AF-embedding of crossed products of AH-algebras by $\mathbb{Z}$ and asymptotic AF-embedding

Abstract

Let A be a unital AH-algebra and let α ∈ Aut(A) be an automorphism. A necessary condition for A X α ∥ being embedded into a unital simple AF-algebra is the existence of a faithful tracial state. If in addition, there is an automorphism K with κ *1 = -id K1 (A) such that α o K and K ° α are asymptotically unitarily equivalent, then A X α 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 K 1 (A), A X α 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 AT-algebra then A X α Z can be embedded into a unital simple AF-algebra if and only if A admits a faithful α-invariant tracial state. Consequently, for any unital simple AT-algebra A, A X α Z can always be embedded into a unital simple AF-algebra. If X is a compact metric space and A: Z 2 → Aut(C(X)) is a homomorphism, then C(X) X Λ Z 2 can be asymptotically embedded into a unital simple AF-algebra provided that X admits a strictly positive A-invariant probability measure. Consequently C(X) X Λ Z 2 is quasidiagonal if X admits a strictly positive A-invariant Borel probability measure.