International Journal of Analysis and Applications Volume 16, Number 1 (2018), 117-124 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-16-2018-117 PERMANENTLY WEAK AMENABILITY OF REES SEMIGROUP ALGEBRAS HASAN HOSSEINZADEH1 AND ALI JABBARI2 1Department of Mathematics, Ardabil Branch, Islamic Azad University, Ardabil, Iran 2Department of Mathematics, Payame Noor University, Tehran, Iran ∗Corresponding author: jabbari al@yahoo.com Abstract. In this paper, we consider n-weak amenability of full matrix algebras and we prove that the Rees semigroup algebra is permanently weakly amenable. 1. Introduction Let A be a Banach algebra, and let X be a Banach A-bimodule. Then a linear map D : A −→ X is a derivation if D(ab) = a ·D(b) + D(a) · b for every a,b ∈ A. Let x ∈ X, and set δx(a) = a · x − x · a for every a ∈ A. Then δx is a derivation; these derivations are inner derivations. The space of continuous derivations from A into X is denoted by Z1(A,X), and the subspace consisting of the inner derivations is N1(A,X); the first cohomology group of A with coefficients in X is H1(A,X) = Z1(A,X)/N1(A,X). A Banach algebra A is weakly amenable if H1(A,A∗) = {0}. For example, the group algebra L1(G) is weak amenable for each locally compact group G [7]. Let k ∈ N; a Banach algebra A is called k-weakly amenable if H1(A,A(k)) = {0}. Dales, Ghahramani and Grønbæk brought the concept of k-weak amenability of Banach algebras [5]. A Banach algebra A is called Received 23rd September, 2017; accepted 7th December, 2017; published 3rd January, 2018. 2010 Mathematics Subject Classification. Primary 43A07, Secondary 46H25. Key words and phrases. amenability; inverse semigroup; Rees semigroup; weak amenability. c©2018 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 117 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-117 Int. J. Anal. Appl. 16 (1) (2018) 118 permanently weakly amenable if H1(A,A(k)) = {0}, for each k ∈ N. In [5], authors showed that for a locally compact group G, L1(G) is n-weakly amenable for all odd numbers n, but for even case this was open. This open problem solved in [4] and a new prove introduced by Zhang [9]. The above mentioned problem open for semigroups and semigroup algebras. For Rees semi group algebras, Mewomo [8], proved that these algebras are (2k+1)-weakly amenable, in this paper, we investigate permanent weak amenability of n×n matrix Banach algebras. Finally, we prove that the Rees semigroup algebras are permanently weak amenable. 2. Characterization of Derivations Consider the algebra Mn of n×n matrices. Let A be a Banach algebra. The Banach algebra Mn(A) is the collection of n×n matrices with components in A. We identify the dual of Mn(A) with Mn(A∗) and we have (a · Λ)ij = n∑ s=1 ajs ·λis, (Λ ·a)ij = n∑ s=1 λsj ·asi, (2.1) for each a = (aij) ∈ Mn(A) and Λ = (λij) ∈ Mn(A∗). Derivations from Mn(A) into Mn(A ∗) is studied in [1]. Set Eij which it is a n×n matrix, such that whose (i,j)th entry is 1 and other entries are 0. For each a ∈ A, the matrix a⊗Eij is a matrix