2006/12/18 by Lin, Huaxin
#46L05 #46L80 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.math/0612529
Let A be a unital AH-algebra and let α∈ Aut(A) be an automorphism. A necessary condition for A\rtimesα\Z being embedded into a unital simple AF-algebra is the existence of a faithful tracial state. If in addition, there is an automorphism κ with κ*1=-\rm idK1(A) such that α∘ κ and κ∘ \af are asymptotically unitarily equivalent, then A\rtimes\af\Z can be embedded into a unital simple AF-algebra. Consequently, in the case that A is a unital AH-algebra (not necessarily simple) with torsion K1(A), A\rtimesα\Z can be embedded into a unital simple AF-algebra if and only if A admits a faithful α-invariant tracial state. We also show that if A is a unital A\T-algebra then A\rtimesα\Z can be embedded into a unital simple AF-algebra if and only if A admits a faithful \af-invariant tracial state. If X is a compact metric space and Λ: \Z2→ Aut(C(X)) is a \hm then C(X)\rtimesΛ\Z2 can be asymptotically embedded into a unital simple AF-algebra provided that X admits a strictly positive Λ-invariant probability measure. Consequently C(X)\rtimesΛ\Z2 is quasidiagonal if X admits a strictly positive Λ-invariant Borel probability measure.