Compactness of Schur-A Multipliers and Haagerup Tensorproduct
aa r X i v : . [ m a t h . OA ] J a n Compactness of Schur A-multipliers and HaagerupTensor Products
Weijiao HeJanuary 22, 2020
Abstract
In this paper we study the connection between Haagerup tensor productand compactness of Schur A -multiplier. In particular, we give a new char-acterization of elementary C ∗ -algebra in terms of completely compact Schur A -multiplier. Schur multipliers, a class of maps generalising the operators of entry-wise (Schur)multiplication on finite matrices, were first abstractly studied by Grothendieck in [3].Since then they have played an important role in operator theory. In the simplestsituation they arise in the following manner: to a (discrete) set X and a function φ : X × X → C , one associates an operator S φ on the space of compact operatorson the Hilbert space ℓ ( X ); if the resulting map is (completely) bounded, we call S φ a Schur multiplier with symbol φ .In [4], Hladnik studied an important class of Schur multipliers: compact Schurmultipliers, i.e the map S φ is a compact operator. Hladnik identified the space ofSchur multipliers with Haagerup tensor product c ⊗ h c .Recently, in [7], McKee, Todorov and Turowska generalised the notion of Schurmultipliers to new setting, on which we will inverstigate the generalization of Hlad-nik’s thorem. This paper is organised as following.In section 2, we give some basic definitions which we used in this paper, includingthe definition of Schur A -multiplier.In section 3, we prove that if either X or Y is not discrete measure space, thenthere is no non-zero compact Schur A -multiplier. By this result, we could restrictour attention to the case X = Y = N .In section 4, we study some properties of Haagerup tensor product. We prove aTheorem which based on the work of Smith [10], Ylinen [12] and Saar (see [6]), to geta complete relationship between Haagerup tensor product and completely compactmaps on C ∗ -algebras, which will be very useful for the study of the compactnessof Schur A -multipliers in the later section. Furthemore, that theorem gives a newcharacterization of the C ∗ -algebra of compact operators of some Hilbert space.1n second 5 and section 6, we study the complete compactness of Schur A -multipliers when A is ∗ -isomorphic to a subalgebra of K ( H ), and we prove a gener-alisation of Hladnik’s Theorem.In Section 7, it contains our main results. We study the relationship betweencomplete compactness and compactness, in the end we use these relations and theresults of previous sections to prove that the generalization of Hladnik’s Theorem istrue if and only if the C ∗ -algebra A if ∗ -isomorphic B ( H ) for some finite dimensionalHilbert space H . For any measure space (
Z, λ ) and Banach space B , we let L ( Z, B ) denote the spaceof all square integrable λ -measurable functions from Z into B . If B = K for someHilbert space K , then L ( Z, K ) is a Hilbert space. Furthermore, for any Hilbertspace K , we denote the space of bounded operators on K by O ( K ), and denote thespace of compact operators on K by O c ( K ) .Let ( X, µ ) and (
Y, ν ) be standard measure spaces (see [7]), H a separable Hilbertspace, and A ⊆ O ( H ) a C ∗ -algebra. We write O c = O c ( L ( X ) , L ( Y )) for the spaceof all compact operators from L ( X, H ) into L ( Y, H ). If k ∈ L ( Y × X, O ( H )) and ξ ∈ L ( X, H ) then for almost all y ∈ Y , the function x → k ( y, x ) ξ ( x ) is weaklymeasurable; moreover Z X k k ( y, x ) ξ ( x ) k dµ ( x ) ≤ k ξ k ( Z X k k ( y, x ) k dµ ( x ) ) . Such functions k will often be referred to as kernels . It follows that the formula( T k ξ )( y ) = Z X k ( y, x ) ξ ( x ) dµ ( x ) ( y ∈ Y ) , defines a (weakly measurable) function T k ξ : Y → H , and a bounded operator T k : L ( Y, H ) → L ( X, H ). Moreover, by [7] we have k T k k ≤ k k k and T k =0 if andonly if k = 0 almost everywhere.If X and Y are operator space, we denote the space of all completely boundedlinear maps from X into Y by CB ( X , Y ) and write CB ( X ) = CB ( X , X ). For thebackground of operator spaces and completely bounded maps, we refer the readerto Section 1.2. In this thesis, if f is a linear map from an operator space X intoan operator space Y , we use f n to denote the corresponding map from M n ( X ) into M n ( Y ).Now we define S ( X × Y, A ) = { T k : k ∈ L ( Y × X, A ) } and note that S ( Y × X, A ) is a dense subspace of the minimal tensor product O c ⊗ A ,thus in particular it is an operator space. A function ϕ : X × Y → CB ( A, O ( H )) willbe called pointwise measurable if, for every a ∈ A , the function ( x, y ) → ϕ ( x, y )( a )2rom X × Y into O ( H ) is weakly measurable ([7]). Let ϕ : X × Y → CB ( A, O ( H ))be a bounded pointwise measurable function. For k ∈ L ( Y × X, A ), let ϕ · k : Y × X → O ( H ) be the function given by( ϕ · k )( y, x ) = ϕ ( x, y )( k ( y, x )) (( y, x ) ∈ Y × X ) . It is easy to show that ϕ · k is weakly measurable and k ϕ · k k ≤ k ϕ k ∞ k k k ([7, Section2]). Let S ϕ : S ( Y × X, A ) → S ( Y × X, O ( H ))be the linear map given by S ϕ ( T k ) = T ϕ · k ( k ∈ L ( Y × X, A )) . Definition 2.1.
