Noninjectivity of the predual bimodule of the measure algebra of infinite discrete groups. (Q1889468)

Banach algebraic amenability in the sense of [\textit{B. E. Johnson}, Cohomology in Banach algebras, Mem. Am. Math. Soc. 127 (1972; Zbl 0256.18014)] is not a very good concept when it comes to dealing with von Neumann algebras: the only amenable von Neumann algebras are the subhomogeneous ones. If one modifies Johnson's definition to take the dual space structure of a von Neumann algebra into account, one obtains the notion of Connes-amenability, which is equivalent to various important properties of von Neumann algebras and allows for the development of a rich theory [see \textit{V. Runde}, Lectures on amenability (Lecture Notes in Mathematics. 1774. Berlin: Springer) (2002; Zbl 0999.46022) for an account]. In [Math. USSR, Sb. 68, No. 2, 555--566 (1990); translation from Mat. Sb. 180, No. 12, 1680--1690 (1989; Zbl 0721.46041)], \textit{A. Ya. Helemskiĭ} showed that a von Neumann algebra is Connes-amenable if and only if its predual bimodule is injective. In [\textit{V. Runde}, Stud. Math. 148, No. 1, 47--66 (2001; Zbl 1003.46028)], the reviewer extended the notion of Connes-amenability to so-called dual Banach algebras: Banach algebras which are dual spaces such that multiplication is separately continuous with respect to the \(w^\ast\)-topology. All von Neumann algebras are dual Banach algebras, as is the measure algebra \(M(G)\) for a locally compact group \(G\). Subsequently, Helemskiĭ raised the question of whether or not the predual bimodule of a general Connes-amenable dual Banach algebra iss necessarily injective (the converse is easily seen to be true). In the paper under review, the author shows that \(c_0(G)\) is an injective Banach \(\ell^1(G)\)-bimodule for a group \(G\) if and only if \(G\) is finite. Since \(\ell^1(G)\) is Connes-amenable if and only if \(G\) is amenable by [\textit{V. Runde}, J. Lond. Math. Soc., II. Ser. 67, No. 3, 643--656 (2003; Zbl 1040.22002)], this settles Helemskiĭ's question in the negative. Some further results concerning so-called normal, virtual diagonals are also proven.