Dimension Growth for C-algebras

We introduce the growth rank of a C∗-algebra — a (N∪ {∞})-valued invariant whose minimal instance is equivalent to the condition that an algebra absorbs the Jiang-Su algebra Z tensorially — and prove that its range is exhausted by simple, nuclear C∗-algebras. As consequences we obtain a well developed theory of dimension growth for approximately homogeneous (AH) C∗-algebras, establish the existence (contrary to expectation) of simple, nuclear, and non-Z-stable C∗algebras which are not tensorially prime, and show the assumption of Z-stability to be particularly natural when seeking classification results for nuclear C∗-algebras via K-theory. The properties of the growth rank lead us to propose a universal property which can be considered inside any class of unital and nuclear C∗-algebras. We prove that Z satisfies this universal property inside a large class of locally subhomogeneous algebras, representing the first uniqueness theorem for Z which does not depend on the classification theory of nuclear C∗-algebras.