that whose (i,j)th entry is a and others entries are 0. Lemma 2.1. Let A be a Banach algebra and let D : A −→ A∗ be a continuous derivation, then D induces a continuous derivation D : Mn(A) −→ Mn(A∗). Moreover, if D is an inner derivation, then D is inner derivation. Proof. Define D : Mn(A) −→ Mn(A∗) by D((a)ij) = (D(aij)) or D((a)ij) = (D(aji)). Clearly, continuity of D implies continuity of D. Similar to argumentation in [6, pp. 17], we have D(ab) = a · D(b) + D(a) · b for every a,b ∈ Mn(A). Thus, D is a module derivation. As well as, if D is inner, by a similar method in proof of Theorem 2.7 of [6], D is inner. � By (2.1), 〈λ⊗Ekl, (Λij) · (aij)〉 = 〈(aij) · (λ⊗Ekl), (Λij)〉 = 〈 n∑ s=1 (asl ·λ⊗Ekl), (Λij)〉 = n∑ s=1 〈asl ·λ, Λks〉 = 〈λ, n∑ s=1 Λks ·asl〉, (2.2) for each λ ∈ A∗, (Λij) ∈ Mn(A∗∗), (aij) ∈ Mn(A) and 0 ≤ k,l ≤ n. Hence, (2.2) implies that ((Λij) · (aij))kl = n∑ s=1 Λks ·asl, (2.3) Int. J. Anal. Appl. 16 (1) (2018) 119 for each (Λij) ∈ Mn(A∗∗), (aij) ∈ Mn(A) and 0 ≤ k,l ≤ n. Similarly ((aij) · (Λij))kl = n∑ s=1 aks · Λsl, (2.4) for each (Λij) ∈ Mn(A∗∗), (aij) ∈ Mn(A) and 0 ≤ k,l ≤ n. By induction on m, for each (aij) ∈ Mn(A) and (λij) ∈ Mn(A(m)) we have ((λij) · (aij))kl = n∑ s=1 λsl ·ask, ((aij) · (λij))kl = n∑ s=1 als ·λks, (2.5) when m is odd and in the case where m is even, we have the following actions: ((λij) · (aij))kl = n∑ s=1 λks ·asl, ((aij) · (λij))kl = n∑ s=1 aks ·λsl. (2.6) Now; we are ready to prove the following Lemma that plays an important role in our main results. Lemma 2.2. Let A be a unital Banach algebra. Then every derivation from Mn(A) into Mn(A (m)) (A(m) is the m-th dual of A) is the sum of an inner derivation and a derivation induced by a derivation from A into A(m). Proof. Let eA be the identity element of A. Suppose that D : Mn(A) −→ Mn(A(m)) is a continuous derivation. For each i,j and k,l, define Dklij : A −→ A (m) by Dklij (a) := (D(a ⊗ Eij))kl, for each a ∈ A. Clearly, Dklij is linear. We prove this Lemma in two cases. Case 1. Let m be an odd positive number. For every a,b ∈ A and each 1 ≤ t ≤ n, we have ( [ D(a⊗Eit) ] · (b⊗Etj))kl = n∑ s=1 (D(a⊗Eit))sl · (b⊗Etj)sk = n∑ s=1 Dslit (a) · bδtsδjk = D tl it(a) · bδjk, and ((a⊗Eit) · [ D(b⊗Etj) ] )kl = n∑ s=1 (a⊗Eit)ls · (D(b⊗Etj))ks = n∑ s=1 aδilδts ·Dkstj (b) = aδil ·D kt tj (b), where δ is the Kronecker’s delta. Then Dklij (ab) = aδil ·D kt tj (b) + D tl it(a) · bδjk. (2.7) Thus, Diiii is a derivation from A into A (m). From (2.5) and (2.7), the following statements hold D jl ij(a) = D il ii(eA) ·a (i 6= l), D ki ij (a) = a ·D kj jj (eA) (j 6= k), (2.8) Int. J. Anal. Appl. 16 (1) (2018) 120 and again by (2.7) and for 1 ≤ i,j, l ≤ n, we have D jj jj (a) = D ij ji(eA) ·a + D ji ij (a) = D ij ji(eA) ·a + D li il(eA) ·a + D jl lj (a) = D ij ji(eA) ·a + D li il(eA) ·a + D ll ll(a) + a ·D jl lj (eA), (2.9) and D ij ji(a) = a ·D ij ji(eA) + D jj jj (a). (2.10) Hence D ij ji(eA) = −D ji ij (eA) for every 1 ≤ i,j ≤ n, and consequently by (2.9), the following relation holds D ji ij (a) = D li il(eA) ·a−a ·D lj