Journal of Sciences, Islamic Republic of Iran 17(1): 67-74 (2006) http://jsciences.ut.ac.ir University of Tehran, ISSN 1016-1104

Arens Regularity and Weak Amenability of Certain Matrix Algebras

R. Alizadeh and G.H. Esslamzadeh*

Faculty of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), 424 Hafez Avenue, 15914 Tehran, Islamic Republic of Iran

Abstract Motivated by an Arens regularity problem, we introduce the concepts of matrix Banach space and matrix Banach algebra. The notion of matrix normed space in the sense of Ruan is a special case of our matrix normed system. A matrix Banach algebra is a matrix Banach space with a completely contractive multiplication. We study the structure of matrix Banach spaces and matrix Banach algebras. Then we investigate Arens regularity and weak amenability of certain matrix algebras which are built on matrix Banach algebras. In particular we show that for such algebras both of Arens regularity and weak amenability problems can be reduced to the same problem for a considerably smaller algebra.

Keywords: Banach algebras; Weak amenability; Arens regularity

Again one might ask, which properties of a C*-algebra 1. Introduction force it to be weakly amenable? Since Arens’ original paper [1], Arens products have The above questions motivated us to consider matrix been used as an important tool in the study of Banach algebras over Banach algebras, which roughly speaking, algebras and their duals. This subject is now well look like operator algebras. Our approach is based on developed and there is a vast literature on it. A result of appropriate weakening of matrix norm conditions of Civin and Yood [6, Theorem 6.1] implies that every operator spaces. Our conditions are strong enough so operator algebras are Arens regular. Unital operator that the class of matrix Banach algebras behave nicely algebras are indeed complete operator spaces with a and weak enough to include algebras with non-operator completely contractive multiplication [5]. On the other norm such as l1, in this class. Indeed we drop the hand there are Arens regular Banach algebras which are assumption of contractivity of Mn module actions from not operator algebras; for example see [8,9]. These facts the definition of matrix normed space in the sense of suggest the following question. Ruan. Which properties of operator algebras imply their Although the abstract operator space theory starts Arens regularity? with the notion of matrix normed space, but due to the The notion of weak amenability was introduced by quantization goal, it moves quickly to the special case of Bade, Curtis and Dales [2]. Since then weak amenability operator spaces; consequently less attention has been of various known Banach algebras has been examined. paid to general matrix normed spaces. Here we consider In particular every C*-algebra is weakly amenable [11]. a class of matrix normed systems which contain the

* E-mail: [email protected]

