Foundations of an equivariant cohomology theory of Banach algebras. I
In several mathematical and physical situations the appearance of algebras is associated with special group actions. Of course, the introduction of an action can drastically change the equivariant cohomology from the classical ones and as those, its purpose is to help in describing and classifying these combined structures providing characterizations of certain classes [cf. \textit{D. Gong}, Mich. Math. J. 39, No. 3, 455-473 (1992; Zbl 0783.55005), \textit{S. Klimek}, \textit{W. Kondracki} and \textit{A. Lesniewski}, K-Theory 4, No. 3, 201-218 (1991; Zbl 0744.46063) and \textit{P. Baum} and \textit{A. Connes}, Lond. Math. Soc.Lect. Note Ser. 135, 1-20 (1988; Zbl 0685.46041)]. In this first part of a treatise, after an equivariant version of the basic definitions has been formulated in the purely algebraic and the Banach settings, special attention is paid to the dual equivariant modules. For a given (locally compact) topological group \(G\) acting continuously on a Banach (or \(C^*\)-)algebra \(\mathcal U\) and coherently on a Banach \({\mathcal U}\)-bimodule \(M\), the equivariant cohomology groups \(H^n_G({\mathcal U}, M)\) are then considered, in particular in lower dimensions, a dimension shift formula is also valid (with some restrictions), and sufficient conditions for the induced natural map \(\Phi: H^n_G({\mathcal U}, M)\to H^n({\mathcal U}, M)\) to be injective and/or surjective are given, providing an isomorphism, beyond the case of a trivial action, when \(G\) is finite with inner action on \(\mathcal U\). Next, the relative cohomology of \textit{M. Gerstenhaber} and \textit{S. D. Schack} [J. Pure Appl. Algebra 43, 53-74 (1986; Zbl 0603.16021)] is settled in the present context, and a chapter is devoted to cohomologically trivial algebras i.e., \(G\)-contractible and \(G\)-amenable. The theorems are as expected so far, modulo the necessary assumptions, which are stronger in the equivariant case. In fact, the ordinary notions still imply the equivariant ones for \(G\) a compact, or amenable group respectively, a \(C^*\)-algebra is contractible if and only if it is \(G\)-contractible, but it is unknown as yet whether a \(G\)-amenable algebra is amenable, for some action of a non-discrete group \(G\).
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