Ideal Amenability of Second Duals of Banach Algebras

Transcription

1 Internatonal Mathematcal Forum, 2, 2007, no. 16, Ideal Amenablty of Second Duals of Banach Algebras M. Eshagh Gord (1), F. Habban (2) and B. Hayat (3) (1) Department of Mathematcs, Faculty of Scences, Semnan Unversty, Semnan, Iran (2) Department of Mathematcs, Isfahan Unversty, Isfahan, Iran (3) Department of Mathematcs, Shahd Behesht Unversty, Tehran, Iran & & Abstract In ths paper we study the deal amenablty of second duals of Banach algebras. We nvestgate relatons between deal amenablty of the second dual of a Banach algebra wth the frst and the second Arens products. Mathematcs Subect Classfcaton: 46H25 Keywords: Ideally amenable, Banach algebra, Dervaton 1 Introducton Let A be a Banach algebra and let A be the second dual algebra of A endowed wth the frst or the second Arens product. Throughout ths paper, the frst and the second Arens product are respectvely denoted by and. These products can be defned by F G = w lm lm ˆf ĝ and F G = w lm lm ˆf ĝ where (f ) and (g ) are nets of elements of A such that f F and g G n w -topology (see [A] and [D-H]). If X s a Banach A-bmodule, then a dervaton from A nto X s a lnear map D such that for every a, b A, D(ab) =D(a).b + a.d(b). If x X and we defne δ x : A X by δ x (a) =

2 766 M. Eshagh Gord, F. Habban and B. Hayat a.x x.a (a A), then δ x s a dervaton. Dervatons of ths form are called nner dervatons. Let A be a Banach algebra and X be a Banach A-bmodule, then X s a Banach A-bmodule f for each a Aand x X and x X we defne ax,x = x,xa, x a, x = x,ax. A Banach algebra A s amenable f every dervaton from A nto every dual A-bmodule s nner, equvalently f H 1 (A,X )={0} for every Banach A- bmodule X, where H 1 (A,X ) s the frst cohomology group of A wth coeffcents n X [J1], [R], [H]. A Banach algebra A s weakly amenable f H 1 (A, A ) = {0} (the specal case of X = A) [B-C-D], [J2]. A Banach algebra A s deally amenable f H 1 (A,I )={0} for every closed two-sded deal I n A. Eshagh-Gord and Yazdanpanah have ntroduced the concept of deal amenablty of Banach algebras n [G-Y], (see [E-H]). 2 Some examples In ths secton, let A be a Banach algebra, X a Banach A bmodule and let Y be a Banach A submodule of X. We gve some examples to show that H 1 (A,X ) = {0} ( H 1 (A,X) = {0}) s not suffcent condton for H 1 (A,Y )={0} ( H 1 (A,Y)={0}). So, there are some examples to show that weak amenablty dose not mply deal amenablty. Example 1. Let A be a non weakly amenable Banach algebra wth bounded left(or rght) approxmate dentty, such that H 2 (A, A )={0}. We know that {0} ker Δ A ˆ A Δ A {0} s an exact sequence of Banach A bmodules, and {0} A (A ˆ A) (ker Δ) {0} s an admssble exact sequence. A s a Banach A submodule of (A ˆ A) and by Theorem of [R] we have H 1 (A, (A ˆ A) ) H 2 (A, A )={0}. Now, let X =(A ˆ A) and Y = A, then H 1 (A,X)={0} but H 1 (A,Y) {0}. Example 2. Let G = PS(2, R), A = X = L 1 (G) and let Y be the augmentaton deal of A. Then we have H 1 (A,Y ) {0} [J-W]. On the other hand A s weakly amenable and so H 1 (A,X )={0}. A s also an example of a weakly amenable Banach algebra whch s not deally amenable.

3 Ideal amenablty of second duals of Banach algebras 767 Example 3. Let G = PS(2, R) and let I be the augmentaton deal of L 1 (G). Suppose that X = A = I # and Y = I. Then H 1 (A,X )={0} H 1 (A,Y ) [G-Y]. Example 4. Let A be a Banach algebra. Then A As a Banach algebra by the followng product and norm: (a, x)(b, y) =(ab, ay + xb) (a, b) = a + b (a, b, x, y A), (a, b A). Let A be a weakly amenable Banach algebra wth a bounded approxmate dentty such that span {ab ba : a, b A}s dense n A. Then by Theorem 5.4 of [Zh], A As weakly amenable. Now let 0 ϕ Ω A, then D : A A {0} A defned by D(a, x) =(0,ϕ(a)ϕ) s a dervaton. Snce D((a, x)(b, y)) = D(ab, ay + xt) = (0,ϕ(ay + xb)ϕ) = (0, (ϕ(a)ϕ(y)+ϕ(x)ϕ(t))ϕ) = (0, (ϕ(x)ϕ)b)+(0,(aϕ(y))ϕ) = (0, ϕ(x)ϕ)(b, y)+(a, x)(0, ϕ(y)ϕ) = D(a, x)(b, y)+(a, x)d(b, y). If D = δ (0,f) for some f A, then we have (0, ϕ(x)ϕ) = D(a, x) = (a, x)(0,f) (0,f)(a, x) = (xf, af) (xf, fa) = (xf fx,af fa). It shows that (0, ϕ(x)ϕ) = D(a, x) = D(0, x) =(0, 0), whch s a contradcton wth ϕ 0. Now let B = X = A A and Y = {0} A. We have H 1 (B,X )= {0} and H 1 (B,Y ) {0}. 3 Ideal Amenablty of Second dual of a Banach algebra In [Gh-L-W] Ghahraman, Loy and Wlls studed some mplcatons of amenablty and weak amenablty of A. In ths paper we study the deal amenablty of the second dual of a Banach algebra.

