Approximate Connes-amenability of dual Banach algebras

Bulletin of the Belgian Mathematical Society, Simon Stevin (Impact Factor: 0.44). 08/2009; 19(2).
Source: arXiv


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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Available from: G. H. Esslamzadeh, Nov 15, 2014
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    ABSTRACT: We investigate the notion of Connes-amenability for dual Banach algebras, as introduced by Runde, for bidual algebras and weighted semigroup algebras. We provide some simplifications to the notion of a $\sigma WC$-virtual diagonal, as introduced by Runde, 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 $l^1(S,\omega)$, we have that $l^1(S,\omega)$ is Connes-amenable (with respect to the canonical predual $c_0(S)$) if and only if $l^1(S,\omega)$ is amenable, which is in turn equivalent to $S$ being an amenable group. This latter point was first shown by Gr{\"o}nb\ae k, 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. Comment: 25 pages
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