Article

# Approximate Connes-amenability of dual Banach algebras

Bulletin of the Belgian Mathematical Society, Simon Stevin (Impact Factor: 0.32). 08/2009;

Source: arXiv

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**ABSTRACT:**Les AA. considèrent un bimodule de Banach X sur une algèbre de Banach 𝔘; ils en étudient les duals successifs X ' ,X '' ,⋯,X (n) ,⋯ et ce qui’ils appellent leur “amenability” faible, c’est-à-dire la nullité du premier groupe de cohomologie de 𝔘 ℌ 1 (𝔘,𝔘 (n) ), dont les coefficients se trouvent dans le n-ième dual de 𝔘 pour la n-“amenability”. La question qui est étudiée avec force détails est l’hérédité de cette propriété, qui dépend de la parité de l’entier n; par ailleurs, ces résultats sont liés à l’existence d’une dérivation continue dans ces espaces successifs. Sont abordés les cas des C * -algèbres, des algèbres de Banach commutatives, des algèbres de groupes, des algèbres d’opérateurs (l’exemple le plus important). Ce très intéressant exposé se caractérise dans sa rédaction par la clarté et la rigueur; cette dernière est confirmée par l’énoncé d’un certain nombre de problèmes ouverts, qui, sans doute, feront l’objet de publications ultérieures.Studia Mathematica 01/1998; 128(1). · 0.55 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We investigate the notion of Connes-amenability for dual Banach algebras, as introduced by Runde, for bidual algebras and weighted semigroup algebras. We provide some simplifications to the notion of a $\sigma WC$-virtual diagonal, as introduced by Runde, especially in the case of the bidual of an Arens regular Banach algebra. We apply these results to discrete, weighted, weakly cancellative semigroup algebras, showing that these behave in the same way as C$^*$-algebras with regards Connes-amenability of the bidual algebra. We also show that for each one of these cancellative semigroup algebras $l^1(S,\omega)$, we have that $l^1(S,\omega)$ is Connes-amenable (with respect to the canonical predual $c_0(S)$) if and only if $l^1(S,\omega)$ is amenable, which is in turn equivalent to $S$ being an amenable group. This latter point was first shown by Gr{\"o}nb\ae k, but we provide a unified proof. Finally, we consider the homological notion of injectivity, and show that here, weighted semigroup algebras do not behave like C$^*$-algebras. Comment: 25 pages08/2005; - [Show abstract] [Hide abstract]

**ABSTRACT:**This paper continues the investigation of the first two authors begun in part I. It is shown that approximate amenability and approximate contractibility are the same properties, as are uniform approximate amenability and amenability. Bounded approximate contractibility and bounded approximate amenability are characterized by the existence of suitable operator bounded approximate identities for the diagonal ideal. Results are given on Banach sequence algebras, Lipschitz algebras and Beurling algebras, and on the crucial role of approximate identities. A new proof is given for a result due to N. Grønbæk on characterizing of amenability for Beurling algebras.Journal of Functional Analysis. 04/2008;

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