A bounded poinwise measurable map ϕ : X × Y → CB ( A, O ( H )) will be called a Schur A -multiplier if the map S ϕ is completely bounded. Equivalently, a bounded pointwise measurable function ϕ : X × Y → CB ( A, O ( H ))is a Schur A -multiplier if and only if the map S ϕ possesses a completely boundedextension to a map from O c ⊗ A into O c ⊗ O ( H ) (which we will still denote by S ϕ ).For the sake of convenience, we will not distinguish Schur A -multiplier ϕ : X × Y → CB ( A, O ( H )) and the corresponding linear map S ϕ : O c ( L ( X ) , L ( Y )) ⊗ A → O c ( L ( X ) , L ( Y )) ⊗ O ( H ) , when we use the terminology ‘Schur A -multiplier’.Another important notion is complete compactness which is defined as follows(see [5]) Definition 2.2. If X and Y are operator spaces, a completely bounded map Φ :
X →Y is called completely compact if for each ǫ > there exists a finite dimensionalsubspace F ⊂ Y such that dist(Φ ( m ) ( x ) , M m ( F )) < ǫ, for every x ∈ M m ( X ) with k x k ≤ for every m ∈ N . Let us recall that a completely bounded linear map which is approximated bya net of linear maps with finite rank in the complete bounded norm is completelycompact [6, Proposition 3.2]. We will use this fact without reference frequently.If X and Y are operator spaces, we denote the set of compact (resp. completelycompact) operators from X into Y by CO ( X , Y ) (resp. CCO ( X , Y )). Remark 2.3.
Let X , Y be operator spaces, ϕ : X → Y be completely bounded linearmap. If
Z ⊂ Y is operator space such that ϕ ( X ) ⊂ Z and there is completely boundedmap h : Y → Z with h ( d ) = d for all d ∈ Z , we define ψ : X → Z by ψ ( a ) = ϕ ( a ) for all a ∈ X , then since the composition of completely compact map and completelybounded map is completely compact, we conclude that ϕ is completely compact ifand only if ψ is completely compact ( the proof is easy consequence of [6, Proposition3.2]) . We will use this fact without reference in this paper. emma 2.4. Let X and Y be operator spaces, ϕ : X → Y a completely boundedlinear map which is not completely compact. If Z is an operator space containing X as a subspace, and f : Z → X is a completely bounded surjective linear map suchthat k f k cb = 1 and f |X = I X (here I X : X → X is the identity operator defined on X ), then ϕ ◦ f is not completely compact.Proof. we have { y ∈ M n ( X ) : k y k ≤ } = f n ( { x ∈ M n ( Z ) : k x k ≤ } ) ,ϕ n ( { y ∈ M n ( X ) : k y k ≤ } ) = ( ϕ ◦ f ) n ( { x ∈ M n ( Z ) : k x k ≤ } ) , by the definition of complete compactness ϕ ◦ f is not completely compact. In this section, we will prove that if (
X, µ ) and (
Y, υ ) are standard measure spaces,then there is no non-trivial compact Schur A-multiplier if either (
X, µ ) or (
Y, υ )is non-atomic. In the following A ⊆ O ( H ) is C ∗ -algebra, and we fix admissibletopologies on X and Y respectively. Lemma 3.1.
Let D be any compact subset of X × Y with ( µ × υ )( D ) > . Thenfor arbitrary positive number ǫ > , there are µ -measurable subset D X of X and υ -measurable subset D Y of Y such that ( µ × υ )(( D X × D Y ) \ D ) < ǫ · ( µ × υ )( D X × D Y ) < ∞ . Furthermore, if ( X, µ )( resp. ( Y, υ )) is non-atomic, there are infinitely many mutu-ally disjoint µ -measurable subsets { D n } n ∈ N of D X (resp. there are infinitely manymutually disjoint υ -measurable subsets { C n } n ∈ N of D Y ), such that ( µ × υ )(( D n × D Y ) \ D ) < ǫ · ( µ × υ )( D n × D Y ) < ∞ , ( resp. ( µ × υ )(( D X × C n ) \ D ) < ǫ · ( µ × υ )( D X × C n ) < ∞ ) for all n ∈ N .Proof. Let 0 < ǫ < δ such that0 < δ < ǫ · ( µ × υ )( D ). By the construction of the product measures (see [2]), thereexists a set { V n × W n : n ∈ N } of disjoint rectangles such that D ⊂ ∪ ∞ n =1 V n × W n and ∞ X n =1 ( µ × υ )( V n × W n ) < ( µ × υ )( D ) + δ. It is easy to see that there is at least one n ∈ N such that( µ × υ )(( V n × W n ) \ D ) < ǫ · µ × υ ( V n × W n ) . X, µ ) is non-atomic. Let n ∈ N be such that1 n · ( µ × υ )( D X × D Y ) < ǫ · ( µ × υ )( D X × D Y ) − ( µ × υ )(( D X × D Y ) \ D ) , (1)and let { E k } nk =1 be disjoint µ -measurable subsets of D X such that µ ( E k ) = n µ ( D X )for each k (for the existence of these sets, see [8, I.4]). Thus we have µ ( D X ) = µ ( ∪ nk =1 E k ) . We claim that there are at least two distinct numbers r, m ∈ N such that( µ × υ )(( E k × D Y ) \ D ) ≤ ǫ · ( µ × υ )( E k × D Y ) , k = r, m. because if this was not true we would have ǫ · ( µ × υ )( D X × D Y ) − ( µ × υ )(( D X × D Y ) \ D ) < ( µ × υ )( E n × D Y )= 1 n · ( µ × υ )( D X × D Y ) , this contradicts (1). This contradiction proved the existence of the two distinctnumbers r and m ∈ N . Now let D = E r . We replace D X by E m and repeated thesame argument, our proof is completed.We list the following two lemmas for reference, their proofs are routine. Lemma 3.2.