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class of matrix normed spaces as a proper subclass. are defined in the following way: This paper is organized as follows. In Section 2 we < f .ab , >=< f , ab > <Δafb, >=<, fba > introduce our terminology. In Section 3, we introduce matrix Banach spaces and investigate their structure. < mf., b >=<, m f . b > =< ma , Δ f > Various examples are also provided in this section. In the Section 4 we investigate weak amenability of the < mnf. , >=< m , nf . > =< nfm Δ > algebra of approximable matrices over commutative When we refer to A** without explicit reference to matrix Banach algebras. In Section 5, we identify the any of Arens products, we mean A** with the first Arens topological centers of the algebra of approximable product. matrices over a matrix Banach algebra. ∗∗ For fixed nA∈ the map mmna . [resp. ∗ ∗ mnma Δ ] is weak− weak continuous, but the map 2. Notations mnma . [mma Δn] is not necessarily Throughout all vector spaces and algebras are over weak∗ − weak∗ continuous, unless n is in A. The first ** , A is a Banach algebra, A-module means Banach A- topological center Z1(A ) is defined by bimodule and dual A-module means the dual X* of an A- ZA(){∗∗ =∈ n A∗∗ : Themap m nm. is module X with its natural A-module structure. 1 a Let X be a vector space. We denote the vector space weak∗∗ -weak continuous} of all m×n matrices on X by Mm×n(X) and Mn×n(X) by = {fnA∈:.=Δ∗∗ nmnmorall mA ∈∗∗} . Mn(X). In particular we denote Mn( ) by Mn. Let 1 ≤ i, ** j ≤ n and x ∈ X . We denote the n×n [resp. × ] The second topological center Z2(A ) is defined ** ** matrix on X whose ij-th entry is x and all other entries similarly. If Z1(A ) = A , then A is called Arens n ∞ ** ** are zero, by x ⊗ E ij [resp. x ⊗ E ij ]. If X is a unital regular. In this case Z2(A ) = A as well. n ∞ n algebra, then we denote 1⊗ E ij [resp. 1⊗ E ij ] by E ij [resp. E ∞ ]. Given BM∈ () X and CM∈ () X, 3. Structure of Matrix Banach Spaces and ij mn× kl× Matrix Banach Algebras the direct sum BCM⊕∈ ()()mk+×+ nl() X is defined by In this section we introduce matrix Banach spaces ⎡⎤B 0 and matrix Banach algebras. Then we study their basic BC⊕=⎢⎥. Let EM∈ () X. By a submatrix mn× properties. ⎣⎦⎢⎥0 C of E we mean a k×l block F in E where 1 ≤ k ≤ m and Definition. Let X be a complex vector space, and 1 ≤ l ≤ n. We denote the set of all matrices EMX∈ n () n ∈ . We say that a norm . on M n ()X is which contain F as a submatrix and zero elsewhere by (i) free-position, if for every x ∈ X and positive E . F n ∗ integers ≤ 1 ≤ ijkl,,,≤ n the equality x ⊗ E ij Let X be a normed space, n ∈ , f ∈ MXn () and n =⊗x E n holds. 1 ≤ i, j ≤ n. Define < f ij , xf>=< , xE⊗>ij x ∈ X . kl ∗ ∗ (ii) monotone, if for every BMX∈ () and every Then f ij ∈ X . Conversely every ()f ij∈ MX n ( ) can n ∗ submatrix F of B , we have BB′ ≤ for all be considered as an element of M n ()X whose actions defined by BE′∈ F .

n (iii) permutation invariant, if for every <>=<>∈(f ),(xfxxM ) , ,( ) (X ). ij ij ∑ ij ij ij n BB, ′∈ Mn () X , where B ′ is obtained by inter- ij,1= changing two rows or two columns of B , the equality ∗ ∗ Therefore we can identify M n (X ) with M n ()X . BB= ′ holds. ∗ Henceforth we consider M n (X ) with the norm which (iv) unitary invariant, if for every unitary matrix ∗ UM∈ n and BMX∈ n () the equalities UB = inherits from M n ()X . The first and second Arens multiplications on A** that BU= B hold. we denote by “.” and “Δ”, respectively, are defined in Clearly every unitary invariant norm is permutation three steps. For ab, ∈ A, f ∈ A ∗ and mn, ∈ A∗∗ , the invariant and every permutation invariant norm is a elements f.a, aΔf, m.f, fΔm of A* and m.n, mΔn of A** free-position norm.

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Definition. Let X be a vector space and for every If . is complete for some n ∈ (and hence all n n ∈ , let . be a norm on M ()X . We say that n n n ∈ ), then ({.})X , is called a matrix Banach n ({.})X , is a matrix normed system if for every n space (MBS). positive integer n , . is a free-position and monotone n Definition. Let A be an algebra and ({.})A, be a norm on M n ()X and for every mn,∈ where n MBS. For every n ∈ , consider M n ()A with the mn> , the inclusion φnm, :()M n XM⎯→ m (X), usual matrix product. If . is an algebra norm on φ()AA=⊕ 0, is an isometry. As in [14] we say that a n ∞ matrix normed system ({.}X , ) satisfies the L M n ()A for every n , that is n condition if for all positive integers mn, and for all BC≤ B C,,∈ B C Mn () A ,∈ n BMX∈ (), CMX∈ () the equality BC⊕ nnn n m mn+ =,max{ B C } holds. We say that a matrix then ({.})A, is called a matrix Banach algebra nm n normed system ({.})X , is a permutation invariant (MBA). We say that a MBA ({.})A, is unital if A is n n matrix normed system, if for every positive integer n , unital and for every positive integer n the equality . is a permutation invariant norm on M ()X . I = 1 holds. n n n n Unitary invariant matrix normed system is defined similarly. Remark 3.2. In our definition of "matrix normed

system" we do not assume contractivity of M n -module Remark 3.1. Suppose ({.})X , is a matrix normed n actions on M n ()X . Indeed by Remark 3.1(iii) and system. Then Ruan’s Theorem [7] [respectively Blecher’s Theorem] a

