Approximate amenability of Banach category algebras with application to semigroup algebras. (English) Zbl 1222.43006

Summary: Let \(C\) be a small category. Then we consider \(\ell ^{1}(C)\) as the \(\ell ^{1}\) algebra over the morphisms of \(C\), with convolution product and also consider \(\ell^{1}(\widehat{C})\) as the \(\ell ^{1}\) algebra over the objects of \(C\), with pointwise multiplication. The main purpose of this paper is to show that approximate amenability of \(\ell ^{1}(C)\) implies of \(\ell^{1}(\widehat{C})\) and clearly this implies that \(C\) has only finitely many objects. Some applications are given, the main one is the characterization of approximate amenability for \(\ell ^{1}(S)\), where \(S\) is a Brandt semigroup, which corrects a result of M. Lashkarizadeh Bami and H. Samea [Semigroup Forum 71, 312–322 (2005; Zbl 1086.43002)].


43A20 \(L^1\)-algebras on groups, semigroups, etc.


Zbl 1086.43002
Full Text: DOI


[1] Dales, H.G.: Banach Algebras and Automatic Continuity. London Math. Soc. Monographs, vol. 24. Clarendon Press, Oxford (2000) · Zbl 0981.46043
[2] Dales, H.G., Loy, R.J., Zhang, Y.: Approximate amenability for Banach sequence algebras. Stud. Math. 177, 81–96 (2005) · Zbl 1117.46030
[3] Duncan, J., Namioka, I.: Amenability of inverse semigroups and their semigroup algebras. Proc. R. Soc. Edinb. A 80, 309–321 (1978) · Zbl 0393.22004
[4] Duncan, J., Paterson, L.T.: Amenability for discrete convolution semigroup algebras. Math. Scand. 66, 141–146 (1990) · Zbl 0748.46027
[5] Ghahramani, F., Loy, R.J.: Generalized notion of amenability. J. Funct. Anal. 208, 229–260 (2004) · Zbl 1045.46029
[6] Ghahramani, F., Loy, R.J., Zhang, Y.: Generalized notions of amenability II. J. Funct. Anal. 254(7), 1776–1810 (2008) · Zbl 1146.46023
[7] Howie, J.M.: An Introduction to Semigroup Theory. Academic Press, San Diego (1976) · Zbl 0355.20056
[8] Lashkarizadeh Bami, M., Samea, H.: Approximate amenability of certain semigroup algebras. Semigroup Forum 71, 312–322 (2005) · Zbl 1086.43002
[9] Paterson, A.L.T.: Amenability. Am. Math. Soc., Providence (1978)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.