The homological essence of Connes amenability: injectivity of the predual bimodule

It is shown that the normal cohomology groups of an operator $C^*$-algebra and, in particular, of a von Neumann algebra are a special case of the standard functor $\operatorname{Ext}$ for Banach bimodules. As a consequence, it is established that Connes amenability of a von Neumann algebra is equivalent to the injectivity (in the sense of “Banach” homological algebra) of the predual bimodule of the algebra. As another consequence, a short proof of the theorem of Johnson, Kadison, and Ringrose on the coincidence of the normal and ordinary (continuous) cohomology is given, in a somewhat strengthened form.
Bibliography: 17 titles.