(i) For every ()x ij∈ MX n () we have, MBS [respectively MBA] is an operator space [respectively operator algebra] only if it is unitary x ≤≤()xx. ∞ kl 1 ij ∑ ij 1 invariant and satisfies L -condition. ij (ii) ({.})X , is complete for some n ∈ , if and Examples 3.3. (i) Let X be a Banach space. If we n equip every M ()X with the l 1 norm, then X turns only if it is complete for every n ∈ . n (iii) It is easy to check that the concepts of matrix into a permutation invariant MBS. But X is not a normed space [14] and unitary invariant matrix normed unitary invariant MBS. This is the greatest MBS max system coincide. To see this let ({.}X , ) be a matrix structure on X . We denote this MBS with X . Also n if X is a Banach space, then we can equip every normed space, n be a positive integer, u be a unitary ∞ M n ()X with the l norm. With this structure X matrix and x be an element of M n (X ). We have turns into a permutation invariant MBS which is not x =≤u∗∗ ux u ux = ux . unitary invariant. This is the least MBS structure on X . nnn n We denote this MBS with X min . Clearly X min satisfies Also ux≤= u x x . Conversely let A condition. nnn ∞ ({.})X , be a unitary invariant matrix normed system. (ii) Let p ∈[1,∞ ) and L . We can equip A with n the operator norm which inherits from A . With this Let n be a positive integer, α be an element of M n structure aA∈ turns into a permutation invariant with the operator norm 1 and x be an element of MBA. M n ()X . Using singular value decomposition, we (iii) Let X be a reflexive Banach space with a conclude that there exist two unitary matrices u and v Schauder basis {x n } whose unconditional constant such thatα =+/()uv2. Now we have equals to one. For every positive integer n , we can n αx =/+/≤/+/+/ux22 vx ux 22 ux vx 2=x . endow with the norm it inherits from X , by taking Similarly x α ≤ x . (,cc , , c ) =+++ cx cx cx 12KKnn[]n 11 22 n

Definition. Let ({.})X , be a matrix normed system. We can equip M with the operator norm . of n n n

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n n B () , where is equipped with the above norm. Lemma 3.7. Let X be a MBS and Y is a Banach b space. Then M ∞ ((BXY, )) is isometrically isomorphic With the order induced by the {}x n , X is a Banach with BK(()) X, Y . In particular MXb ()∗∗=. KX() lattice. So with this structure, is a MBA. ∞ ∞∞

Definition. Let X be a MBS and M ()X be the ∞ 4. Weak Amenability of Approximable Matrices linear space {(x ij ) |,ij ∈ , xij ∈ X}. For In this section we study the weak amenability of BMX∈ ∞ () let n B be the truncation of B to K ∞ ()A . We show that for every permutation invariant M ()X (i.e. B is nn× matrix in the top left corner n n MBA A which satisfies L∞ condition and constructed n of B ) and B be the element n B ⊕ 0 of M ∞ ()X . on a unital commutative weak amenable Banach algebra

Define A , K ∞ ()A is weakly amenable. Theorem 4.1. Let A be a permutation invariant MBA BSupBn=|∈=,∈{}nn limBBMX∞ (). nn which satisfies the L∞ condition. If A is unital

We call every BMX∈ ∞ () for which B < ∞ , a commutative and weakly amenable, then K ∞ ()A is bounded matrix. We denote the space of all bounded weakly amenable. b Proof. Let A be weakly amenable, a∈ A, matrices on X with above norm by M ∞ ()X . An ()f ∈ KA ()∗ and ()aKA∈ (). Then we have element in the closure of the inductive limit ij ∞ ij ∞ lim b ⎯⎯→ M n ()X in M ∞ ()X is called an approximable ∞∞ <()()f ijaaE ij , ⊗kl >=< ()()[ fij , aaE ij ⊗kl ] >=< () fij , matrix. We denote the set of all approximable matrices b ∞∞ in M ()X by K ()X . ∞ ∞ ∞ ∞ []aa⊗ E>= < f , aa >=< fa , a > . ∑∑∑s =1 sk sl sl sk sl sk ss==11 Let A be a MBA and A,BKA∈ ∞ (). It is easy to see that sequence {nnA B} is a norm convergent Thus the following equality holds sequence in K ∞ ()A . We define n

