Higher K-groups for operator systems

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 $\hat{I}´$ as a measure for the spectral gap of the representatives for the K-theory classes. For each $\hat{I}´$ and integer $p\ge 0$ this gives operator system invariants ${\mathcal{V}}_p^{\hat{I}}´(-,n)$, 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 $K_p^{\hat{I}}´$-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 $K_0^{\hat{I}}´$ or $K_1^{\hat{I}}´$. We illustrate our invariants by means of the spectral localizer.

14 pages, 2 figures