Centers of $C^*$-algebras rich in modular ideals

Abstract

We provide, in the spirit of a 1968 paper of C. Delaroche, new conditions under which a $C^*$-algebra has a nonzero center.We also present an example of a (separable) AF algebra with center $\{0\}$ but whose all the primitive ideals are modular, thus answering a question from that paper of Delaroche.