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

Abstract

If $D:A\to X$ is a derivation from a Banach algebra to a contractive, Banach $A$-bimodule, then one can equip $X^{\ast \ast }$ with an $A^{\ast \ast }$-bimodule structure, such that the second transpose $D^{\ast \ast }:A^{\ast \ast }\to X^{\ast \ast }$ is again a derivation. We prove an analogous extension result, where $A^{\ast \ast }$ is replaced by $\F(A)$, the \emph{enveloping dual Banach algebra} of $A$, and $X^{\ast \ast }$ 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 $G$.

arXiv commentv2: AMS-LaTeX, 11pt, 40 pages. Various minor improvements and corrections, including changes to notation and additional references; also new material in Sections 5 and 6. Incorporates referee's revisions. To appear in Mathematica Scandinavica

References in corpus (1)

• Connes-amenability and normal, virtual diagonals for measure algebras, II