Amenability for dual Banach algebras

V. Runde¹

¹ Department of Mathematical Sciences University of Alberta Edmonton, Alberta Canada T6G 2G1

Studia Mathematica, Tome 148 (2001) no. 1, pp. 47-66

Résumé

We define a Banach algebra ${{\mathfrak A}}$ to be dual if ${{\mathfrak A}} = ({{\mathfrak A}}_\ast )^\ast $ for a closed submodule ${{\mathfrak A}}_\ast $ of ${{\mathfrak A}}^\ast $. The class of dual Banach algebras includes all $W^\ast $-algebras, but also all algebras $M(G)$ for locally compact groups $G$, all algebras ${\cal L}(E)$ for reflexive Banach spaces $E$, as well as all biduals of Arens regular Banach algebras. The general impression is that amenable, dual Banach algebras are rather the exception than the rule. We confirm this impression. We first show that under certain conditions an amenable dual Banach algebra is already super-amenable and thus finite-dimensional. We then develop two notions of amenability—Connes amenability and strong Connes amenability—which take the $w^\ast $-topology on dual Banach algebras into account. We relate the amenability of an Arens regular Banach algebra ${\mathfrak A}$ to the (strong) Connes amenability of ${\mathfrak A}^{\ast \ast }$; as an application, we show that there are reflexive Banach spaces with the approximation property such that ${\cal L}(E)$ is not Connes amenable. We characterize the amenability of inner amenable locally compact groups in terms of their algebras of pseudo-measures. Finally, we give a proof of the known fact that the amenable von Neumann algebras are the subhomogeneous ones, which avoids the equivalence of amenability and nuclearity for $C^*$-algebras.

Détail

DOI : 10.4064/sm148-1-5

Keywords: define banach algebra mathfrak dual mathfrak mathfrak ast ast closed submodule mathfrak ast mathfrak ast class dual banach algebras includes ast algebras algebras locally compact groups algebras cal reflexive banach spaces biduals arens regular banach algebras general impression amenable dual banach algebras rather exception rule confirm impression first under certain conditions amenable dual banach algebra already super amenable finite dimensional develop notions amenability connes amenability strong connes amenability which ast topology dual banach algebras account relate amenability arens regular banach algebra mathfrak strong connes amenability mathfrak ast ast application there reflexive banach spaces approximation property cal connes amenable characterize amenability inner amenable locally compact groups terms their algebras pseudo measures finally proof known amenable von neumann algebras subhomogeneous which avoids equivalence amenability nuclearity * algebras