jl(eA) + D ll ll(a). (2.11) Together with (2.9) and (2.10) we have D ji ij (a) = D ij ji(a) −D ij ji(eA) ·a−a ·D ij ji(eA), (2.12) for every a ∈ A. By (2.7) and (2.10) the following equality holds D ij kl(a) = D ij ki(eA) ·a + D ii il (a) = D ij ki(eA) ·a + D ji ij (eA) ·a + D ij jl (a) = D ij ki(eA) ·a + D ji ij (eA) ·a + a ·D ij jl (eA) + D jj jj (a) = D ij ki(eA) ·a + a ·D ij jl (eA) −D ij ji(eA) ·a−a ·D ij ji(eA) + D ij ji(a), (2.13) for every a ∈ A. Then by (2.8), (2.12) and (2.13), we have (D(ars))ij = n∑ k,l=1 D ij kl(akl) = n∑ k=1 D ij ki(eA) ·aki + n∑ l=1 Diiil (ail) = n∑ k=1 D ij ki(eA) ·aki + n∑ l=1 ajl ·D ij jl (eA) −Dijji(eA) ·aji −aji ·D ij ji(eA) + D ij ji(aji) = n∑ k=1 D kj kk(eA) ·aki + n∑ k=1 ajk ·Dikkk(eA) + D ji ij (aji), (2.14) for every (ars) ∈ Mn(A). As well as, (D(EkkEii))ik = n∑ k=1 Dskkk(eA)δsi + n∑ k=1 δksD is ii (eA) = D ik kk(eA) + D ik ii (eA) = 0. This shows that Dikkk(eA) = −D ik ii (eA). Now; for every 1 ≤ k,j ≤ n define Dkj = D kj kk. By the above obtained results we have (D(ars))ij = n∑ k=1 Dkj(eA) ·aki − n∑ k=1 ajk ·Dik(eA) + D ji ij (aji) = ((Drs(eA)) · (ars) − (ars) · (Drs(eA)))ij + D ji ij (aji). (2.15) Int. J. Anal. Appl. 16 (1) (2018) 121 Set D(eA) =   Dl11l(eA) . . . 0 ... Dl22l(eA) ... 0 . . . Dlnnl(eA)   n×n . Then by (2.11) and (2.15) we have D((ars)) = ( Dij(eA) + D(eA) ) · (aij) − (aij) · ( Dij(eA) + D(eA) ) +(Dllll(aij)), where (Dllll(aij)) is a diagonal matrix. Case 2. Now; let m be an even positive number. Then by (2.6) we have ( [ D(a⊗Eit) ] · (b⊗Etj))kl = n∑ s=1 (D(a⊗Eit))ks · (b⊗Etj)sl = n∑ s=1 Dksit (a) · bδtsδjl = D kt it (a) · bδjl, and ((a⊗Eit) · [ D(b⊗Etj) ] )kl = n∑ s=1 (a⊗Eit)ks · (D(b⊗Etj))sl = n∑ s=1 aδikδts ·Dsltj(b) = aδik ·D tl tj(b), for every a,b ∈ A. Then Dklij (ab) = aδik ·D tl tj(b) + D kt it (a) · bδjl. (2.16) Thus, Diiii is a derivation from A into A (m). By (2.6) and (2.16), the following equalities hold D kj ij (a) = D ki ii (eA) ·a (k 6= i), D il ij(a) = a ·D jl jj(eA) (j 6= l), (2.17) and for 1 ≤ i,j, l ≤ n, (2.16) follows Diiii(a) = D ji ji(a) + D ij ij (eA) ·a = D ij ij (eA) ·a + D jl jl(eA) ·a + D li li(a) = D ij ij (eA) ·a + D jl jl(eA) ·a + D ll ll(a) + a ·D li li(eA), (2.18) and D ji ji(a) = D ii ii(a) + D ji ji(eA) ·a, (2.19) for every a ∈ A. Therefore Dijij (eA) = −D ji ji(eA), for every 1 ≤ i,j ≤ n. Then (2.18) implies that D ji ji(a) = D jl jl(eA) ·a−a ·D il il(eA) + D ll ll(a). (2.20) As well as, D ij kl(a) = D ij kj(eA) ·a + D jj il (a) = D ij kj(eA) ·a + a ·D ij il (eA) + D ji ji(a), (2.21) Int. J. Anal. Appl. 16 (1) (2018) 122 for every a ∈ A. By using the relations (2.17) and (2.21), for every (ars) ∈ Mn(A), we have (D(ars))ij = n∑ k,l=1 D ij kl(akl) = n∑ l=1 D ij kj(eA) ·akj + n∑ k=1 D jj il (ail) = n∑ k=1 D ij kj(eA) ·akj + n∑ k=1 ail ·D ij il (eA) + D ji ji(aji) = n∑ k=1 Dikkk(eA) ·akj + n∑ k=1 aik ·D kj kk(eA) + D ji ji(aji). (2.22) Since (D(EkkEii))ik = n∑ k=1 Dkskk(eA)δis + n∑ k=1 δisD sk ii (eA) = D ki kk(eA) + D ik ii (eA) = 0, (2.23) Dkikk(eA) = −D ik ii (eA). Now; define Dkj = D kj kk for every 1 ≤ j,k ≤ n. Then by the above obtained results we have (D(ars))ij = n∑ k=1 Dik(eA) ·akj − n∑ k=1 aik ·Djk(eA) + D ji ji(aji) = ((Drs(eA)) · (ars) − (ars) · (Drs(eA)))ij + D ji ji(aji). (2.24) Similar to Case 1, set D(eA) =   D1l1l(eA) . . . 