Noncommutative continuous functions

By forming completions of families of noncommutative polynomials, we define a notion of noncommutative continuous function and locally bounded Borel function that give a noncommutative analogue of the functional calculus for elements of commutative C*-algebras and von Neumann algebras. These notions give a precise meaning to C*-algebras defined by generator and relations and we show how they relate to many parts of operator and operator algebra theory.