Quasi-product actions of a compact abelian group on a \(C^*\)-algebra (Q1107797)

Let \(\alpha\) be an action of a compact Abelian group on a separable prime \(C^*\)-algebra A, such that also the fixed point subalgebra, \(A^{\alpha}\), is prime. Several conditions on \(\alpha\) are shown to be equivalent, among which are the following: for each \(g\in G\), either \(\alpha_ g=1\) or \(\alpha_ g\) is properly outer; there exists a faithful irreducible representation of A which is also irreducible on \(A^{\alpha};\) there exists a faithful irreducible representation of A which is covariant. An example of a nontrivial action satisfying these conditions is the infinite tensor product action on \(M_{2^{\infty}}=\otimes^{\infty}_{n=1}M_ 2\) obtained from a sequence of nontrivial inner actions on \(M_ 2\), each one appearing infinitely often. In earlier work, this example was shown to be, in a certain sense, typical of nontrivial actions satisfying the third condition. This fact is the key to deducing the first two conditions from the third. The second condition is noteworthy in two respects. First, it involves only the fixed point subalgebra \(A^{\alpha}\subseteq A\), not the action \(\alpha\) itself. (This is not evident in the case of the other two conditions.) Second, while a representation verifying the third condition is required to be covariant, a representation verifying the second condition must in fact be as far as possible from being covariant.