The Gauge Group and Perturbation Semigroup of an Operator System

TL;DR

This paper introduces the concepts of gauge group and perturbation semigroup for operator systems, providing definitions, properties, and an example involving the Toeplitz system, extending previous algebraic frameworks to operator systems.

Contribution

It defines gauge group and perturbation semigroup for operator systems, and analyzes their properties with an explicit example, extending prior algebraic concepts.

Findings

Defined gauge group for operator systems.

Established the perturbation semigroup and its properties.

Computed examples for the Toeplitz system.

Abstract

The perturbation semigroup was first defined in the case of $*$-algebras by Chamseddine, Connes and van Suijlekom. In this paper, we take $\mathcal{E}$ as a concrete operator system with unit. We first give a definition of gauge group $\mathcal{G}(\mathcal{E})$ of $\mathcal{E}$, after that we give the definition of perturbation semigroup of $\mathcal{E}$, and the closed perturbation semigroup of $\mathcal{E}$ with respect to the Haagerup tensor norm. We also show that there is a continuous semigroup homomorphism from the closed perturbation semigroup to the collection of unital completely bounded Hermitian maps over $\mathcal{E}$. Finally we compute the gauge group and perturbation semigroup of the Toeplitz system as an example.