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Extension of derivations, and Connes-amenability of the enveloping dual Banach algebra. (English) Zbl 1328.47041
Summary: 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 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 $$G$$.

##### MSC:
 47B48 Linear operators on Banach algebras 47B47 Commutators, derivations, elementary operators, etc. 46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
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