Classification of AF Flows

Abstract

Abstract. An AF flow is a one-parameter automorphism group of an AF C * -algebra A such that there exists an increasing sequence of invariant finite dimensional sub-C * -algebras whose union is dense in A. In this paper, a classification of C * -dynamical systems of this form up to equivariant isomorphism is presented. Two pictures of the actions are given, one in terms of a modified Bratteli diagram/pathspace construction, and one in terms of a modified K 0 functor.

Classification of Inductive Limits of Outer Actions of ℝ on Approximate Circle Algebras

“…But then x|x| −1 is an eigen-unitary with eigenvalue a in π(B n ). It is easy to see that eigen-unitaries are not possible in a finite dimensional C * -dynamical system (see [1]).…”

“…There are now several results classifying certain kinds of actions of R on C * -algebras (cf., [1][2][3]). In each of these cases, there exists an increasing sequence of homogeneous sub-C * -algebras that are each globally invariant under the action, which is inner when restricted to each of these sub-C * -algebras and that has dense union.…”

“…From Lemma 2.5, we may reduce to the case where both A and B have one minimal direct summand. Cutting down B by a projection in B β with class ϕ( [1])in D(B, β) then brings us to the situation dealt with above.…”

Approximate AF flows

“…It also follows that D(δ α ) ⊃ A n , δ α |A n = ad ih n |A n , and n A n is a core for δ α . Since [h m , h n ] = 0, the generator of this type is called commutative and is studied in Sakai's book [20] (see also [12,2,7]). In particular α is approximately inner in the sense that lim n Ad e ith n (x) = α t (x) uniformly in t on every bounded subset of R for all x ∈ A.…”