Generalized notions of amenability. (English) Zbl 1045.46029

Let \(A\) be a Banach algebra, and let \(E\) be a Banach \(A\)-bimodule. A (bounded) derivation \(D : A \to E\) is said to be inner if there is \(x \in E\) such that \(D = \text{ad}_x\), where \[ \text{ad}_x a = a \cdot x - x \cdot a \qquad (a \in A). \] In his seminal memoir [Cohomology in Banach algebras, Mem. Am. Math. Soc. 127 (1972; Zbl 0256.18014)], B. E. Johnson defined a Banach algebra to be amenable if, for each Banach \(A\)-bimodule \(E\), every derivation \(D : A \to E^\ast\) is inner. The choice of terminology was motivated by the following theorem: a locally compact group \(G\) is amenable if and only if its group algebra \(L^1(G)\) is amenable.
Suppose now that \(A\) is a Banach algebra with the following property: for each Banach \(A\)-bimodule \(E\) and for each derivation \(D : A \to E^\ast\), there is a — not necessarily bounded — net \(( x_\alpha )_\alpha\) in \(E^\ast\) such that \(D a = \lim_\alpha \text{ad}_{x_\alpha} a\) for each \(a \in A\). The authors of the paper under review call such a Banach algebra approximately amenable. It is clear that every amenable Banach algebra is approximately amenable. However, as the authors show, the converse is false. In contrast, a locally compact group \(G\) is amenable if and only if \(L^1(G)\) is approximately amenable so that, for group algebras, the notions of amenability and approximate amenability coincide.
The authors systematically investigate the hereditary properties of approximate amenability, and they establish a characterization of approximate amenability through an appropriate variant of approximate diagonals. In general, the theory of approximately amenable Banach algebras parallels that of amenable Banach algebras, but the results tend to be weaker and not as aesthetically pleasing.
Besides approximate amenability, the authors discuss various other Banach algebraic properties related to (and mostly weaker than) amenability.
The paper concludes with a list of open questions. The reviewer finds the fifth of these questions particularly intriguing: What are the approximately amenable \(C^\ast\)-algebras?
It remains to be seen whether or not the notion of approximate amenability for Banach algebras will turn out to be a fruitful one like that of amenability. It is certainly a property deserving further and deeper investigations.


46H20 Structure, classification of topological algebras
43A20 \(L^1\)-algebras on groups, semigroups, etc.
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
47B47 Commutators, derivations, elementary operators, etc.


Zbl 0256.18014
Full Text: DOI


[1] Archbold, R. J., On the norm of an inner derivation of a \(C^*\)-algebra, Math. Proc. Cambridge Philos. Soc., 84, 273-291 (1978) · Zbl 0388.46038
[2] Bonsall, F. F.; Duncan, J., Complete Normed Algebras (1973), Springer: Springer New York · Zbl 0271.46039
[3] Bowling, S.; Duncan, J., First order cohomology of Banach semigroup algebras, Semigroup Forum, 56, 1, 130-145 (1998) · Zbl 0910.46055
[4] Choi, M. D., A simple \(C^*\)-algebra generated by two finite-order unitaries, Can. J. Math., 31, 867-880 (1979) · Zbl 0441.46047
[5] Chou, C., The exact cardinality of the set of invariant means on a group, Proc. Amer. Math. Soc., 55, 103-106 (1976) · Zbl 0319.43006
[6] Curtis, P. C.; Loy, R. J., The structure of amenable Banach algebras, J. London Math. Soc., 40, 2, 89-104 (1989) · Zbl 0698.46043
[7] Dales, H. G., Banach Algebras and Automatic Continuity (2000), Clarendon Press: Clarendon Press Oxford · Zbl 0981.46043
[8] Dales, H. G.; Ghahramani, F.; Helemskii, A. Ya., Amenability of measure algebras, J. London Math. Soc., 66, 2, 213-226 (2002) · Zbl 1015.43002
[9] Doran, R. S.; Wichman, J., Approximate identities and factorization in Banach modules, Lecture Note in Mathematics, Vol. 768 (1979), Springer: Springer New York · Zbl 0418.46039
[10] Ghahramani, F.; Lau, A. T.-M., Isometric isomorphisms between the second conjugate algebras of group algebras, Bull. London Math. Soc., 20, 342-344 (1988) · Zbl 0628.43002
[11] Ghahramani, F.; Lau, A. T.-M., Weak amenability of certain classes of Banach algebras without bounded approximate identities, Math. Proc. Cambridge Philos. Soc., 133, 357-372 (2002) · Zbl 1010.46048
[12] Ghahramani, F.; Lau, A. T.-M.; Losert, V., Isometric isomorphisms between Banach algebras related to group algebras, Trans. Amer. Math. Soc., 321, 273-283 (1990) · Zbl 0711.43002
[13] Ghahramani, F.; Loy, R. J.; Willis, G. A., Amenability and weak amenability of second conjugate Banach algebras, Proc. Amer. Math. Soc., 124, 1489-1497 (1996) · Zbl 0851.46035
[14] Ghahramani, F.; Runde, V.; Willis, G. A., Derivations on group algebras, Proc. London Math. Soc., 80, 3, 360-390 (2000) · Zbl 1029.22007
[15] Gourdeau, F., Amenability of Lipschitz algebras, Math. Proc. Cambridge Philos. Soc., 112, 581-588 (1992) · Zbl 0782.46043
[16] Haagerup, U., All nuclear \(C^*\)-algebras are amenable, Invent. Math., 74, 305-319 (1983) · Zbl 0529.46041
[17] Helemskii, A. Ya., The Homology of Banach and Topological Algebras (1989), Kluwer: Kluwer Dordrecht · Zbl 0968.46061
[18] Johnson, B. E., Cohomology in Banach algebras, Mem. Amer. Math. Soc., 127 (1972) · Zbl 0246.46040
[19] Johnson, B. E., Approximate diagonals and cohomology of certain annihilator Banach algebras, Amer. J. Math., 94, 685-698 (1972) · Zbl 0246.46040
[20] Johnson, B. E., Non-amenability of the Fourier algebra of a compact group, J. London Math. Soc., 50, 2, 361-374 (1994) · Zbl 0829.43004
[21] Kadison, R. V.; Lance, E. C.; Ringrose, J. R., Derivations and automorphisms of operator algebras, II, J. Funct. Analysis, 1, 204-221 (1967) · Zbl 0149.34501
[22] Lau, A. T.-M.; Paterson, A. L.T., The exact cardinality of the set of topological left invariant means on an amenable locally compact group, Proc. Amer. Math. Soc., 98, 75-80 (1986) · Zbl 0595.43003
[23] Loy, R. J., Identities in tensor products of Banach algebras, Bull. Austral. Math. Soc., 2, 253-260 (1970) · Zbl 0187.38404
[24] Paterson, A. L.T., Amenability (1988), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0648.43001
[25] Reiter, H.; Stegeman, J. D., Classical Harmonic Analysis and Locally Compact Groups (2000), Oxford University Press: Oxford University Press Oxford · Zbl 0965.43001
[26] Runde, V., Lectures on amenability, Lecture Notes in Mathematics, Vol. 1774 (2002), Springer: Springer Berlin · Zbl 0999.46022
[27] Zhang, Y., Nilpotent ideals in a class of Banach algebras, Proc. Amer. Math. Soc., 27, 3237-3242 (1999) · Zbl 0931.46035
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.