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Amenability and weak amenability
of second conjugate Banach algebras


Authors: F. Ghahramani, R. J. Loy and G. A. Willis
Journal: Proc. Amer. Math. Soc. 124 (1996), 1489-1497
MSC (1991): Primary 46H20; Secondary 43A20
DOI: https://doi.org/10.1090/S0002-9939-96-03177-2
MathSciNet review: 1307520
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Abstract | References | Similar Articles | Additional Information

Abstract: For a Banach algebra $\mathfrak {A}$, amenability of $\mathfrak {A}^{**}$ necessitates amenability of $\mathfrak {A}$, and similarly for weak amenability provided $\mathfrak {A}$ is a left ideal in $\mathfrak {A}^{**}$. For $\mathfrak {G}$ a locally compact group, indeed more generally, $L^1(\mathfrak {G})^{**}$ is amenable if and only if $\mathfrak {G}$ is finite. If $L^1(\mathfrak {G})^{**}$ is weakly amenable, then $M(\mathfrak {G})$ is weakly amenable.


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  • 1. R. Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2 (1951), 839--848. MR 13:659f
  • 2. W. G. Bade, P. C. Curtis, Jr., and H. G. Dales, Amenability and weak amenability for Beurling and Lipschitz algebras, Proc. London Math. Soc. 55 (1987), 359--377. MR 88f:46098
  • 3. F. F. Bonsall and J. Duncan, Complete normed algebras, Springer-Verlag, New York, 1973. MR 54:11013
  • 4. G. Brown and W. Moran, Point derivations on $M(G)$, Bull. London Math. Soc. 8 (1976), 57--64. MR 54:5744
  • 5. J. W. Bunce and W. L. Paschke, Derivations on a $C^*$-algebra and its double dual, J. Funct. Anal. 37 (1980), 235--247. MR 81k:46064
  • 6. P. Civin and B. Yood, The second conjugate space of a Banach algebra as an algebra, Pacific J. Math. 11 (1961), 820--847. MR 26:622
  • 7. P. C. Curtis, Jr. and R. J. Loy, The structure of amenable Banach algebras, J. London Math. Soc. (2) 40 (1989), 89--104. MR 90k:46114
  • 8. J. Duncan and S. A. R. Hosseiniun, The second dual of a Banach algebra , Proc. Roy. Soc. Edinburgh Sect. A 19 (1979), 309--325. MR 81f:46057
  • 9. J. Duncan and A. L. T. Paterson, Amenability for discrete convolution semigroup algebras, Math. Scand. 66 (1990), 141--146. MR 91m:43001
  • 10. M. Despi\'{c} and F. Ghahramani, Weak amenability of group algebras of locally compact groups, Canad. Bull. Math. 37 (1994), 165--167. MR 95c:43003
  • 11. J. E. Galé, T. J. Ransford, and M. C. White, Weakly compact homomorphisms, Trans. Amer. Math. Soc. 331 (1992), 815--824. MR 92h:46074
  • 12. F. Ghahramani and A. T.-M. Lau, Isometric isomorphisms between the conjugate algebras of group algebras, Bull. London Math. Soc. 20 (1988), 342--344. MR 89e:43008
  • 13. F. Ghahramani, A. T.-M. Lau, and V. Losert, Isometric isomorphisms between Banach algebras related to locally compact groups, Trans. Amer. Math. Soc. 321 (1990), 273--283. MR 90m:43010
  • 14. F. Gourdeau, Amenability of Banach algebras, Ph.D. thesis, University of Cambridge, 1989. MR 90a:46125
  • 15. N. Grønbæk, Amenability of weighted discrete covolution algebras on cancellative semigroups, Proc. Roy. Soc. Edinburgh Sect. A 110 (1988), 351--360. MR 89k:43004
  • 16. A. Ya. Helemskii, The homology of Banach and topological algebras, Kluwer, Dordrecht, 1989. MR 92d:46178
  • 17. B. E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc. 127 (1972). MR 51:11130
  • 18. ------, Approximate diagonals and cohomology of certain annihilator Banach algebras, Amer. J. Math. 94 (1972), 685--698. MR 47:5598
  • 19. ------, Weak amenability of group algebras, Bull. London Math. Soc. 23 (1991), 281--284. MR 92k:43004
  • 20. A. T.-M. Lau, Continuity of Arens multiplication on the dual space of bounded uniformly continuous functions on locally compact groups and topological semigroups, Math. Proc. Cambridge Philos. Soc. 99 (1986), 273--283. MR 87i:43001
  • 21. A. T.-M. Lau and V. Losert, On the second conjugate algebra of $L^1(\germ G)$ of a locally compact group, J. London Math. Soc. 37 (1988), 464--470. MR 89e:43007
  • 22. A. T.-M. Lau and R. J. Loy, Amenability of convolution algebras, Math. Scand. (to appear).
  • 23. D. R. Sherbert, The structure of ideals and point derivatives in Banach algebras of Lipschitz functions, Trans. Amer. Math. Soc. 111 (1964), 240--272. MR 28:4385
  • 24. S. Wasserman, On tensor products of certain group $C^*$-algebras, J. Funct. Anal. 23 (1976), 239--254.
  • 25. N. J. Young, The irregularity of multiplication in group algebras, Quart. J. Math. Oxford 24 (1973), 59--62. MR 47:9290

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Additional Information

F. Ghahramani
Affiliation: Department of Mathematics and Astronomy, University of Manitoba, Winnipeg, Canada R3T 2N2
Email: ghahram@cc.umanitoba.ca

R. J. Loy
Affiliation: Department of Mathematics, Australian National University, ACT 0200, Australia
Email: loyrmath@durras.anu.edu.au

G. A. Willis
Affiliation: Department of Mathematics, The University of Newcastle, Newcastle 2308, Australia
Email: george@frey.newcastle.edu.au

DOI: https://doi.org/10.1090/S0002-9939-96-03177-2
Keywords: Amenability, locally compact group, second conjugate space
Received by editor(s): June 27, 1994
Received by editor(s) in revised form: October 19, 1994
Communicated by: Theodore Gamelin
Article copyright: © Copyright 1996 American Mathematical Society

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