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- W2950441705 abstract "Let $A$ be a dual Banach algebra with predual $A_ast$ and consider the following assertions: (A) $A$ is Connes-amenable; (B) $A$ has a normal, virtual diagonal; (C) $A_ast$ is an injective $A$-bimodule. For general $A$, all that is known is that (B) implies (A) whereas, for von Neumann algebras, (A), (B), and (C) are equivalent. We show that (C) always implies (B) whereas the converse is false for $A = M(G)$ where $G$ is an infinite, locally compact group. Furthermore, we present partial solutions towards a characterization of (A) and (B) for $B(G)$ in terms of $G$: For amenable, discrete $G$ as well as for certain compact $G$, they are equivalent to $G$ having an abelian subgroup of finite index. The question of whether or not (A) and (B) are always equivalent remains open. However, we introduce a modified definition of a normal, virtual diagonal and, using this modified definition, characterize the Connes-amenable, dual Banach algebras through the existence of an appropriate notion of virtual diagonal." @default.
- W2950441705 created "2019-06-27" @default.
- W2950441705 creator A5046661856 @default.
- W2950441705 date "2003-05-29" @default.
- W2950441705 modified "2023-09-27" @default.
- W2950441705 title "Dual Banach algebras: Connes-amenability, normal, virtual diagonals, and injectivity of the predual bimodule" @default.
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