×

Amenability and weak amenability of second conjugate Banach algebras. (English) Zbl 0851.46035

Proc. Am. Math. Soc. 124, No. 5, 1489-1497 (1996); addendum ibid. 148, No. 10, 4573-4575 (2020).
Summary: 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.

MSC:

46H20 Structure, classification of topological algebras
43A20 \(L^1\)-algebras on groups, semigroups, etc.
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] R. Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2 (1951), 839–848. · Zbl 0044.32601
[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. (3) 55 (1987), no. 2, 359 – 377. · Zbl 0634.46042 · doi:10.1093/plms/s3-55_2.359
[3] Frank F. Bonsall and John Duncan, Complete normed algebras, Springer-Verlag, New York-Heidelberg, 1973. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 80. · Zbl 0271.46039
[4] Gavin Brown and William Moran, Point derivations on \?(\?), Bull. London Math. Soc. 8 (1976), no. 1, 57 – 64. · Zbl 0321.43003 · doi:10.1112/blms/8.1.57
[5] John W. Bunce and William L. Paschke, Derivations on a \?*-algebra and its double dual, J. Funct. Anal. 37 (1980), no. 2, 235 – 247. · Zbl 0433.46052 · doi:10.1016/0022-1236(80)90043-9
[6] Paul Civin and Bertram Yood, The second conjugate space of a Banach algebra as an algebra, Pacific J. Math. 11 (1961), 847 – 870. · Zbl 0119.10903
[7] P. C. Curtis Jr. and R. J. Loy, The structure of amenable Banach algebras, J. London Math. Soc. (2) 40 (1989), no. 1, 89 – 104. · Zbl 0698.46043 · doi:10.1112/jlms/s2-40.1.89
[8] J. Duncan and S. A. R. Hosseiniun, The second dual of a Banach algebra, Proc. Roy. Soc. Edinburgh Sect. A 84 (1979), no. 3-4, 309 – 325. · Zbl 0427.46028 · doi:10.1017/S0308210500017170
[9] J. Duncan and A. L. T. Paterson, Amenability for discrete convolution semigroup algebras, Math. Scand. 66 (1990), no. 1, 141 – 146. · Zbl 0748.46027 · doi:10.7146/math.scand.a-12298
[10] M. Despić and F. Ghahramani, Weak amenability of group algebras of locally compact groups, Canad. Math. Bull. 37 (1994), no. 2, 165 – 167. · Zbl 0813.43001 · doi:10.4153/CMB-1994-024-4
[11] J. E. Galé, T. J. Ransford, and M. C. White, Weakly compact homomorphisms, Trans. Amer. Math. Soc. 331 (1992), no. 2, 815 – 824. · Zbl 0761.46037
[12] F. Ghahramani and A. T. Lau, Isometric isomorphisms between the second conjugate algebras of group algebras, Bull. London Math. Soc. 20 (1988), no. 4, 342 – 344. · Zbl 0628.43002 · doi:10.1112/blms/20.4.342
[13] F. Ghahramani, A. T. Lau, and V. Losert, Isometric isomorphisms between Banach algebras related to locally compact groups, Trans. Amer. Math. Soc. 321 (1990), no. 1, 273 – 283. · Zbl 0711.43002
[14] Frédéric Gourdeau, Amenability of Banach algebras, Math. Proc. Cambridge Philos. Soc. 105 (1989), no. 2, 351 – 355. · Zbl 0717.46042 · doi:10.1017/S0305004100067840
[15] Niels Grønbæk, Amenability of weighted discrete convolution algebras on cancellative semigroups, Proc. Roy. Soc. Edinburgh Sect. A 110 (1988), no. 3-4, 351 – 360. · Zbl 0678.46038 · doi:10.1017/S0308210500022344
[16] A. Ya. Helemskii, The homology of Banach and topological algebras, Mathematics and its Applications (Soviet Series), vol. 41, Kluwer Academic Publishers Group, Dordrecht, 1989. Translated from the Russian by Alan West.
[17] Barry Edward Johnson, Cohomology in Banach algebras, American Mathematical Society, Providence, R.I., 1972. Memoirs of the American Mathematical Society, No. 127. · Zbl 0256.18014
[18] B. E. Johnson, Approximate diagonals and cohomology of certain annihilator Banach algebras, Amer. J. Math. 94 (1972), 685 – 698. · Zbl 0246.46040 · doi:10.2307/2373751
[19] B. E. Johnson, Weak amenability of group algebras, Bull. London Math. Soc. 23 (1991), no. 3, 281 – 284. · Zbl 0757.43002 · doi:10.1112/blms/23.3.281
[20] Anthony To Ming 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), no. 2, 273 – 283. · Zbl 0591.43003 · doi:10.1017/S0305004100064197
[21] Anthony To Ming Lau and Viktor Losert, On the second conjugate algebra of \?\(_{1}\)(\?) of a locally compact group, J. London Math. Soc. (2) 37 (1988), no. 3, 464 – 470. · Zbl 0608.43002 · doi:10.1112/jlms/s2-37.3.464
[22] A. T.-M. Lau and R. J. Loy, Amenability of convolution algebras, Math. Scand. (to appear). · Zbl 0880.46038
[23] Donald R. Sherbert, The structure of ideals and point derivations in Banach algebras of Lipschitz functions, Trans. Amer. Math. Soc. 111 (1964), 240 – 272. · Zbl 0121.10204
[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 Ser. (2) 24 (1973), 59 – 62. · Zbl 0252.43009 · doi:10.1093/qmath/24.1.59
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.