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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 \(G\).

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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