Semiprime ideals in $C^{\ast }$-algebras

Abstract

We show that a not necessarily closed ideal in a $C^{\ast }$-algebra is semiprime if and only if it is idempotent, if and only if it is closed under square roots of positive elements. Among other things, it follows that prime and semiprime ideals in $C^{\ast }$-algebras are automatically self-adjoint. To prove the above, we isolate and study a particular class of ideals, which we call Dixmier ideals. As it turns out, there is a rich theory of powers and roots for Dixmier ideals. We show that every ideal in a $C^{\ast }$-algebra is squeezed by Dixmier ideals from inside and outside tightly in a suitable sense, from which we are able to deduce information about the ideal in the middle.