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Weak amenability of module extensions of Banach algebras

Author: Yong Zhang
Journal: Trans. Amer. Math. Soc. 354 (2002), 4131-4151
MSC (2000): Primary 46H20; Secondary 47B47, 46H10, 46H25, 46H35
Published electronically: June 4, 2002
MathSciNet review: 1926868
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Abstract: We start by discussing general necessary and sufficient conditions for a module extension Banach algebra to be $n$-weakly amenable, for $n = 0,1,2,\cdots$. Then we investigate various special cases. All these case studies finally provide us with a way to construct an example of a weakly amenable Banach algebra which is not $3$-weakly amenable. This answers an open question raised by H. G. Dales, F. Ghahramani and N. Grønbæk.

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  • 1. W. G. Bade, P. C. Curtis and H. G. Dales, Amenability and weak amenability for Beurling and Lipschitz algebras, Proc. London Math. Soc. 55 (1987), 359-377. MR 88f:46098
  • 2. W. G. Bade, H. G. Dales and Z. A. Lykova, Algebraic and strong splittings of extensions of Banach algebras, Mem. Amer. Math. Soc. 137, no. 656, 1999. MR 99g:46059
  • 3. F. F. Bonsall and J. Duncan, Complete normed algebras, Springer-Verlag, Berlin, 1973. MR 54:11013
  • 4. A. Brown, P. R. Halmos and C. Pearcy, Commutators of operators on Hilbert space, Canad. J. Math. 17 (1965) 695-708. MR 34:3311
  • 5. A. Brown and C. Pearcy, Commutators in factors of type III, Canad. J. Math. 18 (1966) 1152-1160. MR 34:1864
  • 6. J. B. Conway, A course in functional analysis, Springer-Verlag, New York, 1985. MR 86h:46001
  • 7. P. C. Curtis Jr., Amenability, weak amenability, and the close homomorphism property for commutative Banach algebras, Function spaces, Edwardsville, IL, 1994, 59-69, Lecture Notes in Pure and Appl. Math., 172, Dekker, New York, 1995. MR 96g:46038
  • 8. P. C. Curtis, Jr., and R. J. Loy, The structure of amenable Banach algebras, J. London Math. Soc. 40 (1989), 89-104. MR 90k:46114
  • 9. H. G. Dales, Banach algebras and automatic continuity, Oxford, New York, 2000. MR 2002e:46001
  • 10. H. G. Dales, F. Ghahramani and N. Grønbæk, Derivations into iterated duals of Banach algebras, Studia Math. 128 (1998), 19-54. MR 99g:46064
  • 11. M. Despic and F. Ghahramani, Weak amenability of group algebras of locally compact groups, Canad. Math. Bull. 37 (1994), 165-167. MR 95c:43003
  • 12. B. E. Forrest and L. W. Marcoux, Weak amenability of triangular Banach algebras, Trans. Amer. Math. Soc. 354 (2002), 1435-1452.
  • 13. F. Ghahramani, R. J. Loy, and G. A. Willis, Amenability and weak amenability of second conjugate Banach algebras, Proc. Amer. Math. Soc. 124 (1996), 1489-1497. MR 96g:46036
  • 14. N. Grønbæk, Weak and cyclic amenability for noncommutative Banach algebras, Proc. Edinburgh Math. Soc. 35 (1992), 315-328. MR 93d:46082
  • 15. N. Grønbæk, Amenability and weak amenability of tensor algebras and algebras of nuclear operators, J. Austral. Math. Soc. Ser. A 51 (1991), 483-488. MR 92f:46062
  • 16. N. Grønbæk, A characterization of weakly amenable Banach algebras, Studia Math. 94 (1989), 149-162. MR 92a:46055
  • 17. U. Haagerup, All nuclear C*-algebras are amenable, Invent. Math. 74 (1983), 305-319. MR 85g:46074
  • 18. P. R. Halmos, Commutators of operators, II, Amer. J. Math. 76 (1954), 191-198. MR 15:538d
  • 19. A. Ya. Helemskii, Banach and locally convex algebras, Oxford University Press, Oxford, 1993. MR 94f:46001
  • 20. E. Hewitt and K. A. Ross, Abstract harmonic analysis II, Springer-Verlag, Berlin, 1970. MR 41:7378
  • 21. B. E. Johnson, Weak amenability of group algebras, Bull. London Math. Soc. 23 (1991), 281-284. MR 92k:43004
  • 22. B. E. Johnson, Derivations from $L\sp 1(G)$ into $L\sp 1(G)$ and $L\sp \infty(G)$, Harmonic analysis, Luxembourg, 1987, 191-198, Lecture Notes in Math., 1359, Springer, Berlin, 1988. MR 90a:46122
  • 23. B. E. Johnson, Cohomology in Banach algebra, Mem. Amer. Math. Soc. 127, 1972. MR 51:11130
  • 24. A. T. Lau and R. J. Loy, Weak amenability of Banach algebras on locally compact groups, J. Funct. Anal. 145 (1997), 175-204. MR 98h:46068
  • 25. V. Losert, A characterization of SIN-groups, Math. Ann. 273 (1985), 81-88. MR 87j:22011
  • 26. P. Milnes, Uniformity and uniformly continuous functions for locally compact groups, Proc. Amer. Math. Soc. 109 (1990), 567-570. MR 90i:22006
  • 27. T. W. Palmer, Banach algebra and the general theory of *-algebras, Vol. I, Cambridge, 1994. MR 95c:46002
  • 28. T. W. Palmer, Classes of nonabelian, noncompact, locally compact groups, Rocky Mt. J. 8 (1978), 683-741. MR 81j:22003
  • 29. C. Pearcy and D. Topping, On commutators in ideals of compact operators, Michigan Math. J. 18 (1971), 247-252. MR 44:2077
  • 30. Yong Zhang, Nilpotent ideals in a class of Banach algebras, Proc. Amer. Math. Soc., 127 (1999), 3237-3242. MR 2000j:46086
  • 31. Yong Zhang, Weak amenability of a class of Banach algebras, Canad. Math. Bull. 44 (2001), 504-508.

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Additional Information

Yong Zhang
Affiliation: Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2

Keywords: $n$-weakly amenable, module, dual module, derivation, operator algebra
Received by editor(s): August 23, 1999
Received by editor(s) in revised form: January 25, 2002
Published electronically: June 4, 2002
Additional Notes: Research supported by NSERC
Article copyright: © Copyright 2002 American Mathematical Society

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