Let ϕ : X × Y → CB ( A, O ( H )) be a Schur-A multiplier such that ϕ ( x, y )( a ) = 0 for ( µ × υ ) -almost ( x, y ) ∈ X × Y for all a ∈ A , then S ϕ = 0 . Lemma 3.3. If ( Z, λ ) is a standard measure space, C = { e ∈ Z : λ ( { e } ) > } , then C is countable and ( Z \ C, λ ) is non-atomic. Proposition 3.4.
Let ( X, µ ) or ( Y, ν ) be non-atomic standard measure space, and ϕ : X × Y → CB ( A, O ( H )) be a compact Schur A-multiplier, then ϕ = 0 for (( µ × ν )) -alomost all ( x, y ) ∈ X × Y .Proof. We prove that if (
X, µ ) is non-atomic, then S ϕ = 0 if ϕ is compact Schur A -multiplier. The other part is proved by the same argument.By Lemma 3.2, there is a ∈ A with k a k = 1 such that for some positive number c >
0, the measure of the set D = { ( x, y ) ∈ X × Y : k ϕ ( x, y )( a ) k > c } (2)is positive. By the Vector-Valued Lusin’s Theorem [11, Corollary B.28], there exists5 compact subset E ⊂ D such that ( µ × υ )( E ) > E into O ( H )defined by ( x, y ) ϕ ( x, y )( a ) , is continuous. Therefore { ϕ ( x, y )( a ) : ( x, y ) ∈ E } is compact subset in O ( H ). Let ǫ be a fixed positive number. By [11, Lemma B.23], there is a function f of theform f = P ni =1 χ A i ⊗ a i , where a i ∈ O ( H ) and A i ⊂ E is measurable, such that k f k ∞ ≤ k ϕ k and k ϕ ( x, y )( a ) − f ( x, y ) k < ǫ (( x, y ) ∈ E ) . By (2) it is easy to see that there is at least one A k such that ( µ × υ )( A k ) > A k is compact.By Lemma 3.1, there are µ -measurable subset D X and υ -measurable subset D Y ,such that ( µ × υ )(( D X × D Y ) \ A k ) < ǫ · ( µ × υ )( D X × D Y ) < ∞ . (3)Now we define the function ϑ : X × Y → O ( H ) in L ( X × Y, O ( H )) by ϑ ( x, y ) = χ D X × D Y ( x, y ) a k (( x, y ) ∈ X × Y ) . Let h ∈ H be such that k h k = 1 and k ϑ ( x, y )( h ) k = k a k ( h ) k ≥ c − ǫ (( x, y ) ∈ D X × D Y ) . (4)Since ( X, µ ) is non-atomic, by Lemma 3.1 there are infinitely many mutuallydisjoint µ -measurable subsets { D n } n ∈ N of D X such that( µ × υ )(( D n × D Y ) \ A k ) < ǫ · µ × υ (( D n × D Y )) < ∞ . Define k n ( x, y ) := aµ ( D n ) · υ ( D Y ) χ D n × D Y ( x, y ) , ξ n ( x ) := hµ ( D n ) χ D n ( x ) (5)Then { k n } n ∈ N ∈ L ( X × Y, O ( H )) and k k n k = k a k = 1 ( n ∈ N ); { ξ n } n ∈ N ⊂ L ( X, H ) and k ξ n k = 1 ( n ∈ N ). So {k T k n k} n ∈ N is bounded. We can complete ourproof by showing that { T ϕ · k n } n ∈ N has no Cauchy subsequence if the given ǫ is smallenough. By (3), (4), Cauchy-Schwarz inequality and that k ( T ϕ · k n − T ϕ · k m )( ξ n ) k ≤k T ϕ · k n − T ϕ · k m k , it is rountine to verify that if ǫ < (1 / · c we have k ( T ϕ · k n − T ϕ · k m ) k ≥ ( Z D Y k Z D n µ ( D n ) · υ ( D Y ) ϑ ( x, y )( h ) dµx k dυ ) − ( Z D Y k Z D n µ ( D n ) · υ ( D Y ) ( ϕ ( x, y )( a ) − ϑ ( x, y ))( h ) dµx k dυy ) > · c, our proof is complete. 6 Haagerup tensor products and completely com-pact maps
Let K be a fixed Hilbert space. We will study the connection between Haageruptensor product and completely compact maps.If { A i } i ∈ I is a collection of C ∗ -algebras, we denote their C (or it is called C ∗ )-direct sum by P ⊕ i ∈ I A i (see [1]).For a C ∗ -algebra A , we follow Fell and Doran [1] to call A elementary C ∗ -algebraif A is ∗ -isomorphic to O c ( H ) for some Hilbert space H ; on the other hand, we call A compact type C ∗ -algebra if A is ∗ -isomorphic to a subalgebra of O c ( H ) for someHilbert space H . If A is compact type C ∗ -algebra, we shall identify A = P ⊕ i ∈ I O c ( H i )for some Hilbert spaces H i ( i ∈ I ) ( [1, Theorem VI.23.3]). Lemma 4.1. If A is a compact-type C ∗ -algebra then the following are equivalent:(i) A is elementary.(ii) For any ̥ ∈ CCO ( A ) , there are families { a i } i ∈ J and { b j } j ∈ J of elements of A such that P j ∈ J a ∗ i a i and P j ∈ J b j b ∗ j are convergent and ̥ ( r ) = X j ∈ J b j r a j ( r ∈ A ) . If these conditions hold, we have
