Noncommutatively compactly supported functions

03/05/2019, 14:00 - 15:00 in MAP/0G/017

Kucerovsky, Dan (University of New Brunswick)


Abstract:

A unital C*-algebra is a noncommutative generalization of the algebra C(X) of continuous functions on a compact topological space; in the nonunital case, it is a generalization of the algebra C0 (X) of continuous functions that vanish at infinity, or more accurately, functions that are each arbitrarily small outside a sufficient large compact set. Let us consider what might be the proper generalization of compactly supported functions to the noncommutative case. The usual approach in noncommutative geometry is to fix some holomorphically closed subalgebra and to use that, but what happens if we need more information about the “compactly supported functions”? The answer has possibly been found by Pedersen, in the form of the Pedersen ideal. However, there are some subtle questions that arise when we take tensor products. One would expect the Pedersen ideal of the tensor product of two C*-algebras to “multiply” in an obvious sense. However, while this might be expected, the situation is complicated by the fact that there is, in general, more than one C*-norm that one can put on the tensor product of C*-algebras. We show that positive elements of a Pedersen ideal of a tensor product can be approximated in a particularly strong sense by sums of tensor products of positive elements. We show that the positive elements of a Pedersen ideal are sometimes stable under Cuntz equivalence. We generalize a result of Pedersen’s by showing that certain classes of completely positive maps take a Pedersen ideal into a Pedersen ideal. This has a range of applications to the structure of tracial cones and related topics, such as the Cuntz-Pedersen space or the Cuntz semigroup. For example, we determine the cone of lower semi-continuous traces of a tensor product in terms of the traces of the tensor factors, in an arbitrary C*-tensor norm.

Mathematical Sciences Research Centre