Approximate Connes-amenability of dual Banach algebras

Abstract

We introduce the notions of approximate Connes-amenability and approximate strong Connes-amenability for dual Banach algebras. Then we characterize these two types of algebras in terms of approximate normal virtual diagonals and approximate $\sigma WC-$virtual diagonals. We investigate these properties for von Neumann algebras, measure algebra and the algebra of $p$-pseudomeasures on locally compact groups. In particular we show that a von Neumann algebra is approximately Connes-amenable if and only if it has an approximate normal virtual diagonal. This is the ``approximate'' analog of the main result of Effros in [10]. We show that in general the concepts of approximate Connes-amenability and Connes-amenability are distinct, but for measure algebras these two concepts coincide. Moreover cases where approximate Connes-amenability of $\mathcal{A}^{**}$ implies approximate Connes-amenability or approximate amenability of $\mathcal{A}$ are also discussed.