Cuntz semigroups of ideals and quotients and a generalized Kasparov Stabilization Theorem

TL;DR

This paper explores the relationships between the Cuntz semigroups of a C*-algebra, its ideal, and quotient, establishing an exactness property and generalizing Kasparov's Stabilization theorem using Hilbert C*-modules.

Contribution

It introduces a relation characterizing equality in the Cuntz semigroup of quotients and generalizes Kasparov's Stabilization theorem through isomorphism of Hilbert modules.

Findings

Cuntz semigroup functor is exact.

Characterization of equality in Cuntz semigroup of quotients.

Generalization of Kasparov's Stabilization theorem.

Abstract

Let A be a C*-algebra and I a closed two-sided ideal of A. We use the Hilbert C*-modules picture of the Cuntz semigroup to investigate the relations between the Cuntz semigroups of I, A and A/I. We obtain a relation on two elements of the Cuntz semigroup of A that characterizes when they are equal in the Cuntz semigroup of A/I. As a corollary, we show that the Cuntz semigroup functor is exact. Replacing the Cuntz equivalence relation of Hilbert modules by their isomorphism, we obtain a generalization of Kasparov's Stabilization theorem.