##
**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.

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.

Reviewer: Volker Runde (Edmonton)

### MSC:

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. |

### Citations:

Zbl 0256.18014
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\textit{F. Ghahramani} and \textit{R. J. Loy}, J. Funct. Anal. 208, No. 1, 229--260 (2004; Zbl 1045.46029)

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