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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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