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

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