Article

Approximate Connes-amenability of dual Banach algebras

(Impact Factor: 0.44). 08/2009; 19(2).
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.

Full-text

Available from: G. H. Esslamzadeh, Nov 15, 2014
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Article: On approximate Connes-amenability of enveloping dual Banach algebras
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ABSTRACT: For a Banach algebra $\mathcal{A}$, we introduce various approximate virtual diagonals such as approximate WAP-virtual diagonal and approximate virtual diagonal. For the enveloping dual Banach algebra $F(\mathcal{A})$ of $\mathcal{A}$, we show that $F(\mathcal{A})$ is approximately Connes-amenable if and only if $\mathcal{A}$ has an approximate WAP-virtual diagonal. Further, for a discrete group $G$, we show that if the group algebra $\ell^1(G)$ has an approximate WAP-virtual diagonal, then it has an approximate virtual diagonal.
Full-text · Article · Jan 2015