Connes-amenability of bidual and weighted semigroup algebras

We investigate the notion of Connes-amenability, introduced by Runde in [10], for bidual algebras and weighted semigroup algebras. We provide some simplifications to the notion of a σ W C-virtual diagonal, as introduced in [13], 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, ω), we have that l1 (S, ω) is Connes-amenable (with respect to the canonical predual c0(S)) if and only if l1 (S, ω) is amenable, which is in turn equivalent to S being an amenable group, and the weight satisfying a certain restrictive condition. This latter point was first shown by Grønbæk in [6], 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.