NOTES ON THE SEPARABILITY OF C*-ALGEBRAS

It is well-known that any abelian C*-algebra can be identified, by Gelfand transform, as C-0(Omega). Here, the spectrum Omega is locally compact Hausdorff in general, and compact exactly when the algebra is unital. In this survey article, we collect some results about the relation between the metrizability of Omega and the separability of C-0(Omega). A brief discussion for the topological structure of the spectrum of a general separable C*-algebra A is also given.