CCO ( A ) = A ⊗ h A .Proof. The implication from (i) to (ii) is [6, Corollary 3.6].(ii) implies (i) :Let A = P ⊕ i ∈ I O c ( H i ) for some collection { H i } i ∈ I of Hilbertspaces, we claim that I is single point.Let H = P ⊕ i ∈ I H i , if we represent any element r of O ( H ) by a matrix ( r i,j ) i,j ∈ I ,where r i,j ∈ O ( H j , H i ), then s ∈ A is a diagonal matrix such that s i,i ∈ O c ( H i ).Furthermore, for any i, j ∈ I we define a map E i,j : O c ( H j , H i ) → O c ( H ) by thefollowing way: for any a ∈ O c ( H j , H i ), E i,j ( a ) is the matrix in O c ( H ) whose allentries are 0 except for the i, j -th entry, which is a . Now we take two distinct points i , i in I , let a ∈ O c ( H i ), b ∈ O c ( H i , H i ), and c ∈ O c ( H i , H i ) be all non-zero.We define Λ : O c ( H ) → O c ( H ) byΛ( r ) = ( E i ,i ( a ) + E i ,i ( b )) r ( E i ,i ( a ) + E i ,i ( c )) ( r ∈ O c ( H )) , then Λ is completely compact by [6, Corollary 3.6], and it is easy to verify thatΛ( A ) ⊂ A . Let ̥ : A → A be defined by ̥ ( r ) = Λ( r ) ( r ∈ A ), by Remark 2.3 ̥ iscompletely compact. But ̥ = φ v for any v ∈ A ⊗ h A because φ v ( O c ( H i )) ⊂ O c ( H i )for any v ∈ A ⊗ A and i ∈ I . This contradiction proved that I is single point, A iselementary. Theorem 4.2. If A is a C ∗ -algebra, then the following are equivalent:(i) A is elementary,(ii) For any ̥ ∈ CCO ( A ) , there are families { a i } J and { b j } j ∈ J of elements of A such that P j ∈ J a ∗ i a i and P j ∈ J b j b ∗ j are convergent and ̥ ( r ) = X j ∈ J b j r a j . If these conditions hold, we have
CCO ( A ) = A ⊗ h A . roof. The implication from (i) to (ii) is [6, Corollary 3.6].(ii) implies (i): In particular, for any u ∈ A , the map x uxu is compact,by [12] there is a faithful ∗ -representation π of A on Hilbert space X such that π ( A ) ⊂ O c ( X ), thus if we identify A with its image in O ( X ), we can consider that A is a norm-closed ∗ -subalgebra of O c ( X ). By Lemma 4.1 (i) holds. By the results of Section 3, the only interesting compact Schur A -multipliers aredefined on X, Y = N , equipped with the counting measure. We assume that A ⊂ O ( H ) for some Hilbert space H , and we can drop the assumption that A isseparable.We identify each T ∈ O ( H ∞ ) with a matrix ( T m,n ) m,n ∈ N , where T m,n ∈ O ( H ).Furthermore, we define the conditional expectation E : O ( ℓ ) → ℓ ∞ by E ( S )( n ) = S n,n , for all matrix S ∈ O ( ℓ ) . For each n ∈ N we define E n : O ( ℓ ) → ℓ ∞ by E n ( S )( k ) = S k,k ( k ≤ n ); E n ( S )( k ) = 0 ( k > n )(for all matrix S ∈ O ( ℓ )). Since ℓ is commutative C ∗ -algebra, E is completelybounded. Therefore the action of Schur A - multiplier ϕ on O c ( ℓ ( N )) ⊗ A can beregarded with S ϕ : O c ( ℓ ( N )) ⊗ A → O c ( ℓ ( N )) ⊗ O ( H ): ( T m,n ) m,n ∈ N ( ϕ ( n, m )( T m,n )) m,n ∈ N . Lemma 5.1.
Let S ϕ be a Schur A -multiplier. If there exist an index set J , andfamilies of { R i } i ∈ J and { S i } i ∈ J ⊂ O ( H ∞ ) such that P i ∈ J R i R ∗ i and P i ∈ J S ∗ i S i areconvergent, and S ϕ ( T ) = X i ∈ J R i T S i , T ∈ O c ( ℓ ( N )) ⊗ A, (6) then P i ∈ J E ( R i ) E ( R i ) ∗ and P i ∈ J E ( S i ) ∗ E ( S i ) are convergent and S ϕ ( T ) = X i ∈ J E ( R i ) T E ( S i ) , T ∈ O c ( ℓ ( N )) ⊗ A. (7) Proof.