[(f ij )(aweaklimf ij )] kl =∗−n sla sk . (1) nn ∑ ABABAB=,lim ∈A. s =1 Similarly we have It is easy to see that for A = ()aij and Bb= ()ij in ∞ n K ()A the equality ABa= ()b holds. ∞ ∑ k =1 ik kj [(afij )( ij )] kl =∗− weaklimafn∑ ls ks . (2) s =1 Remark. Let X be a MBS and Y be a Banach space. ∗ Suppose D :⎯→KA∞∞() KA () is a bounded By identifying M ((BXY, )) with BM(() X, Y) in n n derivation. Now for positive integers ij,,,kl we can the usual way, we can equip BX(,Y) with a MBS kl define Dij by structure. In particular we can equip X ∗ with a MBS kl ∗∞kl structure. DAij :⎯→ ADaDaEij () = [ ( ⊗ij )] kl . Proofs of the following lemmas are not complicated, but for convenience of readers we present their proofs in Using (1) and (2) we can conclude that for every Section 6. ab, ∈ A and for every positive integers ijklm,,,, the following identity holds b Lemma 3.4. If X is a MBS, then M ∞ ()X is a DabDaEkl ( )=⊗ ([∞∞ ][ bE ⊗ ]) Banach space. ij im mj kl ∞∞ ∞ =⊗[Da ( Eim )[ b ⊗ Emj ]] kl +⊗ [[ a Eim ] (3)

Lemma 3.5. Let X be a MBS and ()x ij ∈ MX∞ (). ∞ ml km +⊗Db()]() Emj kl = D im abδδ jk + a il D mj () b n Then ()x ij ∈ KX∞ () if and only if limnijij→∞ (x )= (x ) . Where δ is the Kronecker’s delta. Thus for every

mm positive integer m , Dmm is a bounded derivation which Lemma 3.6. If A is a MBA, then K ∞ ()A is a Banach algebra. is zero by assumption. On the other hand we have the following identity

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n t ∞ Also we have uunk nk= I n for every 1 ≤ km≤. [(())][(Dars ij =⊗ Dlimn∑ ( a rs E rs ))] ij rs,=1 Clearly S spans M . Now set (4) n n ij m = limnrsrs∑ D(). a t rs,=1 gmuDunn= −/1(∑ knk). k =1 From (3) we conclude that for all positive integers 1 ij m ij, and m , By (9) we have g n ≤ D and for ≤, ≤ ,

nnnnn ji mi mj mm D([]) uuni nj = uDuni ()() nj +. Duni u nj Daij ()=−+ Dim (1) aaDjm (1) Dmm () a.

