An algebra of continuous functions as a continuous envelope of its subalgebras

Abstract

To an arbitrary involutive stereotype algebra A the continuous envelope operation assigns its nearest, in some sense, involutive stereotype algebra En_vCA so that homomorphisms to various C*-algebras separate the elements of Env_C A but do not distinguish between the properties of A and those of Env_CA.

If A is an involutive stereotype subalgebra in the algebra C(M) of continuous functions on a paracompact locally compact topological space M, then, for C(M) to be a continuous envelope of A, i.e., Env_CA = C(M), it is necessary but not sufficient that A be dense in C(M). In this note we announce a necessary and sufficient condition for this: the involutive spectrum of A must coincide with M up to a weakening of the topology such that the system of compact subsets in M and the topology on each compact subset remains the same.