### 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 $X^{**}$ 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 $\mathsf{F}(A)$, the *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 $\mathsf{F}(A)$. In particular we show that $\mathsf{F}(A)$ is Connes-amenable if and only if $A$ admits a so-called $\operatorname{WAP}$-virtual diagonal. We show that when $A=L^1(G)$, existence of a $\operatorname{WAP}$-virtual diagonal is equivalent to the existence of a virtual diagonal in the usual sense. Our approach does not involve invariant means for $G$.

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PDFDOI: http://dx.doi.org/10.7146/math.scand.a-22870

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ISSN 0025-5521 (print) ISSN 1903-1807 (online)

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