A generalization of K-theory to operator systems

TL;DR

This paper extends K-theory to operator systems, creating new invariants that generalize classical concepts and are applicable to noncommutative spaces, with potential implications for spectral analysis.

Contribution

It introduces a novel K-theory generalization for operator systems, defining new invariants indexed by matrix size and establishing their properties and reductions to classical K-theory.

Findings

Constructed a direct system with a semigroup structure

Defined the $K_0$-group as a Grothendieck group of the direct limit

Applied the invariant to spectral localizer

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 $K_0$-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.