center of a $C^*$ algebra

Question

Suppose the center of $C^*$ algebra $A$ is 0,does this $C^*$ algebra constructed by simple non-unital $C^*$ algebras.To be more precise,$A$ can be only simple or direct sum of simple non-unital $C^*$ algebras,or tensor product of simple non-unital $C^*$ algebras.

Does there exist other $C^*$-algebras whose center is 0?

Answer 1

A $C^*$-algebra with trivial center need not be simple. Consider $A=C_0(\mathbb R)\otimes \mathcal K=C_0(\mathbb R,\mathcal K)$, where $\mathcal K$ is compact operators on separable Hilbert space. The center of $A$ is trivial, but it is not simple, nor is it of the forms you have described.