Extension of Derivations, and Connes- Amenability of the Enveloping Dual Banach Algebra - Choi, Yemon; Samei, Ebrahim; Stokke, Ross (Mathematica Scandinavica, 2015)
If $D:A → X$ 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 $D^{**}: A^{**} → X^{**}$ 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 $G$.