Connes amenability-like properties (Q2872218)

The fundamental notion of amenable group was introduced by \textit{J. von Neumann} in [Fundam. Math. 13, 333 (1929; JFM 55.0151.02)]. It was extended to Banach algebras by \textit{B. E. Johnson} [Mem. Am. Math. Soc. 127, 96 p. (1972; Zbl 0256.18014)] in such a way that \(\mathrm{L}^1(G)\) is amenable if and only if the group \(G\) is. More recently, some weaker properties have been investigated, such as Connes amenability, see [\textit{V. Runde}, Stud. Math. 148, No. 1, 47--66 (2001; Zbl 1003.46028)], and pseudo-amenability, introduced by \textit{F. Ghahramani} and \textit{Y. Zhang} [Math. Proc. Camb. Philos. Soc. 142, No. 1, 111--123 (2007; Zbl 1118.46046)].NEWLINENEWLINEIn the paper under review, two properties of dual Banach algebras, obtained by further weakening the conditions in the definition of amenability, are discussed, respectively called \(w^*\)-approximate Connes amenability and pseudo-Connes amenability. An important tool throughout the article is the interpretation of the various properties in terms of (not necessarily bounded) diagonals. Elementary properties are established and it is proved that the first of these properties does not hold for weighted Banach sequence algebras \(\ell^1(\omega)\), while the second is preserved under finite \(\ell^1\)-direct sums. Using these two results, the author is able to construct examples showing that the notion of pseudo-Connes amenability is strictly weaker than those of Connes amenability and pseudo-amenability.NEWLINENEWLINERelations between the two properties are also investigated for certain classes of algebras: a dual Banach algebra is \(w^*\)-approximately amenable if and only if its unitization is pseudo-Connes amenable, and the two notions coincide for unital algebras. \(w^*\)-approximate amenability is shown to imply pseudo-Connes amenability under some additional assumptions, such as commutativity. It is currently not known whether or not this implication holds in general. The example of \(\ell^1(\mathbb{N})\) proves that pseudo-Connes amenability does not imply \(w^*\)-approximate amenability.NEWLINENEWLINEAnother problem that remains open at this time consists in deciding if \(w^*\)-approximate Connes amenability, which is by definition a weakening of approximate Connes amenability, introduced by the author and others in [\textit{G. H. Esslamzadeh} et al., Bull. Belg. Math. Soc. - Simon Stevin 19, No. 2, 193--213 (2012; Zbl 1254.46052)], is actually equivalent to it.