A new prime \(C^{\ast}\)-algebra that is not primitive (Q2497936)

A question raised by Dixmier which was open for many decades asked whether there is a non-separable prime \(C^*\)-algebra that is not primitive. The first example of such a \(C^*\)-algebra was produced by \textit{N.\,Weaver} [J.~Funct.\ Anal.\ 203, No.\,2, 356--361 (2003; Zbl 1031.46063)]. Another example of a suitable graph \(C^*\)-algebra by Katsura appeared in 2006. The present short note contains a very brief argument that a certain prime \(C^*\)-algebra related to the so-called McAlister monoid is not primitive.