The convergence of P i ∈ J E ( R i ) E ( R i ) ∗ and P i ∈ J E ( S i ) ∗ E ( S i ) is an easy con-vergence of P i ∈ J R i R ∗ i and P i ∈ J S ∗ i S i . Let R i = ( a ( i ) m,n ) m,n ∈ N , S i = ( b ( i ) m,n ) m,n ∈ N ,where a ( i ) m,n , b ( i ) m,n ∈ O ( H ). Since S ϕ is linear and continuous, the linear span of { E p,q ( a ) : a ∈ A ; p, q ∈ N } is norm-dense in O c ( ℓ ( N )) ⊗ A (here we recall that E p,q ( a ) is the matrix whose entries are all 0 but p, q -th entry is a ), thus in oder to8erify (7), it is sufficient to verify that it holds for E p,q ( a ) for any a ∈ A and p, q ∈ N .By (6) we have ( · is the multiplication of matice) e p,q ⊗ O ( H ) · ( S ϕ ( E p,q ( a )) · e p,q ⊗ O ( H ) = E p,q ( X i ∈ J a ( i ) p,p ab ( i ) p,q ) . (8)On the other hand we have e p ′′ ,q ′′ ⊗ O ( H ) · S ϕ ( E p,q ( a )) · e p ′ ,q ′ ⊗ O ( H ) = 0 (9)for all p ′′ or q ′′ = p, or p ′ or q ′ = q. Now (8) and (9) imply that S ϕ ( E p,q ( a )) = X i ∈ J E ( R i ) E p,q ( a ) E ( S i )Our proof is complete.In the sequel part, we use symbol CS ( A, O ( H )) ( resp. CCS ( A, O ( H ))) to denotethe set of compact ( resp.completly compact ) Schur A -multiplier. Furthermore, wedefine CS ( A ) = CS ( A, A )( resp. CCS ( A ) = CCS ( A, A )).
Remark 5.2.
Recall the discussion in section 1.2, we have CO ( A, O ( H )) ⊂ CS ( A, O ( H )) , and CCO ( A, O ( H )) ⊂ CCS ( A, O ( H )) , Proposition 5.3.
CCS ( O c ( H )) = c ( N , O c ( H )) ⊗ h c ( N , O c ( H )) Proof.
Suppose ϕ is Schur O c ( H )-multiplier. Each T ∈ O c ( ℓ ( N )) ⊗ O c ( H ) maybe identified with a matrix ( T m,n ) m,n ∈ N , where T m,n ∈ O c ( H ) for all m, n ∈ N . ByLemma 4.1 there exist an index set J and { S i } i ∈ J ⊂ O c ( H ∞ ), { R i } i ∈ J ⊂ O c ( H ∞ )such that P i ∈ J R i R ∗ i ∈ J and P i ∈ J S ∗ i S i converge uniformly and S ϕ ( T ) = X i ∈ J R i T S i , T ∈ O c ( H ∞ ) . By Lemma 5.1 we have S ϕ ( T ) = X i ∈ J E ( R i ) T E ( S i ) , T ∈ O c ( ℓ ( N )) ⊗ A. (10)Now it is easy to verify that {E ( R i ) } i ∈ J , {E ( S i ) } i ∈ J are collections of compact op-erators, P i ∈ J E ( R i ) E ( R i ) ∗ and P i ∈ J E ( S i ) ∗ E ( S i ) converge in norm. By [10], let v = P i ∈ J E ( R i ) ⊗ E ( S i ) ∈ c ( N , O c ( H )) ⊗ h c ( N , O c ( H )), we have k S ϕ k cb = k v k h .Conversely, for any v ∈ c ( N , O c ( H )) ⊗ h c ( N , O c ( H )), there are { R ′ k } k ∈ N , { S ′ k } k ∈ N in O c ( H ) such that v = P ∞ k =1 R ′ k ⊗ S ′ k and k P ∞ k =1 R ′ k R ′ k ∗ k < + ∞ , k P ∞ k =1 S ′ k ∗ S ′ k k < + ∞ , then ( φ v ) | ( O c ( H ∞ )) , T ∞ X k =1 R ′ k T S ′ k
9s a completely compact map on O c ( H ∞ ), and it is easy to see that there is Schur O c ( H )-multiplier ϕ such that S ϕ = φ v . Therefore the map C ( N , O c ( H )) ⊗ h C ( N , O c ( H )) → CCS ( O c ( H ( H ))) , v ( φ v ) |O c ( H ∞ )is linear isometry with range CCS ( O c ( H )). Our proof is complete. Theorem 5.4. If A is C ∗ -algebra, then the following two conditions are equivalent:(I) A is elementary;(II) CCS ( A ) = c ( N , A ) ⊗ h c ( N , A ) .If these conditions hold, Schur A -multiplier ϕ is completely compact if and onlyif there are index set J , { a ik } i ∈ J,k ∈ N and { b ik } i ∈ J,k ∈ N ⊂ A such that :(i) P i a ik ( a ik ) ∗ and P i ( b ik ) ∗ b ik are convergent in the norm of A for each k ∈ N ,and X i ∈ J a ik ( a ik ) ∗ , X i ∈ J ( b ik ) ∗ b ik → k → ∞ (ii) for any m, n ∈ N , ϕ ( m, n )( x ) = X i a in x b im ( x ∈ A ) . (11) Proof.
