Abstract Bivariant Cuntz Semigroups

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Date
2017-02-13MFO Scientific Program
Research in Pairs 2015Series
Oberwolfach Preprints;2017,04Author
Antoine, Ramon
Perera, Francesc
Thiel, Hannes
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Show full item recordOWP-2017-04
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 $((Cu(A),Cu(B)))$ should be considered as the target in analogues of the UCT for bivariant theories of Cuntz semigroups.
<br />Abstract bivariant Cuntz semigroups are computable in a number of interesting cases. We explore its behaviour under the tensor product with the Cuntz semigroup of strongly self-absorbing C*-algebras and the Jacelon-Razak algebra. We also show that order-zero maps between C*-algebras naturally define elements
in the respective bivariant Cuntz semigroup.