Comparison maps for tensor products of Cuntz semigroups

For C*-algebras A, B, we study the canonical comparison from the universal tensor product of their Cuntz semigroups to the Cuntz semigroup of their C*-tensor product, τλA,B: Cu(A) ⊗Cu Cu(B) → Cu(A ⊗λ B), λ ∈ {min, max}. For type I C*-algebras A, B, without separability assumptions, injectivity, order reflection, surjectivity and isomorphism are equivalent: they hold exactly when at least one factor has real rank zero. In particular, comparison is neither injective nor surjective for A = B = C([0,1]). This answers, within the type I class, the tensor-comparison question posed by Antoine, Perera and Thiel in 2014 (Problem 6.4.11). For arbitrary A, B, the corresponding spatial comparison for semigroups of closed ideals generated by one positive element is an order embedding. Its image consists of countable joins of I ⊗min J, with I, J singly generated ideals of A, B, respectively. It is therefore an isomorphism exactly when every closed ideal of A ⊗min B is generated by the elementary tensors it contains. This settles the general order-reflection question posed by Antoine, Perera and Thiel in 2015 (Problem 7.2.11). If one factor is purely infinite, both maps τλA,B are order embeddings. For a fixed strongly purely infinite B, the minimal comparison is an isomorphism for every A exactly when B is exact; the maximal comparison has this property exactly when B is nuclear. Finally, suppose A, B are nonzero simple separable unital exact stably finite C*-algebras of real rank zero, both absorb the universal uniformly hyperfinite algebra, and at least one is nuclear and satisfies the universal coefficient theorem. Then both comparison maps are order embeddings, and they are surjective exactly when K1(A) = 0 or K1(B) = 0. In general, the scaled Cuntz semigroups and KK-equivalence classes of the individual factors do not determine surjectivity. Repository: GitHub repository