International Mathematical Forum, 5, 2010, no. 16, 775-779 Approximate n Ideal Amenability of Banach Algebras M. Shadab Amirkabir University of Technology Hafez Ave., P. O. Box 15914, Tehran, Iran shadab.maryam@gmail.aom G. H. Essalmzadeh Shiraz University P. O. Box 71454, Shiraz, Iran Abstract In this paper we introduce approximate n ideal amenability for a Banach algebra A. Let I be a closed two-sided ideal in A, we say A is approximately n I weakly amenable if every bounded derivation D : A I (n) is approximately inner, and, A is approximately n ideally amenable if A is approximately n I weakly amenable for every twosided ideal in A. We study the approximate I weak amenability of a Banach algebra A for some special closed two-sided ideal I. Mathematics Subject Classification: 46H25 Keywords: Approximately inner derivation, approximately n ideally amenable, ideally amenable 1 Introduction Let A be a Banach algebra and X be a Banach A module, that is, X is a Banach space and an A module such that the module operations (a, x) ax and (a, x) xa from A X into X are jointly continuous. The dual space X of X is also a Banach A module by the following module actions: <x,ax >=< xa, x >, < x, x a>=< ax, x > (a A,x X, x X ). In particular, for every n N, the n th dual X (n) of X is a Banach A module, and so for every closed ideal I of A, I is a Banach A module and I (n) is a
776 M. Shadab and G. H. Essalmzadeh dual A module for every n N. Let X be a Banach A module. Then a continuous linear map D : A X is called a derivation if D(ab) =a.d(b)+d(a).b (a, b A). For x X, we define δ x : A X as follows: δ x (a) =a.x x.a (a A). It is easy to show that δ x is a derivation. Such derivations are called inner derivations. We denote the set of continuous derivations from A into X by Z 1 (A,X) and the set of inner derivations by B 1 (A,X). We denote space by H 1 (A,X) and the quotient space by Z 1 (A,X)/B (A,X). The space H 1 (A,X) is called the first cohomology group of A with coefficients in X. A is called amenable if every derivation from A into every dual A module is inner. A is called weakly amenable if, H 1 (A, A )={0}. A Banach algebra A is called n weakly amenable if, H 1 (A, A (n) )={0}. A derivation D : A X is called approximately inner if there exists a net (x ) X such that D(a) = lim (a.x x.a) (a A). A Banach algebra A is called approximately amenable if for each Banach A module X every bounded derivation D : A X is approximately inner. A is called approximately weakly amenable if every bounded derivation D : A A is approximately inner. A is called approximately n weak amenable if any continuous derivation D : A A (n) is approximately inner. Further, A Banach algebra A is called ideally amenable if H 1 (A,I )={0}, for every closed ideal I of A. This definition was introduced by Eshaghi Gordji and Yazdanpanah in [4]. In this paper we introduce approximate n ideal amenability of Banach algebra. 2 Approximate n ideal amenability Definition 2.1. Let A be a Banach algebra, n N and I be a closed two-sided ideal in A. Then A is approximately n I weakly amenable if any continuous derivation D : A I (n) is approximately inner, A is approximately n ideally amenable if A is approximately n I weakly amenable for every closed twosided ideal in A and A is permanently approximately ideally amenable if A is approximately n I weakly amenable for every closed two-sided ideal I in A and for each n N. Theorem 2.2. Let A be a Banach algebra and I be a closed ideal of A. For each n N, ifa is approximately (n +2) I weakly amenable then A is approximately n I weakly amenable.
Approximate n ideal amenability 777 Proof. Let D Z 1 (A,I (n) ). Since J : I (n) I (n+2) is an A module homomorphism, then D can be viewed as an element of D Z 1 (A,I (n+2) ), and so there exists a net (Λ ) I (n+2) with D(a) = lim a.λ Λ.a (a A). Let P : I (n+2) I (n) be the natural projection. Then D(a) =P (D(a)) = lim a.p (Λ ) P (Λ ).a (a A). So D is an approximately inner derivation. Thus A is approximately n I weakly amenable. Corollary 2.3. Let A be a Banach algebra, and n N. IfA is approximately (n +2) ideally amenable then A is approximately n ideally amenable. Theorem 2.4. Let A be a Banach algebra and I be a closed two-sided ideal in A with a bounded approximate identity. If A is approximately n ideally amenable (or permanently approximately ideally amenable), then I is approximately n ideally amenable (or permanently approximately ideally amenable). Proof. Let J be a closed two-sided ideal in I. It is easy to see that J is an ideal in A. Let D Z 1 (I,J (n) ). By [7, Proposition 2.1.6], D can be extended to D Z 1 (A,J (n) ). So there is a net (x ) J (n) such that D(a) = lim(a.x x.a) (a A). Then D(i) = D(i) = lim (i.x x.i) (i I). So D is approximately inner. Corollary 2.5. Let A be a Banach algebra with a bounded approximate identity and M(A) be the multiplier algebra of A. IfM(A) is approximately n ideally amenable (or permanently approximately ideally amenable), then A is approximately n ideally amenable (or permanently approximately ideally amenable). Let A be the unitization of A. We know that A is amenable if and only if A is amenable [1]. If A is weakly amenable then A is weakly amenable and the weak amenability of A does not imply the weak amenability of A [6]. Gordji and Yazdanpanah have shown that A is ideally amenable if and only if A is ideally amenable [4]. Also, Esslamzadeh and Shojaee have shown that if A has a bounded approximate identity then A is approximately weakly amenable if and only if A is approximately wakly amenable [2]. In the following, we will take the same result for approximate n ideal amenability. Proposition 2.6. Let A be a Banach algebra, and n N. Then the following assertions hold. (i) If A is approximately n ideally amenable, then A is approximately n ideally amenable. (ii) If A is approximately (2n 1) ideally amenable, then A is approximately (2n 1) ideally amenable.
