Abstract bivariant Cuntz semigroups

Abstract

We show that abstract Cuntz semigroups form a closed symmetric monoidal category. Thus, given Cuntz semigroups $S$ and $T$, there is another Cuntz semigroup $[[S,T]]$ playing the role of morphisms from $S$ to $T$. Applied to C${}^{\ast }$-algebras $A$ and $B$, the semigroup $[[\mathrm{Cu}(A),\mathrm{Cu}(B)]]$ should be considered as the target in analogues of the UCT for bivariant theories of Cuntz semigroups. Abstract bivariant Cuntz semigroups are computable in a number of interesting cases. We also show that order-zero maps between C${}^{\ast }$-algebras naturally define elements in the respective bivariant Cuntz semigroup.

29 pages. This paper has been published in International Mathematics Research Notices in a revised form subsequent to editorial input. Material on these pages is copyright Oxford University Press. The previous version of this paper (arXiv:1702.01588v1) has been split in two parts, of which this is the first

References in corpus (2)

• Completely positive maps of order zero
• Tensor products and regularity properties of Cuntz semigroups

Cited by in corpus (1)

• MF traces and the Cuntz semigroup