The Gauge Group and Perturbation Semigroup of an Operator System

Abstract

The perturbation semigroup was first defined in the case of ∗-algebras by Chamseddine, Connes, and van Suijlekom. In this paper, we take 𝓔 as a concrete operator system with a unit. We first define the gauge group 𝒢(𝓔) of 𝓔. After that, we define the perturbation semigroup of 𝓔 and the closed perturbation semigroup of 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 𝓔. Finally, we compute the gauge group and perturbation semigroup of the Toeplitz system as an example.