The Cuntz semigroup of a C*-algebra is an important invariant in the
structure and classification theory of C*-algebras. It captures more
information than K-theory but is often more delicate to handle. We
systematically study the lattice and category theoretic aspects of Cuntz
semigroups.
Given a C*-algebra A, its (concrete) Cuntz semigroup Cu(A) is an object
in the category Cu of (abstract) Cuntz semigroups, as introduced by Coward,
Elliott and Ivanescu. To clarify the distinction between concrete and abstract
Cuntz semigroups, we will call the latter Cu-semigroups.
We establish the existence of tensor products in the category Cu and study
the basic properties of this construction. We show that Cu is a symmetric,
monoidal category and relate Cu(A⊗B) with Cu(A)⊗Cu​Cu(B) for
certain classes of C*-algebras.
As a main tool for our approach we introduce the category W of
pre-completed Cuntz semigroups. We show that Cu is a full, reflective
subcategory of W. One can then easily deduce properties of Cu from
respective properties of W, e.g. the existence of tensor products and
inductive limits. The advantage is that constructions in W are much easier
since the objects are purely algebraic.
We also develop a theory of Cu-semirings and their semimodules. The Cuntz
semigroup of a strongly self-absorbing C*-algebra has a natural product giving
it the structure of a Cu-semiring. We give explicit characterizations of
Cu-semimodules over such Cu-semirings. For instance, we show that a
Cu-semigroup S tensorially absorbs the Cu-semiring of the Jiang-Su
algebra if and only if S is almost unperforated and almost divisible, thus
establishing a semigroup version of the Toms-Winter conjecture.Comment: 195 pages; revised version; several proofs streamlined; some results
corrected, in particular added 5.2.3-5.2.