The implication from ( I ) to ( II ) was proved in the previous proposition.( II ) implies ( I ): This is the combination of Remark 5.2 and Proposition 4.2.Therefore if ( I ) or ( II ) holds, we may identify A = O c ( H ) for some Hilbert space H . Let ϕ : N × N → CB ( A ) be a given Schur A -multiplier. We prove that ϕ iscompletely compact if and only if ( i ) and ( ii ) hold.If ( i ) and ( ii ) hold, then the second part of ( i ) implies that k a ik k = k a ik ( a ik ) ∗ k → k a ik k = k ( b ik ) ∗ b ik k →
0) for each fixed i if k → ∞ .Thus we may define R i (resp. S i ) ∈ c ( N , O c ( H )) by ( R i ) k,k = a ik (resp. ( S i ) k,k = b ik ). Now ( i ) implies that P i ∈ J R i R ∗ i and P i ∈ J S ∗ i S i are convergent in c ( N , O c ( H )), then we conclude that P i R i ⊗ h S i is in c ( N , O c ( H )) ⊗ h c ( N , O c ( H )), and ( ii ) implies that S ϕ ( T ) = X i R i T S i ( T ∈ O c ( ℓ ) ⊗ O c ( H )) , so by Theorem 5.3 S ϕ is completely compact map.Now suppose that ϕ is completely compact Schur O c ( H )-multiplier, then there is v = P k ∈ N R k ⊗ h S k in c ( N , O c ( H )) ⊗ c ( N , O c ( H )) such that S ϕ = ( φ v ) | ( O c ( H ∞ )).Furthermore, P i ∈ N R i R ∗ i and P i ∈ N S ∗ i S i are convergent in the norm of O ( H ∞ ), weconclude that P i ∈ N R i R ∗ i , P i ∈ N S ∗ i S i ∈ C ( N , O c ( H )). We define a ik = ( R i ) k,k and b ik = ( S i ) k,k for each i, k ∈ N , then it is easy to verify that { a ik } i,k ∈ N and { b ik } i,k ∈ N satisfy (1) and (2). C ∗ -algebra In this section, we will use the results of last section to study the compactness ofSchur A -multiplier where A is compact-type C ∗ -algebra. We identify A = P ⊕ i ∈ I O c ( H i )for some collection { H i } i ∈ I of Hilbert spaces, take H = P ⊕ i ∈ I H i and let f : O c ( H ) → A be the canonical projection, thus f is completely positive. We shall say that thepair ( H, f ) is associated to A . 10 emma 6.1. Let A be compact-type C ∗ -algebra, ( H, f ) its associated pair. If ϕ : N × N → CB ( A ) is Schur A -multiplier, then there is a Schur O c ( H ) -multiplier ψ : N × N → CB ( O c ( H )) such that S ψ | ( O c ( ℓ ( N )) ⊗ A ) = S ϕ . Moreover, S ϕ is(completely) compact if and only if S ψ may be chosen to be (completely) compactmap.Proof. Since f is completely bounded, by [7, Theorem 2.6] ρ : N × N → CB ( O c ( H ) , A )defined by ρ ( n, m ) = f is Schur zmathcalO c ( H )-multiplier. For each m, n let us define ψ ( n, m ) : O c ( H ) →O c ( H ) by ψ ( n, m )( a ) = ( ϕ ( n, m ) ◦ ρ ( n, m )) ( a ) ( a ∈ O c ( H )) . Then S ψ ( T ) = S ϕ ◦ S ρ ( T ) for all T ∈ O c ( ℓ ) ⊗ O c ( H ), S ψ : O c ( ℓ ) ⊗ O c ( H ) →O c ( ℓ ) ⊗ O c ( H ) is Schur O c ( H )-multiplier whose range is contained in O c ( ℓ ) ⊗ A and S ψ ( T ) = S ϕ ( T ) for all T ∈ O c ( ℓ ) ⊗ A . Now since S ρ is completely bounded,we conclude that if S ϕ is (completely) compact then S ψ is (completely) compact.Conversely, if S ψ is (completely) compact, since S ψ ( T ) = S ϕ ( T ) for all T ∈ O c ( ℓ ) ⊗ A and that there is completely positive map id ⊗ f : O c ( ℓ ) ⊗O c ( H ) → O c ( ℓ ) ⊗ A whichis identity map on O c ( ℓ ) ⊗ A , we conclude that S ϕ is (completely) compact.By Lemma 6.1 and Theorem 5.4 together we have: Theorem 6.2. If A is compact type C ∗ -algebra, then there is Hilbert space H suchthat the following are equivalent:(i) ϕ is in C . C . S (A);(ii) there are index set J and { a ik } i ∈ J,k ∈ N and { b ik } i ∈ J,k ∈ N ⊂ O c ( H ) such that:(1) P i a ik ( a ik ) ∗ and P i ( b ik ) ∗ b ik are convergent in the norm of O ( H ) for each k ∈ N and P i ∈ J a ik ( a ik ) ∗ , P i ∈ J ( b ik ) ∗ b ik → as k → ∞ ;(2) for any m, n ∈ N , ϕ ( m, n )( x ) = X i a in x b im , x ∈ A. Remark 6.3.
If we compare the previous theorem with Theorem 5.4, we noticedthat in condition (ii) { a ik } i ∈ J,k ∈ N and { b ik } i ∈ J,k ∈ N can be chosen from A if and onlyif A is ∗ -isomorphic to O c ( K ) for some Hilbert space K . In this section, we prove some results by aid of which we could identify compactnessand complete compactness in some cases. We fix Ω to be a compact Hausdorff space, C (Ω) to be the space of all continuous complex-valued functions on Ω.The proofs of the following two lemmas are standard and we ommit them: Lemma 7.1.
