Download PDF - Derivations into iterated duals of Banach algebrasAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Derivations into iterated duals of Banach algebras

Authors 1, 2, 3

Affiliations

  1. Department of Pure Mathematics, University of of Leeds, Leeds LS2 9JT, England
  2. Department of Mathematics, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada
  3. Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark

Abstract

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., ℋ1(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.

Bibliography

  1. C. A. Akemann, The dual space of an operator algebra, Trans. Amer. Math. Soc. 126 (1967), 286-302.
  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.
  3. F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer, Berlin, 1973.
  4. P. Civin and B. Yood, The second conjugate space of a Banach algebra as an algebra, Pacific J. Math. 11 (1961), 847-870.
  5. A. Connes, On the cohomology of operator algebras, J. Funct. Anal. 28 (1978), 248-252.
  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.
  7. M. Despič and F. Ghahramani, Weak amenability of group algebras of locally compact groups, Canad. Math. Bull. 37 (1994), 165-167.
  8. J. Diestel and J. J. Uhl, Jr., Vector Measures, Math. Surveys 15, Amer. Math. Soc., 1977.
  9. J. Duncan and S. A. Hosseinium, The second dual of a Banach algebra, Proc. Roy. Soc. Edinburgh Sect. A 84 (1979), 309-325.
  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.
  11. F. Ghahramani, Automorphisms of weighted measure algebras, ibid., 144-154.
  12. F. Ghahramani and J. P. McClure, Module homomorphisms of the dual modules of convolution Banach algebras, Canad. Math. Bull. 35 (1992), 180-185.
  13. N. Grønbæk, Commutative Banach algebras, module derivations, and semigroups, J. London Math. Soc. (2) 40 (1989), 137-157.
  14. N. Grønbæk, A characterization of weakly amenable Banach algebras, Studia Math. 94 (1989), 150-162.
  15. N. Grønbæk, Weak and cyclic amenability for non-commutative Banach algebras, Proc. Edinburgh Math. Soc. 35 (1992), 315-328.
  16. N. Grønbæk, Morita equivalence for self-induced Banach algebras, Houston J. Math. 22 (1996), 109-140.
  17. N. Grønbæk, B. E. Johnson and G. A. Willis, Amenability of Banach algebras of compact operators, Israel J. Math. 87 (1994), 289-324.
  18. U. Haagerup, All nuclear C*-algebras are amenable, Invent. Math. 74 (1983), 305-319.
  19. A. Ya. Helemskiĭ, The Homology of Banach and Topological Algebras, Kluwer, 1989.
  20. B. E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc. 127 (1972).
  21. B. E. Johnson, Weak amenability of group algebras, Bull. London Math. Soc. 23 (1991), 281-284.
  22. B. E. Johnson, private communication.
  23. W. B. Johnson, Factoring compact operators, Israel J. Math. 9 (1971), 337-345.
  24. H. Kamowitz and S. Sheinberg, Derivations and automorphisms of L^1(0,1), Trans. Amer. Math. Soc. 135 (1969), 415-427.
  25. D. Lamb, The second dual of certain Beurling algebras, preprint.
  26. T. W. Palmer, Banach Algebras and the General Theory of *-Algebras, Vol. I, Algebras and Banach Algebras, Cambridge Univ. Press, 1994.
  27. G. Pisier, Factorization of Linear Operators and Geometry of Banach Spaces, CBMS Regional Conf. Ser. in Math. 60, Amer. Math. Soc., 1986.
  28. J. Pym, Remarks on the second duals of Banach algebras, J. Nigerian Math. Soc. 2 (1983), 31-33.
  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.
  30. S. Sakai, C*-Algebras and W*-Algebras, Springer, New York, 1971.
  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.
  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).
  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.
  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.
  35. M. Wodzicki, Vanishing of cyclic homology of stable C*-algebras, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), 329-334.
  36. N. J. Young, The irregularity of multiplication in group algebras, Quart. J. Math. Oxford Ser. (2) 24 (1973), 59-62.
Studia Mathematica
1998/128/1
Export to PBN, Opens in new tab
Pages:
19-54
Main language of publication
English
Received
1997-02-03
Accepted
1997-08-18
Published
1998
Exact and natural sciences