Uniformity in $C^*$-algebras

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.