Approximate Connes-amenability of dual Banach algebras

TL;DR

This paper introduces and characterizes approximate Connes-amenability for dual Banach algebras, establishing key equivalences and distinctions, especially for von Neumann and measure algebras, extending Effros's classical results.

Contribution

It defines approximate Connes-amenability and characterizes it via approximate virtual diagonals, extending the theory of amenability in dual Banach algebras and von Neumann algebras.

Findings

A von Neumann algebra is approximately Connes-amenable iff it has an approximate normal virtual diagonal.

For measure algebras, approximate Connes-amenability and Connes-amenability coincide.

Approximate Connes-amenability of the double dual can imply properties for the original algebra.

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.