# Journal of Operator Theory

Volume 64, Issue 1, Summer 2010 pp. 155-169.

The Cuntz semigroup of ideals and quotients and a generalized Kasparov Stabilization Theorem**Authors**: Alin Ciuperca (1), Leonel Robert (2), and Luis Santiago (3)

**Author institution:**(1) Department of Mathematics, University of Toronto, Toronto, ON, M5S 2E4, Canada

(2) Fields Institute, Toronto, ON, M5T 3J1, Canada

(3) Department of Mathematics, University of Toronto, Toronto, ON, M5S 2E4, Canada

**Summary:**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.

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