$C^*$-algebras generated by three projections

TL;DR

This paper demonstrates how certain $C^*$-algebras can be generated by three specific projections, extending to matrix algebras and particular classes like purely infinite simple and AF-algebras.

Contribution

It introduces a method to generate matrix algebras over $C^*$-algebras using three mutually unitarily equivalent and almost orthogonal projections, generalizing previous results.

Findings

Matrix algebras over $C^*$-algebras can be generated by three projections.

Conditions are provided for specific classes of $C^*$-algebras to be generated by these projections.

The results connect generator properties with algebraic structures like purely infinite and AF-algebras.

Abstract

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 $k\ge \de(n)=\min\big\{k\in\mathbb N\,|\,(k-1)(k-2)\ge 2n\big\}$. Then combining this result with recent works of Nagisa, Thiel and Winter on the generators of $C^*$--algebras, we show that for a $C^*$-algebra $\aa$ 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 $C^*$--algebras and $AF$--algebras, we give some conditions that make them be generated by three mutually unitarily equivalent and almost mutually orthogonal projections.