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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. The authors systematically study the lattice and category theoretic aspects of Cuntz semigroups. Given a C^*-algebra A, its (concrete) Cuntz semigroup mathrm{Cu}(A) is an object in the category mathrm{Cu} of (abstract) Cuntz semigroups, as introduced by Coward, Elliott and Ivanescu. To clarify the distinction between concrete and abstract Cuntz semigroups, the authors call the latter mathrm{Cu}-semigroups. The authors establish the existence of tensor products in the category mathrm{Cu} and study the basic properties of this construction. They show that mathrm{Cu} is a symmetric, monoidal category and relate mathrm{Cu}(Aotimes B) with mathrm{Cu}(A)otimes_{mathrm{Cu}}mathrm{Cu}(B) for certain classes of C^*-algebras. As a main tool for their approach the authors introduce the category mathrm{W} of pre-completed Cuntz semigroups. They show that mathrm{Cu} is a full, reflective subcategory of mathrm{W}. One can then easily deduce properties of mathrm{Cu} from respective properties of mathrm{W}, for example the existence of tensor products and inductive limits. The advantage is that constructions in mathrm{W} are much easier since the objects are purely algebraic.

C*-algebras. --- Tensor products. --- Tensor algebra. --- Semigroups.

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