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*-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.