On Local AH algebras

Abstract

We show that every unital amenable separable simple $C^{\ast }$-algebra with finite tracial rank which satisfies the UCT has in fact tracial rank at most one. We also show that unital separable simple $C^{\ast }$-algebrass which are "tracially" locally AH with slow dimension growth are $\mathcal{Z}$-stable. As a consequence, unital separable simple $C^{\ast }$-algebras which are locally AH with no dimension growth are isomorphic to a unital simple AH-algebra with no dimension growth.

Significant revision. It has now 91 pages

References in corpus (8)

• Invariant means and finite representation theory of C*-algebras
• Approximate Homotopy of Homomorphisms from $C(X)$ into a Simple $C^{\ast }$-algebra
• Localizing the Elliott conjecture at strongly self-absorbing C*-algebras
• Stability in the Cuntz semigroup of a commutative C*-algebra
• Homomorphisms from AH-algebras
• Strict comparison and Z-absorption of nuclear C*-algebras
• Localizing the Elliott Conjecture at Strongly Self-absorbing C*-algebras --An Appendix
• Nuclear dimension and Z-stability of pure C*-algebras

Cited by in corpus (4)

• Homomorphisms into a simple Z-stable C*-Algebras
• High-dimensional Z-stable AH algebras
• Asymptotic unitary equivalence in $C^{\ast }$-algebras
• Kishimoto's Conjugacy Theorems in simple $C^{\ast }$-algebras of tracial rank one