Algebraic analogs of the Connes spectrum

AbstractIn this paper we obtain algebraic analogs of certain known results on crossed products of von Neumann algebras and of C∗-algebras. Of particular interest here is the Connes spectrum, introduced by A. Connes in his classification of factors of type III. In addition, we are concerned with certain von Neumann algebra results contained in the work of Connes, Takesaki, and Nakagami and with certain, more algebraic, C∗-algebra results from the work of Olesen, Pederson, and Kishimoto. Let G be a finite group and let A be a G-graded ring with 1. Then there exists an extension ring of A, determined by this structure, called the smash product of A by G∗. This overring comes from the theory of Hopf algebras and we denote it by A # G∗. In the first half of this paper, we introduce and study the Connes and strong Connes subgroups of G and we relate them to the ideal structure of A # G∗. In particular we obtain criteria for A # G∗ to be prime or simple. In a different direction, suppose A is a K-algebra and G is a finite abelian group of K-automorphisms of A. If the field K contains appropriate roots of unity, then the skew group ring AG has a natural smash product structure A # Ĝ∗ where Ĝ=Hom(G,K′) is the dual group of G. The problem is to explicitly determine the Connes subgroup of Ĝ in terms of the action of G on A. This is achieved in the second half of the paper under the additional assumption that A is a prime ring. A more detailed discussion of the operator algebra sources of our results will be given in the Appendix