Since every bounded derivation from A into A ∗ is Thus zero, then nntnn Du()nj+ uDu ni () ni u nj Daji ()=− Dmi (1) aaDmj (1). (5) ij im jm n ntn = u nj ([uuni nj ]) D ([ uuni nj ]). Also we have Divide both sides of the above identity by m and ∞∞ ∞∞ sk is sum over i while keeping j fixed to obtain 0[(==+DEkk E ii )] ik Dkk (1)δδ si ksD ii (1) ∑∑ nn n m ss==11 Du()nj −=−[gn unj ][] ugnj n . Since {uuni nj} i =1 is ik ik another listing of the elements of S and S spans M , =+DDkk (1)ii (1) n then and hence D ()nnuguuguM= [−∈ ] . ik ik nn n DDkk (1)=− ii (1) . (6) Let gg= (), then for 1 ≤,ij ≤ n we have From (4) and (6) we conclude that nnkl ∞∞nn∞ n DE()ij =− ( gnkl ) E ij E ij ( g nkl ) (10) [D((a ))]= limDaij ( ) rs ij n ∑ kl,= 1 kl kl n n Also from (8) we have =+lim [Daij (1) aDij (1) nk∑∑k1==ikil 1 jljl n ij ij ij [(())]DE∞ = lim [ D (1)δ −−+DaaDDaji(1) ji ji ji (1)ji ( ji )] (7) ij rs n∑ ks kr k =1 (11) n kj = lim [Da (1) sr nk∑ k1= kki −+δskrkDD(1)] rssr (δ ). n ++aDik (1)] Dji ( a ). ∑ k =1 jk kk ij ji From (10) and (11) the following equalities hold for If for every positive integer ij, we define every 1 ≤ ij,≤ n kj D kj= D kk , then from (7) we conclude that Dg(1) = 1 ≤≠ ij ≤n ⎪⎧ ij nij ⎨ (12) Dggijnji (1) = −≤,≤1 . [(Da (rs ))] ij= [( D rs (1))( a rs ) ⎩⎪ ij nii njj (8) ji From (12) we can conclude that for all positive −+()((1))]aDrs rs ij Da ij (). ji integers mn, where mn> the following equality On the other hand for every positive integer n we holds can consider M n as a subalgebra of M n ()A and equip g nij= gij mij ,≤1 ≠ ≤n . it with the norm which inherits from M n ()A . Let

Su={nnm1 ,..., u} be the set of elements of M n with We set g ij= g nij for ij,,n ∈ and ij≠ . Since exactly one nonzero entry which is 1 or -1, in every row the sequence {(diag gn11,..., g nnn , 00) , ,... } is a bounded and column. Since A is permutation invariant and sequence with the bound D , then it has a satisfies L∞ condition, the following identity holds. subsequence that converges to an element ∗ t uu==≤≤11 km. (9) {(diag g11,..., g nn ,... )} in K ∞ ()A in the weak* nk n nk n

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∗ ∗∗ b ∗∗ topology. Thus g = (g ij ) is an element of K ∞ ()A . KA∞∞()= MA ( ).

From (8) and (12) we conclude that the equality ∗ Now Let aA∈ , f ∈ A , ()aKAij ∈ ∞ (), Da()=−,∈ gaaga K∞ () A holds. Therefore K ∞ ()A is ∗ ∗ ∗∗∗∗ ∗∗ weakly amenable. □ ()bKAij ∈ ∞ () and ()()mnij,∈ ij KA∞ (). Then we Using the argument of the above Theorem we can have prove the following Lemma. <(b∗∞ )(aaE ),⊗ > ij ij kl

Lemma 4.2. Let A be a unital, commutative and =<()()[baaE∗∞ , ⊗ ] >=< (), b∗ permutation invariant MBA which satisfies the equality ij ij kl ij

diag ()λλ,... = 1 for every positive integer n and ∞∞ 1 n n []aa⊗ E∞∗>= ∑∑ss==11sk sl sl sk every λ1,...,λn ∈{11} − , . If A is weakly amenable, then ∞ K ()A is weakly amenable. = . ∞ ∑ s =1 sl sk Let X be a reflexive Banach space with a Schauder Thus basis {}x n whose unconditional constant is equal to one. If we equip with the MBS structure of Example ∗∗n [(b ij)(a ij )] kl =∗− weak limn bsla sk . (1) b ∑ s =1 3.2(III), then M ∞ () and BX() are isometrically Also we have isomorphic as Banach spaces and K ∞ () and K ()X are isometrically isomorphic as Banach algebras. ∗∗ ∗ ∞ <,⊗>()()mbaEij ij kl The following result follows immediately from [10, ∗∗∗ ∞ ∗∗ Theorem 4.5] but still it is worth to be mentioned here, =<()()[]mbaEij , ij ⊗kl >=< () mij , because our approach is totally different. ∞∞ []ba∗∞⊗ E >= ∑∑ss==11ks ls ls sk Theorem 4.3. Let X be a reflexive Banach space with ∞ a Schauder basis whose unconditional constant is equal =< mb∗∗ ∗ , a > . ∑ s =1 ls ks to one. Then K ()X is weakly amenable. Proof. We equip with the MBS structure of the Thus Example 3.3(III). Clearly (.), satisfies the n n [(mb∗∗∗ )( )]=∗− weak lim mb∗∗∗. (2) conditions of the Lemma 4.2 and hence the weak ij ij kl n ∑ s =1 ls ks amenability of K ∞ ( ) can be earned from the weak Therefore amenability of . But by the above statement, ∗∗ ∗∗ ∞ <,⊗>()()mnfEij ij kl KX()= K∞ () and hence K ()X is weakly amenable. □ ∗∗ ∗∗ ∞ ∗∗ =<()()[]mnaEij , ij ⊗ks >=< () mij ,

