Duality for automorphisms on a compact C*-dynamical system

Abstract

When considering an action α of a compact group G on a C*-algebra A, the notion of an α-invariant Hilbert space in A has proved extremely useful [1, 4, 8, 14, 17, 18]. Following Roberts [13] a Hilbert space in (a unital algebra) A is a closed subspace H of A such that x*y is a scalar for all x, y in H. For example if G is abelian, and α is ergodic in the sense that the fixed point algebra Aα is trivial, then A is generated as a Banach space by a unitary in each of the spectral subspaceswhich are then invariant one dimensional Hilbert spaces. If G is not abelian, then Hilbert spaces (which are always assumed to be invariant) do not necessarily exist, even for ergodic actions. For non-ergodic actions, it is also desirable to relax the requirement to x*y being a constant multiple of some positive element of Aα. More generally, if γ is a finite dimensional matrix representation of G and n is a positive integer, we define to be the subspacewhere d is the dimension d(γ) of γ, and Mnd denotes n×d complex matrices. (Usually we will denote the extended action of αg to αg ⊗ 1 on A⊗Mnd again by αg.) Let .