A generalization of K-theory to operator systems

2024/09/04 by van Suijlekom, Walter D. · 2 citations
#FOS: Mathematics #Functional Analysis (math.FA) #K-Theory and Homology (math.KT) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2409.02773

Abstract

We propose a generalization of K-theory to operator systems. Motivated by spectral truncations of noncommutative spaces described by C^*-algebras and inspired by the realization of the K-theory of a C^*-algebra as the Witt group of hermitian forms, we introduce new operator system invariants indexed by the corresponding matrix size. A direct system is constructed whose direct limit possesses a semigroup structure, and we define the K₀-group as the corresponding Grothendieck group. This is an invariant of unital operator systems, and, more generally, an invariant up to Morita equivalence of operator systems. For C^*-algebras it reduces to the usual definition. We illustrate our invariant by means of the spectral localizer.

Cited by

• Spectral localizers in KK-theory
• The odd spectral localiser via asymptotic morphisms and quasi-projections

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