A generalization of K-theory to operator systems

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 K₀-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.