Banach Algebra Dynamical Systems And Φ-groups Of Involutive Algebras

The theory of C*-dynamical systems and the crossed product C*-algebras is very important in the study of Κ-theory of group C*-algebras, it is reflected in the research of Baum-Connes’conjecture, which is the central field of non-commutative geometry. This paper generalizes the theory of C*-dynamical systems and the crossed product C*-algebras to the general Banach algebras. The theory of systems of n-subspaces or n-projections is on the structure of subspaces or multi-projections. This paper applied this theory on modules to construct an abelian group, called Φ-group, which is a gen-eralization of the classical Κ-group, and this new group has a direct relationship with the classification of systems of n-subspaces. This paper is arranged as follows:Chapter1is the introduction, we recall the theory of C*-dynamical systems and the crossed product C*-algebras, and the theory of systems of n-subspaces, and give the motivations of our study. In the end we list the main results of this paper.In Chapter2, we introduce some preliminaries which are used in the thesis, in-cluding the theory of C*-dynamical systems, the crossed product C*-algebras, and the theory of systems of n-subspaces and the construction of Κ-theory.In Chapter3, after defining the Banach algebra dynamical systems we construct the crossed product Banach algebras through some class of covariant representations. In the construction of the crossed product Banach algebras, it is clear to see a signifi-cant difference between the C*-dynamical systems and the Banach algebras dynamical systems. There is a lack of some good propositions of C*-dynamical systems. Then we study the representations of the crossed product Banach algebras, and construct the one-to-one correspondence between the covariant representations and the integrated representations in a weak form, in which we mainly use the theory of multipliers of Banach algebras. In the last section we study the regular representations of Banach algebra dynamical systems, and construct the reduced crossed products. The whole process also reflects the generality of Banach algebra dynamical systems with respect to the C*-dynamical systems. In the end of this chapter, we give a theorem which claims that when the group is compact, the reduced crossed product Banach algebra coincides with the crossed product one.In Chapter4, we generalize the theory of n-subspaces to n-submodules, from which we construct the Φ-group which can be regarded as a generalization of Κ-theory. We then study the propositions of this group, and it only has part of propositions of the K-group. In the end we compute the Φ-group for commutative systems of n-subspaces, thus Φc(C), from which we find the direct relationship between the Φ-group and the classification of systems of n-subspaces.