Uniformity in $C^*$-algebras

TL;DR

This paper develops a uniform structure on the set of representations of separable $C^*$-algebras, introduces a noncommutative modulus of continuity, and explores uniform continuity of algebra morphisms.

Contribution

It introduces a novel uniform structure on representations of $C^*$-algebras and defines a noncommutative modulus of continuity, advancing the understanding of uniform properties in noncommutative analysis.

Findings

Defined a uniform structure on representation sets

Introduced a noncommutative modulus of continuity

Proved equivalence of two notions of uniform continuity for morphisms

Abstract

We introduce a notion of a uniform structure on the set of all representations of a given separable, not necessarilly commutative $C^*$-algebra $\mathfrak{A}$ by introducing a suitable family of metrics on the set of representations of $\mathfrak{A}$ and investigate its properties. We define the noncommutative analogue of the notion of the modulus of continuity of an element in $C^*$-algebra and we establish its basic properties. We also deal with morphisms of $C^*$-algebras by defining two notions of uniform continuity and show their equivalence.