Compact abelian groups of automorphisms of simpleC *-algebras

If is a continuous representation of a compact abelian groupG as *-automorphisms of a prime or simpleC ^*-algebraA we study how algebraic properties of the fixed-point algebraA ₀ are reflected in properties of the spectrum of and outerness of ₁,tG. For example, ifA is prime, ()=Sp() if and only ifA ₀ is prime, and the same with simple (instead of prime) whenG is finite. In the latter case the relative commutant ofA ₀ inA is trivial if and only if all ₁l are outer, and ifG is cyclic of prime order all the conditions above are equivalent. This paper replaces the earlier versions 12] and 9].