Traces and Pedersen ideals of tensor products of non-unital C*-algebras
C ∗ -algebra, tensor product
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. 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 semicontinuous traces of a tensor product in terms of the traces of the tensor factors, in an arbitrary C*-tensor norm. 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. We provide theorems that in many cases compute the Cuntz semigroup of a tensor product.
Ivanescu, C. and Kucerovski, D. (2019). Traces and Pedersen ideals of tensor products of non-unital C*-algebras. New York Journal of Mathematics, 25, 423-450. https://arxiv.org/abs/1811.04430v2
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