Higher K-groups for operator systems

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 δ as a measure for the spectral gap of the representatives for the K-theory classes. For each δ δ and integer p ≥ 0 p ≥ 0 this gives operator system invariants V p δ ( − , n ) 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 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 K_0^δ or K 1 δ K_1^δ . We illustrate our invariants by means of the spectral localizer.