Let X and Y be operator spaces, Ψ ∈ O c ( X , Y ) . If (Φ α ) α ∈ A ⊂ B ( Y ) is a net with sup α ∈ A k Φ α k < ∞ and Φ ∈ B ( Y ) such that Φ α ( a ) → Φ( x ) for all x ∈ X . Then k Φ α ◦ Ψ − Φ ◦ Ψ k → α ∈ A . emma 7.2. For any ( f i,j ) ∞ i,j =1 ∈ O ( ℓ ) ⊗ min C (Ω) , we have k ( f i,j ) ∞ i,j =1 k = sup {k ( f i,j ( ω )) ∞ i,j =1 k : ω ∈ Ω } , where the norm of the right hand-side is taken from O ( ℓ ) . In particular, k ( f i,j ( ω )) ∞ i,j =1 k ≤k ( f i,j ) ∞ i,j =1 k . Lemma 7.3. If A is a C ∗ -algebra which is ∗ -isomorphic to a subalgebra of M n ⊗ C (Ω) for some n ∈ N and compact space Ω , then compact linear map from A into A iscompletely compact.Proof. Let ϕ : A → A be a compact linear map. We identify A as a C ∗ -subalgebraof M n ⊗ C (Ω). Since M n ⊗ C (Ω) is nuclear, let { ψ m : M n ⊗ C (Ω) → M n ⊗ C (Ω) } m ∈ I be a net of finite-rank completely positive maps such that ψ m ( a ) → a for all a ∈ M n ⊗ C (Ω) provided m → ∞ . Then we have ψ m ◦ ϕ ( a ) → ϕ ( a ) for all a ∈ A . Since {k ψ m ◦ ϕ k} is bounded, by Lemma 7.1 we have k ψ m ◦ ϕ − ϕ k →
0. Therefore { ψ m ◦ ϕ } is Cauchy net in O ( A, M n ⊗ C (Ω)). But CB ( A, M n ⊗ C (Ω)) = O ( A, M n ⊗ C (Ω)), byOpen Mapping Theorem we conclude that { ψ ◦ ϕ } is Cauchy net in CB ( A, M n ⊗ C (Ω))as well, it is easy to verify that ψ m ◦ ϕ → ϕ completely. Since each ψ m ◦ ϕ is of finiterank, ϕ is completely compact. Proposition 7.4.
Let V be an operator space, and n ∈ N be fixed number. Let ϕ i,j : V → C (Ω) be completely bounded maps ( i, j = 1 , . . . , n ), we define S ϕ : M n ( V ) → M n ( C (Ω)) by S ϕ (( v i,j ) ni,j =1 ) = ( ϕ i,j ( v i,j )) ni,j =1 . Then k S ϕ k cb = k S ϕ k .Proof. Let D n be the subalgebra of all the diagonal matrices in M n . We will use theidea of the proof of [9, proposition 8.6] . For any C ∈ D n , we have S ϕ ( C ( v i,j ) ni,j =1 ) = C ( ϕ i,j ( v i,j )) ni,j =1 ,S ϕ (( v i,j ) ni,j =1 C ) = ( ϕ i,j ( v i,j )) ni,j =1 C where we define the multiplication between C and elements of M n ( V ) or M n ( C (Ω))by the multiplication of matrices. Since S ϕ is D n -bimodule map, by the similarargument of [9, Proposition 8.6] we can prove that S ϕ is completely bounded.Now we may get the following proposition immediately: Proposition 7.5.
Let V be an operator space, A a C ∗ -algebra which is ∗ -isomorphicto a subalgebra of M n ⊗ C (Ω) for some n ∈ N . Let ϕ i,j : V → A be a boundedlinear map (so it is completely bounded automatically) for each i, j ∈ N , if S ϕ : O c ( ℓ ) ⊗ V → O c ( ℓ ) ⊗ A is a bounded linear map which satisfies S ϕ (( v i,j ) ni,j =1 ) = ( ϕ i,j ( v i,j )) ni,j =1 , then S ϕ is completely bounded and k S ϕ k cb = k S ϕ k . Combine Lemma 7.3 and Proposition 7.5 we get:12 roposition 7.6. If A is a C ∗ -algebra which is ∗ -isomorphic to a subalgebra of M n ⊗ C (Ω) for some n ∈ N and compact space Ω , then the compact Schur A -multipliers are completely compact. By modifying the proof of [5], it is not hard to prove:
Lemma 7.7.
Let H be a Hilbert space with infinite dimension, I and index setwhich has the same cardinal number of H when we regard H as a set. Let us selectan arbitrary pairwise orthogonal family { H i } i ∈ I of finite-dimensional subspaces of H such that P i ∈ I H i = H and that there is a subset { i k } k ∈ N ⊂ I such that dim ( H k ) >k , then there exists a linear map ϕ : O c ( H ) → O c ( H ) such that ϕ ( O ( H k )) ⊂ O ( H k ) and that ϕ is completely bounded, compact, but not completely compact. The lemma is similar to the previous:
Lemma 7.8.
Let H be a Hilbert space with infinite dimension, I and index setwhich has the same cardinal number of H when we regard H as a set. Let usselect an arbitrary pairwise orthogonal family { H i } i ∈ I of finite-dimensional subspacesof H such that P i ∈ I H i = H and that there is a subset { i k } k ∈ N ⊂ I such that dim ( H k ) > k , then there exists a linear map θ : P ⊕ i ∈ I O ( H i ) → P ⊕ i ∈ I O ( H i ) such that θ ( O ( H i )) ⊂ O ( H i ) and that θ is completely bounded, compact, but not completelycompact. Proposition 7.9.
