Flat dimension growth for C∗-algebras

Simple and nuclear C∗-algebras which fail to absorb the Jiang–Su algebra tensorially have settled many open questions in the theory of nuclear C∗-algebras, but have been little studied in their own right. This is due partly to a dearth of invariants sensitive to differences between such algebras. We present two new real-valued invariants to fill this void: the dimension–rank ratio (for unital AH algebras), and the radius of comparison (for unital and stably finite algebras). We establish their basic properties, show that they have natural connections to ordered K-theory, and prove that the range of the dimension–rank ratio is exhausted by simple algebras (this last result shows the class of simple, nuclear and non-Z-stable C∗-algebras to be uncountable). In passing, we establish a theory of moderate dimension growth for AH algebras, the existence of which was first supposed by Blackadar. The minimal instances of both invariants are shown to coincide with the condition of being tracially AF among simple unital AH algebras of real rank zero and stable rank one, whence they may be thought of as generalised measures of dimension growth. We argue that the radius of comparison may be thought of as an abstract version of the dimension–rank ratio.