∞∞ []nf∗∗⊗ E ∞ >= 5. Topological Center of Approximable ∑∑ss==11sl ks ks sl Matrices ∞ =< mn∗∗ ∗∗ , f > . In this section we study the topological centers of the ∑ s =1 ks sl algebra of approximable matrices. Recall that a MBS A ∗∗ So the first Arens product on M ()A can be is approximable if K ()AMA= b (). The dual of an n ∞∞ expressed by the following identity. operator space is an example of an approximable MBS. n [(mn∗∗∗∗ )( )]=∗− weak lim mn∗∗∗∗. (3) Theorem 5.1. Let A be a MBA with approximable ij ij kl n∑ ks sl s =1 ∗∗ b ∗∗ dual. Then ZKii(())∞∞ A= M (() ZA ), i =,12 . proof. Since A ∗ is approximable then we have Similarly we have

∗∗b ∗ K ∞∞∞()AMAKA== () (). Hence

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∞ ∗n ∗ ⎧(ia )[( )Δ=∗−Δ ( b )] weak lim ab with L condition. Then K ∞ ()A is Arens regular if and ⎪ ij ij kl n ∑ s =1 ls ks only if A ∗ is approximable. ⎪ n (ii )[( b∗∗∗ )Δ=∗−Δ ( m )] weak lim b∗∗∗ m (4) ⎨ ij ij kl n ∑ s =1 sl sk ⎪ Corollary 5.5. For every unital operator algebra A , the n ⎪(iii )[( m∗∗ )Δ=∗−Δ ( n ∗∗ )] weak lim m∗∗ n ∗∗ . Banach algebra K ()A is Arens regular. ⎩⎪ ij ij kl n ∑ s =1 ks sl ∞ From (3) and (iii) of (4) we conclude that ∗∗ b ∗∗ 6. Proofs of Lemmas 3.4 to 3.7 Z(iiKA∞∞ ( ) )= MZA ( ( )). □ ∞ Proof of Lemma 3.4. Let {}B nn=1 be a Cauchy Corollary 5.2. Let A be a unital MBA with b sequence in M ∞ ()X and set xBijn= () n ij . Then by approximable dual. Then A is Arens regular if and only Remark 3.1(i) for arbitrary positive integers ij, , the if K ∞ ()A is Arens regular. sequence {x } is Cauchy and hence there is a ijn ∞ Theorem 5.3. Let A be a unital MBA with L Bx= ()ij ∈ MX∞ () such that limnx ijn= x ij . Now fix ∗ 2 condition. If K ∞ ()A is Arens regular then A is n > 0 and choose m > 0 such that x − 0 and positive integers ∑ ij ijm1 m ij,=1 kk12< < ... < knn < k+1 < ... such that

where M is a upper bound for {B n }. Therefore kknn+1 ()ff−≥ () ε ij ij b ()x ij ∈ MX∞ (). Let ε > 0 and consider N 1 > 0 such for every n . Now for every positive integer n there that for every mN≥ the inequality BB− 1 such that

⎧ k n +1 BB− <−() BB +ε/4. Also choose ()()aaijn= ijn NN12 N 1 ⎪ N 2 ⎪ N ()0a = NN> such that 2 +>−ε /2kknn+1 () ff (). for m≥ N we have ⎩ ij ij ijn ij ij BB−≤−+ BB B − B we have mNNm11

m <−()44BB +/+/εε ∞ NN21 lim lim<> (fa ), ( )[ E ]= 0 N 2 m n ij ijn ∑ i =1 ii < ()BB−+− ( B B ) +ε/2 NNNNN221 and NN22

