Approximate Connes-amenability of dual Banach algebras

Details Approximate Connes-amenability of dual Banach algebras Approximate Connes-amenability of dual Banach algebras

2009-08-25

arxiv.org/abs/0908.3566v2

Mathematics

Functional Analysis

19 pages. This is a totally restructured version of the former one which includes some further results on measure algebras and

Scientific paper

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 and measure algebras of 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 [E. G. Effros, Amenability and virtual diagonals for von Neumann algebras, J. Funct. Anal. 78 (1988), 137-153]. We show that in general the concepts of approximate Connes-ameanbility and Connes-ameanbility are distinct, but for measure algebras these two concepts coincide. Moreover cases where approximate Connes-amenability of $\A^{**}$ implies approximate Connes-amenability or approximate amenability of $\A$ are also discussed.

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