FS-property for \(C^*\)-algebras (Q2701615)

Recall that a \(C^*\)-algebra \(A\) has real rank zero, i.e., invertible selfadjoint elements are dense in the set \(A_{sa}\) of all selfadjoint elements of \(A\), if and only if \(A\) has the FS-property, i.e., the set of selfadjoint elements with finite spectrum is dense \(A_{sa}\), as proved by Brown and Pedersen. In this paper, it is shown by examples that for separable unital \(C^*\)-algebras \(A\) and \(C\) of real rank zero with \(C\) nuclear, the minimal \(C^*\)-tensor product \(A\otimes C\) may have real rank strictly higher than zero even if \(A\) is assumed to be nuclear or exact.