A Survey on Non-commutative Dimension Theories

We develop an abstract theory of noncommutative dimension theories, and we show that the common theories fit into this setting, in particular the real and stable rank, the topological dimension, the decomposition rank and the nuclear dimension. Then we show how to compute or estimate the dimension theories of certain classes of C-algebras, in particular subhomogeneous and type I C-algebras. We prove some connections between the low dimensional cases in the different theories for type I algebras. As an application, we show that type I algebras with stable rank one have torsion-freeK0-groups. Our techniques alow to reduce essentially every question about dimension theories to the case of separableC-algebras. We will present applications of this by generalizing interesting results of Winter and Lin from the separable to the general case. The theory of C-algebras is often considered as non-commutative topology, which is justified by the natural duality between unital, commutative C-algebras and the category of compact, Hausdorff spaces. Given this fact, one tries to transfer concepts from commutative topology toC-algebras, and we will focus on the theory of dimension. In fact there are different non-commutative dimension theories (often called rank), and we will give an abstract setting to study these theories. The low-dimensional case of these theories is of most interest. One reason is that the low-dimensional cases often agree for different theories. But more importantly, low dimension (in any dimension theory) is considered as a regularity property. Such regularity properties are very important for proving the Elliott conjecture, which predicts that separable, nuclear, simple C-algebras are classified by their Elliott invariant, a tuple consisting of ordered K-theory, the space of traces and a pairing between the two. There are counterexamples to this general form of the conjecture, but if one restricts to certain subclasses that are regular enough, then the conjecture has been verified. It is interesting that many regularity properties are statements about a certain dimension theory. The obvious example is the requirement of real rank zero, which implies that the C-algebra has many projections. In this setting many classification results have been obtained, although real rank zero is not enough to verify the Elliott conjecture. Date: October 21, 2011. 2010Mathematics Subject Classification. Primary 46L05, 46L85, ; Secondary 46M20,