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Abstract

We introduce a notion of uniform structure on the set of all representations of a given separable, not necessarily 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 a $C^*$-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.