Higher K-groups for operator systems

2024/11/05 by van Suijlekom, Walter D. · 1 citation
#FOS: Mathematics #Functional Analysis (math.FA) #K-Theory and Homology (math.KT) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2411.02981

Abstract

We extend our previous definition of K-theoretic invariants for operator systems based on hermitian forms to higher K-theoretical invariants. We realize the need for a positive parameter δ as a measure for the spectral gap of the representatives for the K-theory classes. For each δ and integer p ≥ 0 this gives operator system invariants \mathcal V_p^δ(-,n), indexed by the corresponding matrix size. The corresponding direct system of these invariants has a direct limit that possesses a semigroup structure, and we define the K_p^δ-groups as the corresponding Grothendieck groups. This is an invariant of unital operator systems, and, more generally, an invariant up to Morita equivalence of operator systems. Moreover, there is a formal periodicity that reduces all these groups to either K₀^δ or K₁^δ. We illustrate our invariants by means of the spectral localizer.

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• Spectral localizers in KK-theory

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