Connes-amenability of bidual and weighted semigroup algebras

We investigate the notion of Connes-amenability, introduced by Runde in [14], for bidual algebras and weighted semigroup algebras. We provide some simplifications to the notion of a σWC-virtual diagonal, as introduced in [10], 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(S, ω), we have that l(S, ω) is Connes-amenable (with respect to the canonical predual c0(S)) if and only if l (S, ω) is amenable, which is in turn equivalent to S being an amenable group. This latter point was first shown by Grönbæk in [5], 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. 2000 Mathematics Subject Classification: 22D15, 43A20, 46H25, 46H99 (primary), 46E15, 46M20, 47B47.