On approximate Connes-amenability of enveloping dual Banach algebras

TL;DR

This paper investigates the approximate Connes-amenability of enveloping dual Banach algebras, establishing equivalences with approximate virtual diagonals and exploring implications for group algebras.

Contribution

It introduces new concepts of approximate virtual diagonals and characterizes approximate Connes-amenability of enveloping dual Banach algebras.

Findings

F(A) is approximately Connes-amenable iff A has an approximate WAP-virtual diagonal.

For discrete groups G, approximate WAP-virtual diagonal implies approximate virtual diagonal for ℓ¹(G).

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.