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Weak amenability of module extensions of Banach algebras
Author(s):
Yong
Zhang
Journal:
Trans. Amer. Math. Soc.
354
(2002),
4131-4151.
MSC (2000):
Primary 46H20;
Secondary 47B47, 46H10, 46H25, 46H35
Posted:
June 4, 2002
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Abstract:
We start by discussing general necessary and sufficient conditions for a module extension Banach algebra to be  -weakly amenable, for  . 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  -weakly amenable. This answers an open question raised by H. G. Dales, F. Ghahramani and N. Grønbæk.
References:
-
- 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
 into  and  , 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
Email:
zhangy@cc.umanitoba.ca
DOI:
10.1090/S0002-9947-02-03039-8
PII:
S 0002-9947(02)03039-8
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
Posted:
June 4, 2002
Additional Notes:
Research supported by NSERC
Copyright of article:
Copyright
2002,
American Mathematical Society
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