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arXiv · 1702.01588

Abstract bivariant Cuntz semigroups

Ramon Antoine·Francesc Perera·Hannes Thiel
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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$^*$-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$^*$-algebras naturally define elements in the respective bivariant Cuntz semigroup.

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Ramon Antoine, Francesc Perera, Hannes Thiel. 2018-11-21. Abstract bivariant Cuntz semigroups. https://doi.org/10.1093/imrn%2Frny143

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