On the Borel structure ofC*-algebras

We provide a method of embedding aC*-algebra in aC*-algebra called its -envelope. is contained in the enveloping algebra of but is generally much smaller, and if is commutative with identity then can be identified with the algebra of bounded Baire functions on the spectrum of. The main result is to completely determine the structure of for all separable G. C. R. algebras. This provides a good basis for a non-commutative theory of probability.We should like to thankJ. T. Lewis, G. W. Mackey andR. J. Plymen, who have given us considerable encouragement and insight into the quantum mechanical relevance of the ideas developed here.