
TL;DR
This paper proves that certain simple, unital, amenable $C^*$-algebras with finite tracial rank have at most tracial rank one, and establishes conditions under which they are ${ m Z}$-stable or isomorphic to AH-algebras.
Contribution
It demonstrates that unital amenable simple $C^*$-algebras with finite tracial rank satisfying the UCT have tracial rank at most one, and characterizes when they are ${ m Z}$-stable or isomorphic to AH-algebras.
Findings
Algebras with finite tracial rank satisfying UCT have tracial rank ≤ 1.
Tracially locally AH algebras with slow dimension growth are ${ m Z}$-stable.
Locally AH algebras with no dimension growth are isomorphic to AH-algebras.
Abstract
We show that every unital amenable separable simple -algebra with finite tracial rank which satisfies the UCT has in fact tracial rank at most one. We also show that unital separable simple -algebrass which are "tracially" locally AH with slow dimension growth are -stable. As a consequence, unital separable simple -algebras which are locally AH with no dimension growth are isomorphic to a unital simple AH-algebra with no dimension growth.
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Taxonomy
TopicsAdvanced Operator Algebra Research · Advanced Topics in Algebra · Noncommutative and Quantum Gravity Theories
