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

TL;DR

This paper establishes new conditions for $C^*$-algebras to have a nonzero center and provides an example of an AF algebra with trivial center despite all primitive ideals being modular, addressing a question from Delaroche's 1968 work.

Contribution

It introduces novel criteria for the presence of a nonzero center in $C^*$-algebras and constructs an example of an AF algebra with trivial center but modular primitive ideals.

Findings

New conditions for nonzero centers in $C^*$-algebras

An example of a separable AF algebra with zero center

All primitive ideals of the example are modular

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.