Foundations of an Equivariant Cohomology Theory for Banach Algebras, II
✍ Scribed by Kjeld Knudsen Jensen
- Publisher
- Elsevier Science
- Year
- 1999
- Tongue
- English
- Weight
- 506 KB
- Volume
- 147
- Category
- Article
- ISSN
- 0001-8708
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✦ Synopsis
We continue the investigation of the equivariant cohomology theory for Banach algebras defined in [KJe]. This theory is an equivariant version of the continuous cohomology theory for Banach and Operator algebras defined, described, and developed in
In this paper, we study the relation between the equivariant cohomology groups and the standard cohomology groups of the crossed product algebra. The background for these considerations is the K-theory result of Julg [PJu] that the equivariant K-theory of a C*-algebra is isomorphic to the K-theory of the crossed product algebra. Furthermore, we investigate the relation between the equivariant cohomology theory defined in [KJe] and the equivariant cohomology theories defined in [DGo, K-K-L].
The paper [KJe] is organized in three chapters, Chapters I, II, and III, and Appendix A. Therefore, we have chosen to organize this paper in three chapters (and an appendix) numbered IV, V, and VI. References to [KJe] are given as references to the number (e.g., Definition I.2.2 or II.3.4) in that paper, but without [KJe]. That is, a reference to II.3.4 will be a reference to [KJe], while a reference to IV.2.3 is a reference to this paper. We now sketch the content of this paper.
In Section 1 of Chapter IV, the construction of the covariance algebra, as exhibited in [D-K-R], is given. We apply the same construction to a Banach equivariant module; the resulting object is called the covariance module. As the terminology suggests, the covariance module is a Banach module over the covariance algebra. However, it is important to note that the covariance module cannot (in general) be constructed if M is a dual equivariant module that is not a Banach equivariant module. When G is a