A generalization of K-theory to operator systems

Walter D. van Suijlekom

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arXiv · 2409.02773

A generalization of K-theory to operator systems

Walter D. van Suijlekom

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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.

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Walter D. van Suijlekom. 2024-09-04. A generalization of K-theory to operator systems. https://arxiv.org/abs/2409.02773

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