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Higher K-groups for operator systems

Walter D. van Suijlekom

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.

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 as a measure for the spectral gap of the representatives for the K-theory classes. For each and integer this gives operator system invariants , 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 -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 or . We illustrate our invariants by means of the spectral localizer.

Paper Structure

This paper contains 9 sections, 13 theorems, 40 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background on operator systems
  3. $\delta$-gapped K-theory for unital operator systems: $K_0^\delta$ and $K_1^\delta$
  4. The $K_0^\delta$-groups
  5. $K_1^\delta$ for unital operator systems
  6. Higher $\delta$-gapped K-groups and formal periodicity
  7. Application to the spectral localizer
  8. Example: spectral localizer on the circle
  9. Clifford algebras

Key Result

Proposition 1

Let $\phi: E \to B(\mathcal{H})$ be a ucp map. Then

Figures (1)

  • Figure 1: The negative and positive eigenvalues for the spectral localizer $L_{\kappa,N} = L_{\kappa,N,s=0}$ on the circle for $m=1, \kappa= 1, N=3$ (left) and $m=2, \kappa= 0.1, N=3$ (right). The red (diamond-shaped) dots indicate the surplus of positive eigenvalues as compared to the negative ones.

Theorems & Definitions (24)

  • Proposition 1
  • Definition 2
  • Proposition 3
  • Corollary 4
  • Corollary 5
  • Definition 6
  • Proposition 7
  • Proposition 8
  • Proposition 9: Stability of $K_0^\delta$
  • Example 10
  • ...and 14 more