Representations of non-commutative Banach algebras by continuous functions

A main result of the Gelfand theory states that any commutative semi-simple Banach algebra is isomorphic to an algebra of continuous complex-valued functions on a Hausdorff compact. In the present paper we extend this result to a class of non-commutative algebras. We introduce and compare several concepts of continuity of functions taking values in Banach algebras which differ from point to point and show that they, in a certain sense, coincide. Finally, we give applications to algebras of singular integral and convolution operators and to algebras of approximation methods.