Extension of derivations, and Connes-amenability of the enveloping dual Banach algebra

TL;DR

This paper extends derivations from Banach algebras to their enveloping duals, characterizes Connes-amenability of the enveloping dual algebra, and links it to WAP-virtual diagonals without using invariant means.

Contribution

It generalizes derivation extension results to enveloping dual Banach algebras and provides new characterizations of their Connes-amenability.

Findings

F(A) is Connes-amenable iff A admits a WAP-virtual diagonal.

Existence of a WAP-virtual diagonal for L^1(G) is equivalent to a virtual diagonal.

The approach avoids using invariant means for G.

Abstract

If $D:A \to X$ is a derivation from a Banach algebra to a contractive, Banach $A$-bimodule, then one can equip $X^{**}$ with an $A^{**}$-bimodule structure, such that the second transpose $D^{**}: A^{**} \to X^{**}$ is again a derivation. We prove an analogous extension result, where $A^{**}$ is replaced by $\F(A)$, the \emph{enveloping dual Banach algebra} of $A$, and $X^{**}$ by an appropriate kind of universal, enveloping, normal dual bimodule of $X$. Using this, we obtain some new characterizations of Connes-amenability of $\F(A)$. In particular we show that $\F(A)$ is Connes-amenable if and only if $A$ admits a so-called WAP-virtual diagonal. We show that when $A=L^1(G)$, existence of a WAP-virtual diagonal is equivalent to the existence of a virtual diagonal in the usual sense. Our approach does not involve invariant means for $G$.