0 ... D2l2l(eA) ... 0 . . . Dnlnl(eA)   n×n . Now; by applying (2.20) and (2.24) the following holds D((ars)) = ( Dij(eA) + D(eA) ) · (aij) − (aij) · ( Dij(eA) + D(eA) ) +(Dllll(aij)). Hence proof is complete. � Weak amenability and (2k + 1)-weak amenability of Mn(A) considered in [3, 8]. Now; by above Lemma we have the following result: Theorem 2.1. Let A be a unital Banach algebra. Then A is permanently weakly amenable if and only if Mn(A) is permanently weakly amenable. Proof. Let Mn(A) be permanently weakly amenable and let D : A −→ A(k) be a continuous derivation, k ∈ N. Then by Lemma 2.1, D induces a continuous derivation D : Mn(A) −→ Mn(A(k)). Hence, by our assumption D is inner and Lemma 2.1, implies that D is inner. Conversely, suppose that A is permanently weakly amenable. Let D : Mn(A) −→ Mn(A(k)) be a contin- uous module derivation, k ∈ N. Then by Lemma 2.2, it is equal to the sum of an inner derivation and a Int. J. Anal. Appl. 16 (1) (2018) 123 derivation induced by a derivation from A into A(k). Since A is permanently weakly module, D is equal to sum of two inner derivations. Thereby, Mn(A) is permanently weakly module amenable. � Example 2.1. Let G be a discrete group. Then by [4], [5] and Theorem 2.1, Mn(` 1(G)) is permanently weakly amenable. Example 2.2. Let A be a unital C∗-algebra. Then Mn(A) is permanently weakly amenable. 3. Rees semigroup algebras Let G be a group, and m,n ∈ N; the zero adjoined to G is o. A Rees semigroup has the form S = M(G,P,m,n); here P = (aij) ∈ Mn,m(G) is the collection of n × m matrices with components in G. For x ∈ G, 1 ≤ i ≤ m and 1 ≤ j ≤ n, let (x)ij be the element of Mm,n(Go) with x in the (i,j)-th place and o elsewhere. As a set, S consists of the collection of all these matrices (x)ij. Multiplication in S is given by the formula (x)ij(y)kl = (xajky)il (x,y ∈ G, 1 ≤ i,k ≤ m, 1 ≤ j, l ≤ n). It is known that S is a semigroup. Now; consider the semigroup Mo(G,P,m,n), where the elements of this semigroup are those of M(G,P,m,n), together with the element o, identified with the matrix that has o in each place (so that o is the zero of Mo(G,P,m,n)), and the components of P are belong to Go. The matrix P is called the sandwich matrix in each case. The semigroup Mo(G,P,m,n) is a Rees matrix semigroup with a zero over G. We write Mo(G,P,n) for Mo(G,P,n,n) in the case where m = n. As well as, P is called regular if every row and column contains at least one entry in G. The semigroup Mo(G,P,m,n) is regular as a semigroup if and only if the sandwich matrix P is regular. According to [6] we have the following equalities as Banach spaces `1(S) = Mo(`1(G),P,m,n) = M(`1(G),P,m,n) ⊕Cδ0. Bowling and Duncan proved that for any Rees semigroup S, `1(S) is weakly amenable [3, Theorem 2.5] and after them Mewomo in [8], proved that `1(S) is (2k + 1)-weakly amenable where S = Mo(G,P,n), for k,n ∈ N. Now; we are completing them works as follows: Theorem 3.1. Let S = Mo(G,P,n), n ∈ N. Then `1(S) is permanently weakly amenable. Proof. It is sufficient we show that `1(S) is (2k)-weakly amenable, for k ∈ N. For any locally compact group G, `1(G) is permanently weakly amenable ( [4, pp. 3179] and [5, Theorem 4.1]). Theorem 2.1 implies that Mn(` 1(G)) is (2k)-weakly amenable. Since Mn(` 1(G)) = `1(S), `1(S) is (2k)-weakly amenable. � Let S be a semigroup. The weak amenability of `1(S) is considered by Blackmore in [2]. He proved that `1(S) to be weakly amenable whenever S is completely regular, in the sense that, for each s ∈ S, there exists Int. J. Anal. Appl. 16 (1) (2018) 124 t ∈ S with sts = s and st = ts. Suppose that S has a zero o. Then S is o-simple if S[2] 6= {o} and the only ideals in S are {o} and S. The semigroup S is called completely o-simple if it is o-simple and contains a primitive idempotent. Corollary 3.1. Let S be an infinite, completely o-simple semigroup with finitely many idempotents. Then `1(S) is permanently weakly amenable. Proof. By Corollary 4.2 of [8], it suffices to show that `1(S) is (2k)-weakly amenable, for k ∈ N. The semigroup S is isomorphic as a semigroup to a regular Rees matrix semigroup with a zero Mo(G,P,n), n ∈ N [6, Theorem 3.13]. Now; apply Theorem 3.1. � Acknowledgment. The authors wish to thank Professor Yong Zhang for pointing out the reference [9] in response to an earlier version of this article. This work is supported by a grant from the Ardabil Branch, Islamic Azad University. Thus; the first author would like to express his deep gratitude to the Ardabil Branch, Islamic Azad University, for financial supports. References [1] R. Alizadeh and G. Esslamzadeh, The structure of derivations from a full matrix algebra into its dual, Iranian J. Sci. Tec, Trans. A, 32(2008), 61–64. [2] T. D. Blackmore, Weak amenability of discrete semigroup algebras, Semigroup Forum, 55(1997), 196–205. [3] S. Bowling and J. Duncan, First order cohomology of Banach semigroup algebras, Semigroup Forum, 56(1998), 130–145. [4] Y. Choi, F. Ghahramani, Y. Zhang, Approximate and pseudo-amenability of various classes of Banach algebras, J. Funct. Anal., 256(2009), 3158-3191. [5] H. G. Dales, F. Ghahramani and N. Grønbæk, Derivatios into iterated duals of Banach algebras, Studia Math., 128(1)(1998), 19–54. [6] H. G. Dales, A. T-M. Lau and D. Strauss, Banach algebras on semigroups and their compactifications, Memoirs Amer. Math. Soc., 205, American Mathematical Society, Providence, 2010. [7] B. E. Johnson, Weak amenability of group algebras, Bull. London Math. Soc., 23(1991), 281-284. [8] O. T. Mewomo, On n-weak amenability of Rees semigroup algebras, Proc. Indian Acad. Sci., 118(4)(2008), 547–555. [9] Y. Zhang, 2m-Weak amenability of group algebras, J. Math. Anal. Appl., 396(2012), 412-416. 1. Introduction 2. Characterization of Derivations 3. Rees semigroup algebras Acknowledgment References