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 $δ$ as a measure for the spectral gap of the representatives for the K-theory classes. For each $δ$ and integer $p \geq 0$ 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 $K_0^δ$ or $K_1^δ$. We illustrate our invariants by means of the spectral localizer.