[Paper Review] Cuntz semigroups of ideals and quotients and a generalized Kasparov Stabilization Theorem

Alin Ciuperca, Leonel Robert+1 more|ArXiv.org|Oct 31, 2007

Advanced Operator Algebra Research13 references3 citations

TL;DR

This paper establishes a generalized Kasparov stabilization theorem using the Cuntz semigroup framework for C*-algebras and their ideals. By analyzing Hilbert C*-modules and Cuntz equivalence, it proves that two modules over a C*-algebra become isomorphic after stabilization with $ l_2(I) $ if and only if their quotients modulo $ I $ are isomorphic, extending Kasparov's original result to the non-simple case and proving exactness of the Cuntz semigroup functor for ideals.

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.

Motivation & Objective

• To understand the relationship between the Cuntz semigroups of a C*-algebra $ A $, its ideal $ I $, and the quotient $ A/I $.
• To characterize when two elements in $ Cu_s(A) $ become equal in $ Cu_s(A/I) $, using a relation involving $ [M]I $ and $ [N]I $.
• To generalize Kasparov’s stabilization theorem by replacing Cuntz equivalence with isomorphism of Hilbert modules.
• To prove that the Cuntz semigroup functor is exact for $ π: A \to A/I $, under $ \sigma $-unitality assumptions on $ I $.
• To extend the result to equivariant settings with compact group actions, proving an equivariant stabilization theorem.

Proposed method

• Uses the Hilbert C*-module picture of the Cuntz semigroup, where elements are represented by isomorphism classes of countably generated modules.
• Defines $ [M]I := [MI] $, the image of a module $ M $ under the ideal $ I $, to relate $ Cu_s(A) $ and $ Cu_s(I) $.
• Establishes a key inequality: $ Cu_s(\pi)([M]) \leq Cu_s(\pi)([N]) $ iff $ [M] + [N]I \leq [N] + [M]I $, characterizing equality in the quotient.
• Applies Kasparov’s stabilization theorem to show $ Cu_s(\pi) $ restricts to an isomorphism between $ Cu_s(A) + [l_2(I)] $ and $ Cu_s(A/I) $.
• Adapts the Mingo-Phillips proof strategy for Kasparov’s theorem, using averaging over compact groups to construct equivariant lifts.
• Proves a new equivariant stabilization theorem by constructing equivariant operators via group averaging and dense range approximations.

Experimental results

Research questions

• RQ1 When do two Hilbert C*-modules over a C*-algebra $ A $ have isomorphic quotients modulo a $ \sigma $-unital ideal $ I $?
• RQ2 What is the precise condition in terms of the Cuntz semigroup for two elements of $ Cu_s(A) $ to map to the same element in $ Cu_s(A/I) $?
• RQ3 Can Kasparov’s stabilization theorem be generalized beyond the stable rank one case to arbitrary C*-algebras and ideals?
• RQ4 Is the Cuntz semigroup functor exact for short exact sequences of C*-algebras?
• RQ5 Can the stabilization result be extended to the equivariant setting with compact group actions?

Key findings

• Two elements $ [M], [N] \in Cu_s(A) $ satisfy $ Cu_s(\pi)([M]) = Cu_s(\pi)([N]) $ if and only if $ [M] + [N]I = [N] + [M]I $, providing a complete characterization in the quotient.
• The Cuntz semigroup functor is exact: the sequence $ 0 \to Cu_s(I) \to Cu_s(A) \to Cu_s(A/I) \to 0 $ is exact as an ordered semigroup sequence.
• The map $ Cu_s(\pi) $ restricted to $ Cu_s(A) + [l_2(I)] $ is an isomorphism onto $ Cu_s(A/I) $, generalizing Kasparov’s stabilization theorem.
• For $ M/MI \cong N/NI $ as $ A/I $-modules, there exists an isomorphism $ \Phi: M \oplus l_2(I) \to N \oplus l_2(I) $ inducing the quotient map.
• In the equivariant case with compact $ G $, if $ \phi: M/MI \to N/NI $ is an equivariant isomorphism, then there exists an equivariant isomorphism $ \Phi: M \oplus L_2(G, l_2(I)) \to N \oplus L_2(G, l_2(I)) $ lifting $ \phi $.
• The construction relies on averaging over $ G $ to produce equivariant operators and uses a $ G $-continuous version of the stabilization theorem for $ G $-modules.