ν ( B ), inother words, elements of Γ( µ, ν ) have marginals µ, ν . Then µ ⊕ ν is definedto be the set Γ( µ, ν ). Thus in general, the direct sum of probability measuresis no longer uniquely determined. NALYTIC LIFTS OF OPERATOR CONCAVE FUNCTIONS 19
Notice that Γ( µ, ν ) is nonempty, since the product measure µ × ν ∈ Γ( µ, ν ). Also for any γ ∈ Γ( µ, ν ) we have that supp( γ ) ⊆ supp( µ ) × supp( ν ).We may regard operator means of finitely supported measures as a sequenceof functions satisfying the following. Definition 4.3 (Operator mean of discrete probability measures) . For each0 < n ∈ N and Hilbert space E let F n : P ( E ) n P ( E ) be an operatormonotone free function. Then we say that the sequence of functions F n isan operator mean if it satisfies the following1) For a permutation σ ∈ S n , F n ( X , . . . , X n ) = F n ( X σ (1) , . . . , X σ ( n ) );2) For 0 < k ∈ N , F nk ( X , . . . , X | {z } k times , . . . , X n , . . . , X n | {z } k times ) = F n ( X , . . . , X n ).In order to simplify notation, the subscript n will often be omitted, simplywriting F = F n for an operator mean.Notice that given an operator mean F = F n , it is automatically defined fordiscrete probability measures with rational weights by grouping together therepeated variables and applying 1), 2). It is known that operator concavityfor a free function F : P ( E ) n P ( E ) is equivalent to operator monotonicitywhich in turn is equivalent to (6) by an argument similar to the one inProposition 2.1, for more details see [20]. Proposition 4.3.
An operator mean F n : P ( E ) n P ( E ) preserves directsums of discrete probability measures with rational weights in the sense ofDefinition 4.2.Proof. Without loss of generality, let µ = P ni =1 1 n δ X i , ν = P ki =1 1 k δ Y i begiven. Then any µ ⊕ ν is supported on the set supp( µ ) × supp( ν ) = { X i ⊕ Y j : i ∈ { , . . . , n } , j ∈ { , . . . , k }} . Then using the direct sum invariance of F and grouping elements by 2), we obtain that F ( µ ⊕ ν ) = F ( µ ) ⊕ F ( ν ). (cid:3) In order to study operator means of general probability measures, insteadof considering the restrictive set of functions F : P ∞ ( P ( E )) P ( E ) weconsider first free functions of random variables, that is F : ( L ([0 , , λ ) + ⊗ P ( E )) P ( E ). Let S ([0 , , λ ) denote the set of simple functions on [0 , S ([0 , , λ ) is norm-dense in L p ([0 , , λ ) for 1 ≤ p ≤ + ∞ and the sameis true for S ([0 , , λ ) + ⊗ P ( E ) in L p ([0 , , λ ) + ⊗ P ( E ). Theorem 4.4.
Assume that F : S ([0 , , λ ) + ⊗ P ( E ) P ( E ) is free functionthat satisfies (6) . Then for each ≤ p ≤ + ∞ there exists a unique ˆ F p : L p ([0 , , λ ) + ⊗ P ( E ) P ( E ) extending F .Proof. In essence F can be regarded as a sequence of free functions indexedby 0 < n ∈ N for each Hilbert space E , that is F n : P ( E ) n P ( E ) satisfyingthe assumptions of Proposition 3.5. Then we can apply Corollary 3.7 so wehave (14) for each F n . Each F n is thus norm-continuous by Theorem 3.10, so F : S ([0 , , λ ) + ⊗ P ( E ) P ( E ) is relative norm-continuous with respect to the norm topology of L p ([0 , , λ ) + ⊗ P ( E ). Then, it admits a unique norm-continuous extension ˆ F p : L p ([0 , , λ ) + ⊗ P ( E ) P ( E ), since S ([0 , , λ ) + ⊗ P ( E ) is norm dense in L p ([0 , , λ ) + ⊗ P ( E ). (cid:3) Proposition 4.5.