Let A be a compact-type C ∗ -algebra such that all compact com-pletely bounded linear map ϕ : A → A is completely compact, then A is ∗ -isomorphicto P ⊕ k ∈ I M n k , n k ≤ N for some N ∈ N , where I is an index set.Proof. We assume A= P ⊕ k ∈ I O c ( H k ), H = P ⊕ k ∈ I H k . So there are three cases:(1) H k is of infinite dimension for some k ;(2) Each H k is of finite-dimension but { dim( H k ) } k ∈ I is unbounded, so it is con-venient to assume that A = P ⊕ k ∈ I M n k , and there is a subset { n k i } of { n k : k ∈ I } such that i ≦ n k i for all i ∈ N ;(3) A= P ⊕ k ∈ I M n k , n k ≤ N for some N ∈ N .We need to prove that (1) and (2) are not true.(1) is failed: Suppose H k is of infinite-dimension. By Lemma 7.7, there is a map ϕ : O c ( H k ) → O c ( H k ) which is completely bounded, compact but not completelycompact. Let g : P ⊕ i ∈ I O c ( H i ) → O c ( H k ) be the canonical extension of ϕ , that is, g |O c ( H k )= ϕ and g | P ⊕ i = k O c ( H i ) = 0, then g is compact, completely bounded, butby Lemma 2.4 it is easy to see that g is not completely compact. But g can beregarded as a compact, completely bounded but not completely compact linear mapfrom P ⊕ i ∈ N O c ( H i ) into P ⊕ i ∈ N O c ( H i ), so (1) is failed.(2) is failed: In this case, let B = P ⊕ i ∈ N M i . By Lemma 7.7, there is a completelybounded linear map ϕ : B → B satisfing ϕ ( M i ) ⊂ M i which is compact but notcompletely compact. But B = P ⊕ i ∈ N M i is a norm-closed ∗ -subalgebra of A = P ⊕ k ∈ I M n k (since i ≤ n k i , M i ⊂ M n ki ), and there is conditional expectation E from13 to B . So by Lemma 2.4 it is easy to see that ϕ ◦ E : P ⊕ k ∈ I M n k → B is compactand completely bounded, but it is not completely compact. Furthermore, ϕ ◦ E canbe regarded as a linear map from A into A which is compact, completely boundedbut not completely compact, so (2) is failed. Proposition 7.10.
Let A= P ⊕ k ∈ I M n k such that n k ≤ N for some N ∈ N , thenfor any C ∗ -algebra B , the linear map ϕ : B → A is compact if and only if ϕ iscompletely compact.Proof. Since A is ∗ -isomorphic to a C ∗ -subalgebra of M N ⊗ C ( I ∪{∞} ), and I ∪{∞} is compact space, then the statement is an easy consequence of Proposition 7.6.Now we could summarize a theorem as following: Theorem 7.11. If A is a compact-type C ∗ -algebra, then the following is equivalent:(i)Any compact completely bounded bounded linear map ϕ : A → A is completelycompact; (ii) A= P ⊕ k ∈ I M n k , n k ≤ N for some N ∈ N . If these conditions hold,then for any C ∗ -algebra B , if linear map ϕ : B → P ⊕ k ∈ I M n k is compact, then it iscompletely compact. Now let us go back to study Schur A -multiplier defined on N × N . Theorem 7.12. If A ⊂ O c ( H ) is C ∗ -algebra, then CS ( A ) = CCS ( A ) if and only if A = P ⊕ k ∈ I M n k such that n k ≤ N for some N ∈ N .Proof. This is the combination of Proposition 7.5 and Theorem 7.11 because for A ⊂ O c ( H ), A = P ⊕ k ∈ I M n k implies that A is ∗ -isomorphic to a subalgebra of M n ⊗ C (Ω) for some Ω and n . Corollary 7.13. If A is a finite dimension C ∗ -algebra, then CS ( A ) = CCS ( A ) . Combine Theorem 7.12 and Proposition 5.3, we get [4, Proposition 5]:
Corollary 7.14. CS ( C ) = c ( N , C ) ⊗ h c ( N , C ) . Theorem 7.15. If A is a C ∗ -algebra, the following two conditions are equivalent:(i) A is ∗ -isomorphic to O ( H ) for some finite dimensional Hilbert space;(ii) CS ( A ) = c ( N , A ) ⊗ h c ( N , A ) .Proof. The implication from (i) to (ii) is the combination of Theorem 5.4 and The-orem 7.12.(ii) implies (i): By Remark 5.2, for any u ∈ A , the map from A into A definedby x uxu is compact, then by Ylinen [12] A is compact-type, thus A has thefollowing form: A = ⊕ X i ∈ I O c ( H i ) , where each H i is Hilbert space, and by Theorem 6.2, for any v ∈ C ( N , A ) ⊗ h C ( N , A ), φ v is completely compact Schur A -multiplier, so condition (ii) impliesthat CS ( A ) ⊂ CCS ( A ) , c ( N , A ) ⊗ h c ( N , A ) = CS ( A ) = CCS ( A ) . Therefore, by Theorem 7.12 A = P ⊕ k ∈ I M n k , n k ≤ N for some N ∈ N . On the otherhand, by Theorem 5.4, A is elementary C ∗ -algebra, hence there is at most one n k isnon-zero, (i) is proved. Acknowledgement
My sincere thanks to my advisor Professor Ivan Todorov for his guidance during thiswork. I would also like to thank Dr.Stanislav Shkarin, he inspired me to investigatethe proof of Lemma 3.1.Mathematical Sciences Research Centre, Queen’s University Belfast, Belfast,BT7 1NN, United KingdomEmail: [email protected]
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