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We define a Banach algebra A to be dual if $A = (A_\ast)^\ast$ for a closedsubmodule $A_\ast$ of $A^\ast$. The class of dual Banach algebras includes all$W^\ast$-algebras, but also all algebras M(G) for locally compact groups G, allalgebras L(E) for reflexive Banach spaces E, as well as all biduals of Arensregular Banach algebras. The general impression is that amenable, dual Banachalgebras are rather the exception than the rule. We confirm this impression. Wefirst show that under certain conditions an amenable dual Banach algebra isalready super-amenable and thus finite-dimensional. We then develop two notionsof amenability - Connes-amenability and strong Connes-amenability - which takethe $w^\ast$-topology on dual Banach algebras into account. We relate theamenability of an Arens regular Banach algebra A to the (strong)Connes-amenability of $A^{\ast\ast}$; as an application, we show that there arereflexive Banach spaces with the approximation property such that L(E) is notConnes-amenable. We characterize the amenability of inner amenable locallycompact groups in terms of their algebras of pseudo-measures. Finally, we givea proof of the known fact that the amenable von Neumann algebras are thesubhomogeneous ones which avoids the equivalence of amenability and nuclearityfor $C^\ast$-algebras.

Amenability for dual Banach algebras