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On approximate Connes-amenability of enveloping dual Banach algebras. (English) Zbl 1375.46034

A dual Banach algebra \(A\) is called approximately Connes amenable if, for every normal dual Banach \(A\)-bimodule \(X\), every \(w^{*}\)-continuous derivation \(D:A\rightarrow X\) is approximately inner, that is, there exists a net \((x_{\alpha})\subseteq X\) such that \(D(b)=\lim b\cdot x_{\alpha}-x_{\alpha}\cdot b\) for every \(b\in A\).
Let \(A\) be a Banach algebra and let \(X\) be a Banach \(A\)-bimodule \(X\). An element \(x\in X\) is called weakly almost periodic if the module actions \[ A\rightarrow X,\quad a\mapsto a\cdot x,\quad a\mapsto x\cdot a\qquad(a\in A) \] are weakly compact. The set of all weakly almost periodic elements of \(X\) is denoted by \(\mathrm{WAP}(X)\). For a Banach algebra \(A\), denote by \(F(A)\) the dual Banach algebra \(\mathrm{WAP}(A^{*})^{*}\), which is called enveloping dual Banach algebra associated to \(A\). [Y.-M. Choi et al., Math. Scand. 117, No. 2, 258–303 (2015; Zbl 1328.47041)]
The authors define a new virtual diagonal, called approximate \(\mathrm{WAP}\)-virtual diagonal. This is a net \((M_{\alpha})\subseteq F_{A}(A\otimes_{\pi}A)\) such that \[ a\cdot M_{\alpha}-M_{\alpha}\cdot a\rightarrow 0,\quad \Delta_{\mathrm{WAP}}(M_{\alpha})\rightarrow e\quad (a\in A). \] As the main result of the paper, we have the following.
Theorem. Let \(A\) be a Banach algebra. Then \(F(A)\) is approximately Connes amenable if and only if \(A\) has an approximate \(\mathrm{WAP}\)-virtual diagonal.

MSC:

46H20 Structure, classification of topological algebras
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)

Citations:

Zbl 1328.47041
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