$C^{\ast }$-algebras generated by three projections

Abstract

In this short note, we prove that for a $C^{\ast }$-algebra $\mathring{a}$ generated by $n$ elements, $M_k(\widetilde{\stackrel{˚}{a}})$ is generated by $k$ mutually unitarily equivalent and almost mutually orthogonal projections for any $k\ge \de(n)=\min\big\{k\in\mathbb N\,|\,(k-1)(k-2)\ge 2n\big\}$. Then combining this result with recent works of Nagisa, Thiel and Winter on the generators of $C^{\ast }$--algebras, we show that for a $C^{\ast }$-algebra $\mathring{a}$ generated by finite number of elements, there is $d\ge 3$ 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 $C^{\ast }$--algebras and $AF$--algebras, we give some conditions that make them be generated by three mutually unitarily equivalent and almost mutually orthogonal projections.

10 pages

References in corpus (2)

• A Note on Approximately Divisible C${}^{\ast }$-algebras
• Completeness of $n$--tuple of projections in $C^{\ast }$--algebras