K-theoretic rigidity and slow dimension growth

Details K-theoretic rigidity and slow dimension growth K-theoretic rigidity and slow dimension growth

2009-10-12

arxiv.org/abs/0910.2061v2

Mathematics

Operator Algebras

15 pages, to appear in Invent. Math

Scientific paper

Let A be an approximately subhomogeneous (ASH) C*-algebra with slow dimension growth. We prove that if A is unital and simple, then the Cuntz semigroup of A agrees with that of its tensor product with the Jiang-Su algebra Z. In tandem with a result of W. Winter, this yields the equivalence of Z-stability and slow dimension growth for unital simple ASH algebras. This equivalence has several consequences, including the following classification theorem: unital ASH algebras which are simple, have slow dimension growth, and in which projections separate traces are determined up to isomorphism by their graded ordered K-theory, and none of the latter three conditions can be relaxed in general.

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