$C^*$-algebras generated by three projections

In this short note, we prove that for a $C^*$-algebra $aa$ generated by $n$ elements, $M_{k}(tilde{aa})$ is generated by $k$ mutually unitarily equivalent and almost mutually orthogonal projections for any $kge de(n)=minbig{kinmathbb N,|,(k-1)(k-2)ge 2nbig}$. Then combining this result with recent works of Nagisa, Thiel and Winter on the generators of $C^*$--algebras, we show that for a $C^*$-algebra $aa$ generated by finite number of elements, there is $dge 3$ such that $M_d(tilde A)$ is generated by three mutually unitarily equivalent and almost mutually orthogonal projections. Furthermore, for certain separable purely infinite simple unital $C^*$--algebras and $AF$--algebras, we give some conditions that make them be generated by three mutually unitarily equivalent and almost mutually orthogonal projections.