Foundations of an equivariant cohomology theory for Banach algebras. II
The author continues the investigation of the equivariant cohomology theory for Banach algebras [for part I see ibid. 117, No. 1, 52-146 (1996; Zbl 0857.46050)]. In this paper the author studies the relation between the equivariant cohomology groups and the standard cohomology groups of the crossed product algebra. This paper is organized in three chapters numbered IV, V, and VI. In Chapter IV, the construction of the covariance algebra as exhibited in the paper [\textit{S. Doplicher}, \textit{D. Kastler} and \textit{D. W. Robinson}, Commun. Math. Phys. 3, 1-28 (1966; Zbl 0152.23803)], is given. The same construction applies to a Banach covariance module, i.e. a Banach module over the covariant algebra. When \(G\) is a locally compact group acting on \({\mathfrak A}\), the notation \({\mathfrak A}^G_1\) of the previous paper for the covariance algebra is used; the covariance module constructed from a Banach \(G\)-\({\mathfrak A}\)-module \({\mathcal M}\) is denoted by \({\mathcal M}^G_1\). An action of the group \(G\) on the covariance algebra \({\mathfrak A}^G_1\) and the covariance module \({\mathcal M}^G_1\) is introduced. It is proved that with these action \({\mathcal M}^G_1\) is an equivariant \({\mathfrak A}^G_1\)-module. The author considers the question when a covariance algebra is amenable. In Chapter V, the relation between equivariant cohomology and various cohomology theories of the covariance algebra with coefficients in the covariance module is considered. In Chapter VI, there are considered the equivariant cohomology theories defined in the papers [\textit{S. Klimek}, \textit{W. Kondracki} and \textit{A. Lesniewski}, K-Theory 4, No. 3, 201-218 (1991; Zbl 0744.46063); \textit{D. Gong}, Mich. Math. J. 39, No. 3, 455-473 (1992; Zbl 0783.55005)]. It is proved that if the group \(G\) is finite or the algebra is finite-dimensional, the equivariant theories of the previous papers are special cases of the theory developed in part I of the author's paper.
scientific article; zbMATH DE number 1417608
scientific article