778 M. Shadab and G. H. Essalmzadeh Proof. (i) Let A be approximately n ideally amenable, and I be a closed ideal of A. Let D Z 1 (A,I (n) ). It is easy to show that I is an ideal of A. We define D : A I (n) by D(a + ) =D(a), (a A, C). Then D Z 1 (A,I (n) ). Since A is approximately n ideally amenable, then D is approximately inner, and hence D is approximately inner. (ii) Let A be approximately (2n 1) ideally amenable and I be a closed ideal of A. First we know that A is approximately (2n 1) weakly amenable. Then by [2, Proposition 3.2(ii)], A is approximately (2n 1) weakly amenable. Thus A is approximately (2n 1) I weakly amenable whenever I = A. Let I A. Then 1 I and I is an ideal of A. If D Z 1 (A,I (n) ), then D(1) = 0 and D Z 1 (A,I (n) ). Hence D is approximately inner. 3 Approximate n L 1 (G) amenability for M(G) In this section we find some conditions for approximate n L 1 (G) weak amenability of M(G). Theorem 3.1. Let A be a Banach algebra and I be a closed two-sided ideal in A with a bounded approximate identity. If I is approximately n weak amenable, then A is approximately n I weakly amenable. Proof. Let D Z 1 (A,I (n) ) and ι : I A be the embeding map. D ι Z 1 (I,I (n) ) and so there exists (x ) I (n) such that Then D τ(a) = lim (a.x x.a) (a I). Since I has a bounded approximate identity, I 2 = I and for i, j I, a Awe have < ij, D(a) >=< i,jd(a) >=< i,d(ja) D(j)a > = lim < i, ja.x x.ja > lim < ai, j.x x.j > = lim <ija,x > lim < aij, x > = lim < ij, a.x x.a > (a A). Therefore D(a) = lim a.x x.a. Hence D is approximately inner. Then M(G) is approxi- Corollary 3.2. Let G be a locally compact group. mately n L 1 (G) weakly amenable. Proof. By [3, 5] L 1 (G) is permanently weakly amenable and so is permanently approximately weakly amenable, hence by Theorem 3.1, M(G) is approximately n L 1 (G) weakly amenable.
Approximate n ideal amenability 779 Corollary 3.3. Let G be a compact topological group. Then L 1 (G) is approximately n L 1 (G)-weakly amenable. Proof. Since G is compact, then L 1 (G) is an ideal in L 1 (G). Therefore by Theorem 3.1, the result follows. Let G be a locally compact group with left invariant Haar measure λ. In the following corollary, S(G) denotes a Segal subalgebra of L 1 (G). Corollary 3.4. Let G be a locally compact SIN-group and let S(G) be a Segal algebra on G. Then M(G) is approximately n S(G) weakly amenable. Proof. Since G is a SIN-group, then S(G) has a central approximate identity which is bounded in L 1 norm [3] and by [3, Theorem 5.7] S(G) is approximately permanently weakly amenable, hence by Theorem 3.1, the result follows. References [1] H. G. Dales, B anach algebras and automatic continuity, Clarendon Press, Oxford, 2000. [2] G. H. Esslamzadeh and B. Shojaee, Approximate weak amenability of Banach algebras, preprint. [3] Y. Choi, F. Ghahramani and Y. Zhang, Approximate and pseudoamenability of various classes of Banach algebras, J. Funct. Anal. 256 (2009), 3158-3191. [4] M. E. Gordji and T. Yazdanpanah, Derivationa into duals of ideals of Banach algebras, Proc. Indian Acad. Sci. 114 (4) (2004), 339-408. [5] M. E. Gordji and S. A. Hosseiniun, I deal amenability of Banach algebebras on locally compact groups, Proc. Indian. Acad. Sci. Vol. 115, No. 3 (2005), 319-325. [6] B. E. Johnson and M. C. White, A non-weakly amenable augmentation ideal, To appear. [7] V. Runde, Lectures on Amenability, Springer-Verlag, Berlin, Hedinberg, New York, 2001. Received: August, 2009