Dimension growth for $C^*$-algebras

Details Dimension growth for $C^*$-algebras Dimension growth for $C^*$-algebras

2005-09-07

arxiv.org/abs/math/0509159v2

Mathematics

Operator Algebras

26 pages; minor revisions; to appear in Adv. Math

Scientific paper

We introduce the growth rank of a C*-algebra, a (N \cup {\infty})-valued invariant which measures how far an algebra is from absorbing the Jiang-Su algebra Z tensorially. We prove that its range is exhausted by simple nuclear C*-algebras, and obtain in the process a well developed theory of unbounded dimension growth for approximately homogeneous (AH) algebras. Another consequence of the range result is the existence of a simple, nuclear, and non-Z-stable C*-algebra which is not tensorially prime. The properties of the growth rank suggest a universal property which may be considered inside any class of unital and nuclear C*-algebras. We prove that Z satisfies this property inside a class of locally subhomogeneous algebras.

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