Article

Approximate Connes-amenability of dual Banach algebras

Bulletin of the Belgian Mathematical Society, Simon Stevin (Impact Factor: 0.32). 08/2009;
Source: arXiv

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 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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