  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.
