Derivations into iterated duals of Banach algebras

Dales, H. ; Ghahramani, F. ; Grønbæek, N.

Studia Mathematica, Tome 129 (1998), p. 19-54 / Harvested from The Polish Digital Mathematics Library

Résumé

We introduce two new notions of amenability for a Banach algebra A. The algebra A is n-weakly amenable (for n ∈ ℕ) if the first continuous cohomology group of A with coefficients in the n th dual space $A^{(n)}$ is zero; i.e., $ℋ^1(A,A^{(n)}) = {0}$. Further, A is permanently weakly amenable if A is n-weakly amenable for each n ∈ ℕ. We begin by examining the relations between m-weak amenability and n-weak amenability for distinct m,n ∈ ℕ. We then examine when Banach algebras in various classes are n-weakly amenable; we study group algebras, C*-algebras, Banach function algebras, and algebras of operators. Our results are summarized and some open questions are raised in the final section.

Détail

Publié le : 1998-01-01

Zbl 0903.46045

EUDML-ID : urn:eudml:doc:216473

Bibliographie

[00000] [1] C. A. Akemann, The dual space of an operator algebra, Trans. Amer. Math. Soc. 126 (1967), 286-302. | Zbl 0157.44603

[00001] [2] W. G. Bade, P. C. Curtis and H. G. Dales, Amenability and weak amenability for Beurling and Lipschitz algebras, Proc. London Math. Soc. (3) 55 (1987), 359-377. | Zbl 0634.46042

[00002] [3] F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer, Berlin, 1973. | Zbl 0271.46039

[00003] [4] P. Civin and B. Yood, The second conjugate space of a Banach algebra as an algebra, Pacific J. Math. 11 (1961), 847-870. | Zbl 0119.10903

[00004] [5] A. Connes, On the cohomology of operator algebras, J. Funct. Anal. 28 (1978), 248-252.

[00005] [6] I. G. Craw and N. J. Young, Regularity of multiplication in weighted group and semigroup algebras, Quart. J. Math. Oxford Ser. (2) 25 (1974), 351-358. | Zbl 0304.46027

[00006] [7] M. Despič and F. Ghahramani, Weak amenability of group algebras of locally compact groups, Canad. Math. Bull. 37 (1994), 165-167. | Zbl 0813.43001

[00007] [8] J. Diestel and J. J. Uhl, Jr., Vector Measures, Math. Surveys 15, Amer. Math. Soc., 1977.

[00008] [9] J. Duncan and S. A. Hosseinium, The second dual of a Banach algebra, Proc. Roy. Soc. Edinburgh Sect. A 84 (1979), 309-325. | Zbl 0427.46028

[00009] [10] J. F. Feinstein, Weak (F)-amenability of R(X), in: Conference on Automatic Continuity and Banach Algebras, Proc. Centre Math. Anal. Austral. Nat. Univ. 21, Austral. Nat. Univ., Canberra, 1989, 97-125.

[00010] [11] F. Ghahramani, Automorphisms of weighted measure algebras, ibid., 144-154.

[00011] [12] F. Ghahramani and J. P. McClure, Module homomorphisms of the dual modules of convolution Banach algebras, Canad. Math. Bull. 35 (1992), 180-185. | Zbl 0789.43001

[00012] [13] N. Grønbæk, Commutative Banach algebras, module derivations, and semigroups, J. London Math. Soc. (2) 40 (1989), 137-157. | Zbl 0632.46046

[00013] [14] N. Grønbæk, A characterization of weakly amenable Banach algebras, Studia Math. 94 (1989), 150-162.

[00014] [15] N. Grønbæk, Weak and cyclic amenability for non-commutative Banach algebras, Proc. Edinburgh Math. Soc. 35 (1992), 315-328. | Zbl 0760.46043

[00015] [16] N. Grønbæk, Morita equivalence for self-induced Banach algebras, Houston J. Math. 22 (1996), 109-140. | Zbl 0864.46026

[00016] [17] N. Grønbæk, B. E. Johnson and G. A. Willis, Amenability of Banach algebras of compact operators, Israel J. Math. 87 (1994), 289-324. | Zbl 0806.46058

[00017] [18] U. Haagerup, All nuclear C*-algebras are amenable, Invent. Math. 74 (1983), 305-319. | Zbl 0529.46041

[00018] [19] A. Ya. Helemskiĭ, The Homology of Banach and Topological Algebras, Kluwer, 1989.

[00019] [20] B. E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc. 127 (1972). | Zbl 0256.18014

[00020] [21] B. E. Johnson, Weak amenability of group algebras, Bull. London Math. Soc. 23 (1991), 281-284. | Zbl 0757.43002

[00021] [22] B. E. Johnson, private communication.

[00022] [23] W. B. Johnson, Factoring compact operators, Israel J. Math. 9 (1971), 337-345. | Zbl 0236.47045

[00023] [24] H. Kamowitz and S. Sheinberg, Derivations and automorphisms of L^1(0,1), Trans. Amer. Math. Soc. 135 (1969), 415-427. | Zbl 0172.41703

[00024] [25] D. Lamb, The second dual of certain Beurling algebras, preprint.

[00025] [26] T. W. Palmer, Banach Algebras and the General Theory of *-Algebras, Vol. I, Algebras and Banach Algebras, Cambridge Univ. Press, 1994. | Zbl 0809.46052

[00026] [27] G. Pisier, Factorization of Linear Operators and Geometry of Banach Spaces, CBMS Regional Conf. Ser. in Math. 60, Amer. Math. Soc., 1986.

[00027] [28] J. Pym, Remarks on the second duals of Banach algebras, J. Nigerian Math. Soc. 2 (1983), 31-33. | Zbl 0572.46044

[00028] [29] C. J. Read, Discontinuous derivations on the algebra of bounded operators on a Banach space, J. London Math. Soc. (2) 40 (1989), 305-326. | Zbl 0722.46020

[00029] [30] S. Sakai, C*-Algebras and W*-Algebras, Springer, New York, 1971.

[00030] [31] Yu. V. Selivanov, Biprojective Banach algebras, Izv. Akad. Nauk SSSR 43 (1979), 1159-1174 (in Russian); English transl.: Math. USSR-Izv. 15 (1980), 381-399.

[00031] [32] M. V. Sheĭnberg, A characterization of the algebra C(Ω) in terms of cohomology groups, Uspekhi Mat. Nauk 32 (5) (1977), 203-204 (in Russian).

[00032] [33] D. R. Sherbert, The structure of ideals and point derivations in Banach algebras of Lipschitz functions, Trans. Amer. Math. Soc. 111 (1964), 240-212. | Zbl 0121.10204

[00033] [34] S. Watanabe, A Banach algebra which is an ideal in the second dual algebra, Sci. Rep. Niigata Univ. Ser. A 11 (1974), 95-101. | Zbl 0359.46034

[00034] [35] M. Wodzicki, Vanishing of cyclic homology of stable C*-algebras, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), 329-334. | Zbl 0652.46052

[00035] [36] N. J. Young, The irregularity of multiplication in group algebras, Quart. J. Math. Oxford Ser. (2) 24 (1973), 59-62. | Zbl 0252.43009