Projective Hilbert Modules and Sequential Approximation

2023/01/10 by Lawrence G. Brown, Brown, Lawrence G., Huaxin Lin +1
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Banach Space Theory #Advanced Operator Algebra Research #FOS: Mathematics #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.2301.04247

openalex publication_date 2023/01/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that, when A is a separable C*-algebra, every countably generated Hilbert A-module is projective (with bounded module maps as morphisms). We also study the approximate extensions of bounded module maps. In the case that A is a σ-unital simple C*-algebra with strict comparison and every strictly positive lower semicontinuous affine function on quasitraces can be realized as the rank of an element in Cuntz semigroup, we show that the Cuntz semigroup is the same as unitarily equivalent class of countably generated Hilbert A-modules if and only if A has stable rank one.

Related

• Cuntz semigroups of C*-algebras of stable rank one and projective Hilbert modules
• Isomorphism of Hilbert modules over stably finite C*-algebras
• Hilbert C^*-modules over Σ^*-algebras
• Extensions of Hilbert C*-modules: classification in simple cases
• Strict comparison and stable rank one