4 768 M. Eshagh Gord, F. Habban and B. Hayat Theorem 3.1. Let A be a commutatve Banach algebra. Suppose that A s deally amenable. If one of the followng assertons holds: () A s a left deal n A. () A s a dual algebra. () A s Arens regular and every dervaton from A nto ts dual s weakly compact. Then A s deally amenable. Proof. Let A be deally amenable, then t s weakly amenable. If one of the (), () or () holds, then by Theorem 3.2 of [Gh-L-W], Theorem 2.2 of [Gh-L] and Corollary 7.5 of [D-G-V], A s weakly amenable. Snce A s commutatve, then A s deally amenable [G-Y]. Theorem 3.2. Let A be a Banach algebra. Consder A endowed wth the frst Arens product. If A s an deal of A wth a bounded approxmate dentty, then deal amenablty of A mples deal amenablty of A. Proof. Let I be a closed two sded deal of A. Then I s an deal n A. Now let D : A I be a dervaton, then there s a bounded dervaton D : A I whch s an extenson of D [R]. Snce A s deally amenable, then there exsts f I such that D = δ f. Thus D = δ f and A s deally amenable. For a Banach algebra A, let A op be the Banach algebra wth underlyng Banach space same as A and wth product gven by a b = ba. Lemma 3.3. Let A be a Banach algebra. Then A s deally amenable f and only f A op s deally amenable. Proof. It s clear that I s an deal of A f and only f I s an deal of A op.now let A be deally amenable and let D : A op I be a contnuous dervaton, where I s a closed deal of A op. Snce for every a, b A, D(ab) =D(b a) and I s a closed deal of A, then D : A I s a contnuous dervaton. But A s deally amenable, so there s ξ I such that D(a) =a.ξ ξ.a = (a ξ ξ a). Ths means that A op s deally amenable. The converse s smlar. Lemma 3.4. Let A, B be Banach algebras and let θ : A Bbe a contnuous somorphsm. If A s deally amenable, then B s deally amenable.

5 Ideal amenablty of second duals of Banach algebras 769 Proof. It s evdent. Theorem 3.5. Let A be a Banach algebra wth a contnuous ant-somorphsm ϕ : A A. Then (A, ) s deally amenable f and only f (A, ) s deally amenable. Proof. Let (A, ) be deally amenable and let ϕ : A Abe a contnuous ant-somorphsm. Take F, G Aand let (f ), (g ) be nets n A such that weak lm ˆf = F and w lm ĝ = G. Let ϕ :(A, ) (A, ) be the second adont of ϕ. We have ϕ (F G) = w lm w lm ϕ ( ˆf ĝ ) = w lm w lm (ϕ(f )ϕ(g )) = w lm w lm ϕ (ĝ )ϕ ( ˆf ) = ϕ (G) ϕ (F ). Then ϕ s a contnuous somorphsm from (A, ) onto(a, ) op and so by lemma 3.4, (A, ) op s deally amenable. Now, by lemma 3.3, (A, ) s deally amenable. The converse s smlar. Corollary 3.6. Let A be a commutatve Banach algebra. Then (A, ) s deally amenable f and only f (A, ) s deally amenable. Proof. If A s a commutatve Banach algebra then the dentty map s a contnuous ant-somorphsm on A. Now apply theorem 3.5. Acknowledgement. The second author would lke to thank the Unversty of Isfahan for ts fnancal support. References [A] R. Arens, The adont of a blnear operaton, Proc. Amer. Math. Soc. 2(1951), [B-C-D] W. G. Bade, P. G. Curts and H. G. Dales, Amenablty and weak amenablty for Beurlng and Lpschts algebra, Proc. London Math. Soc. (3) 55 (1987),

6 770 M. Eshagh Gord, F. Habban and B. Hayat [D] H.G.Dales, Banach algebras and automatc contnuty, Oxford, New York, [D-R-V] H. G. Dales, A.Rodrguez-Palacos and M. V. Velasco, The second transpose of a dervaton, J. London Math. Soc. (2) 64 (2001) [D-H] J. Duncan and S. A. Hossenun, The second dual of Banach algebra, Proc. Roy. Soc. Ednburgh Set. A 84 (1979), [E-H] M. Eshagh Gord, S. A. R. Hossenun, Ideal Amenablty of Banach Algebras on Locally Compact Groups, Proc. Indan Acad.Sc. Vol. 115 No.3, (2005), [G-Y] M. E. Gorg, T. Yazdanpanah, Dervatons nto duals of deals of Banach algebras, proc. Indan Acad.Sc. Vol. 114 No.4, (2004), [Gh-L] F. Ghahraman, J. Laal, Amenablty and topologcal centers of the second duals of Banach algebras, Bul. Austral. Math. soc. Vol. 65 (2002) [Gh-L-W] F. Ghahraman, R. J. Loy and G. A. Wlls, Amenablty and weak amenablty of second conugate Banach algebras, Proc. Amer. Math. Soc. Vol. 124, Num. 5 (1996). [H] A. Ya. Helmesk, The homology of Banach and topologcal algebras. Kluwer, (1989). [J1] B. E. Johnson, Cohomology n Banach algebras, Mem. Amer. Math. Soc. 127 (1972). [J2], Weak amenablty of group algebras, Bull. Lodon Math. Soc. 23 (1991), [J-w] B. E. Johnson and M. C. Whte, A non-weakly amenable augmentaton deal, To appear. [R] V. Runde, Lectures on amenablty, Sprnger-Verlage Berln Hednberg New York,(2001). [Zh] Y. Zhang, Weak amenablty of a module extentons of Banach algebras, Trans. Amer. Math. soc. Vol354, Number 10, (2002) Receved: July 19, 2006