Higher K-groups for operator systems

TL;DR

This paper develops higher K-theoretical invariants for operator systems using spectral gap measures, extending previous definitions and providing a new framework for classifying operator systems up to Morita equivalence.

Contribution

It introduces a new family of higher K-theoretic invariants for operator systems based on spectral gaps and constructs their Grothendieck groups, extending prior work.

Findings

Defined operator system invariants $ ext{V}_p^ ext{delta}$ for each spectral gap parameter and matrix size.

Constructed a direct limit with a semigroup structure leading to $K_p^ ext{delta}$ groups.

Illustrated invariants using the spectral localizer.

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 $\delta$ as a measure for the spectral gap of the representatives for the K-theory classes. For each $\delta$ and integer $p \geq 0$ this gives operator system invariants $\mathcal V_p^\delta(-,n)$, 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 $K_p^\delta$-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 $K_0^\delta$ or $K_1^\delta$. We illustrate our invariants by means of the spectral localizer.