N 2 m ∞ < xx− +−BB +/ε 2 lim lim<> (fa ), ( )[ E ] ∑ ij,=1 ij ijN NN1 n m ij ijn ∑ i =1 ii 1

. n ij ijn b Therefore M ∞ ()A is a Banach space. □ Since for every n we have

|<()()fa , >|=||>ε / 2 ij ijn ij ij ijn Proof of Lemma 3.5. Let ()x ij ∈ KX∞ () and T n be a lim sequence in ⎯⎯→ M n ()X such that limTxnij= ( ) . then limn<, (fa ij ) ( ijn ) >≠ 0 . Hence by 1.5.11(f) of [13] For every positive integer n , let k be the least K ∞ ()A is not Arens regular. □ n positive integer that []T nij= 0 for all ij,> kn . Now let Corollary 5.4. Let A be a unital Arens regular MBA

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isomorphic with the BK(()) X, Y via the following ε > 0 and choose N ∈ such that ()xTij−

nm ∞ References +⊗∑∑[]xEij ij . ijn==+11 1. Arens R. The adjoint of a bilinear operation. Proc. Amer. Math. Soc., 2: 839-848 (1951). mm ∞ nm ∞ 2. Bade W.G., Curtis P.C., and Dales H.G. Amenability and But []x ⊗E and []x ⊗E ∑∑in=+11 j = ij ij ∑∑ijn==+11ij ij weak amenability for Beurling and Lipschitz algebras. m Proc. London Math. Soc., 55(3): 359-377 (1987). are submatrices of [(x ij )−T N ] . So 3. Bhatia R. Matrix Analysis. GTM 169, Springer Verlag, Berlin (1997). mn ()xxij−≤ () ij 2() xTij −< N ε 4. Blanco A. On the weak amenability of A(X) and its relation with the approximation property. Journal of and hence limn (x )= (x ) . The converse statement Functional Analysis, 203: 1-26 (2003). nijij→∞ 5. Blecher D.P., Ruan Z.J., and Sinclair A.M. A is clear. □ characterization of operator algebras. J. Funct. Anal., 188: 188-201 (1990). Proof of Lemma 3.6. Let EaFb=(),= () ∈ KA (), 6. Civin P. and Yood B. The second conjugate space of a rs rs ∞ Banach algebra as an algebra. Pacific J. Math., 11(3): ε > 0 be given and n ∈ . For 1 ≤,ij ≤ n, choose 847-870 (1961). ∞ 2 7. Effros E.G. and Ruan Z.J. Operator spaces. London Math. mij such that abik kj < ε/ n, for every s ≥ mij . ∑ ks= 1 Soc. Monographs, new series, No. 23, Oxford Univ. Press Setting mmaxm=|≤,{1 ijn≤}, we have (2000). ij 8. Esslamzadeh G.H. Banach algebra structure and amenability of a class of matrix algebras with [(ab )( )]− [mm ( a ) ( b )] applications. J. Funct. Anal., 161: 364-383 (1999). n ij ij n ij ij n 9. Forrest B.E. and Marcoux L.W. Weak amenability of

∞ m triangular Banach algebras. Trans. Amer. Math. Soc., 354: =−()()ab ab ni∑∑kk==11kkjnikkj 1435-1452 (2002). n 10. Gronbaek N., Johnson B.E., and Willis G.A. Amenability ∞ of Banach algebras of compact operators. Israel J. Math., = ni()abkkj ∑ km=+1 n 87: 289-324 (1994). 11. Haagerup U. All nuclear C*-algebras are amenable. ∞ ≤ n ab Invent. Math., 74: 305-319 (1983). ∑ ij,=1 ∑ km=+1 ik kj 1 12. Lindenstrauss J.and Tzafriri L. Classical Banach Spaces, I. Springer Verlag, Berlin, Heidelberg (1977).

b Proof of Lemma 3.7. Let ()fMBXYij ∈,∞ (( )). Define,

n f ((xfxxKij ))=,∈ limn∑ ij ( ij ) ( ij )∞ (X ). ij,=1

b It is easy to see that M ∞ ((BXY, )) is isomertically

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