Let F : ( S ([0 , , λ ) + ⊗ P ( E )) P ( E ) be an operatormonotone free function. Then F satisfies (6) .Proof. The same argument as in the proof of Theorem 2.4 applies for f := F : ( S ([0 , , λ ) + ⊗ P ( E )) P ( E ) where S ([0 , , λ ) + ⊗ P ( E ) is substituted forboth P ( C ) k and CP ( E ) k , so we get that for each ( Y, X ) ∈ co mat (hypo( f ))( E )we have that Y ≤ f ( X ). Note that in Theorem 2.4 dim( E ) < + ∞ isassumed, but actually does not affect its proof. The claim that for each( Y, X ) ∈ co mat (hypo( f ))( E ) we have that Y ≤ f ( X ) combined with theimplication ’ ⇐ ’ of Proposition 3.5 proves that F satisfies (6). A similarargument to this can also be found in [20]. (cid:3) By the density of S ([0 , , λ ) + in L ([0 , , λ ) + we immediately obtain: Corollary 4.6.
Let F : ( L ([0 , , λ ) + ⊗ P ( E )) P ( E ) be an operatormonotone free function. Then F satisfies (6) . Theorem 4.7.
Assume that the sequence of functions F n : P ( E ) n P ( E ) for < n ∈ N is an operator mean of discrete probability measures. Then ituniquely extends into a stochastic order preserving function ˆ F : P ∞ ( P ( E )) P ( E ) .Proof. Pairs of probability measures admit Skorokhod representations thatare order preserving by Theorem 4.2. Then through the Skorokhod repre-sentation we obtain a lift ˆ F : S ([0 , , λ ) + ⊗ P ( E ) P ( E ) representing thesequence of functions F n : P ( E ) n P ( E ), such that ˆ F is operator mono-tone. Then by Proposition 4.5 ˆ F satisfies (6), so existence and uniqueness ofthe extension ˆ F : L ∞ ([0 , , λ ) + ⊗ P ( E ) P ( E ) follows from Theorem 4.4.Then Corollary 3.7 applies to the translated function G ( X ) := ˆ F ( X + I )where I ( t ) := I E for all t ∈ [0 , L G ( X ) ≥ X + I ) ∈ L ∞ ([0 , , λ ) + ⊗ P ( E ), so it follows that 0 ≤ λ in (15),thus L G ( X ) is order preserving. By the maximal characterization of theSchur complement in Theorem 3.8 it follows that it is also order preserving.Thus ˆ F is also order preserving, i.e. operator monotone. Then its restrictionˆ F : P ∞ ( P ( E )) P ( E ) preserves the stochastic order. (cid:3) Corollary 4.8.
Let F : P ∞ ( P ( E )) P ( E ) be a stochastic order preservingfree function. Then there exists an operator monotone free function ˆ F : L ∞ ([0 , , λ ) + ⊗ P ( E ) P ( E ) that represents F and ˆ F ( X + I ) is of theform (15) where ≤ λ ∈ ( L ∞ ([0 , , λ ) ⊗ B ( H E )) ∗ and I ( t ) := I E for all t ∈ [0 , . In particular ˆ F ( X + I ) is given by (16) .Proof. Through the Skorokhod representation Theorem 4.2 we obtain thelift ˆ F : L ∞ ([0 , , λ ) + ⊗ P ( E ) P ( E ) which by Corollary 4.6 satisfies (6). NALYTIC LIFTS OF OPERATOR CONCAVE FUNCTIONS 21
Then Corollary 3.7 applies to the translated function G ( X ) := ˆ F ( X + I ).By (a) of Corollary 3.7 L G ( X ) ≥ X + I ) ∈ L ∞ ([0 , , λ ) + ⊗ P ( E ),it follows that 0 ≤ λ in (15) and (16) holds. (cid:3) Acknowledgment
This work was partly supported by the grant TUDFO/47138-1/2019-ITM of the Ministry for Innovation and Technology, Hungary; the Hungar-ian National Research, Development and Innovation Office NKFIH, GrantNo. FK128972; and the National Research Foundation of Korea (NRF)grant funded by the Korea government (MEST) No.2015R1A3A2031159,No.2016R1C1B1011972 and No.2019R1C1C1006405.
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