Some Remarks on C^*-Dynamical Systems with a Compact Abelian Group

Abstract

A C^*-dynamical systems consisting of a C^*-algebra \mathfrak{A} and an action α of a compact abelian group G as a group of automorphisms of \mathfrak{A} is investgated.
An explicit structure of the C^*-crossed product C^*(\mathfrak{A};α) of \mathfrak{A} by α is given in terms of the spectral subspaces \mathfrak{A}^α(p), p∈\hat{G} of \mathfrak{A}.
If \mathfrak{A} has a strictly positive element and if the colsed ideal of the fixed poind algebra \mathfrak{A}^α generated by \mathfrak{A}^α(p)^*\mathfrak{A}^α(p) is \mathfrak{A}^α itself for any p∈\hat{G}, then C^*(\mathfrak{A};α) is shown to be stably isomorphic to \mathfrak{A}^α⊗\mathscr{C}(L²(G)).
For the Connes-Olesen invariant Γ(α), it is shown that p∈Γ(α) if and only if \hat{α}_p(I)I≠(0) for any non-zero closed ideal I of C^*(\mathfrak{A};α) where \hat{α} is the action of \hat{G} on C^*(\mathfrak{A};α) dual to the action α on \mathfrak{A}.
The relative commutant of \mathfrak{A}^α in \mathfrak{A} is shown to be commutative if G=T¹ or Z/(p) and to be of type I if G is finite or product of T¹ with a finite group.