Distance between unitary orbits of normal elements in simple C*-algebras of real rank zero

Let $x, y$ be two normal elements in a unital simple C*-algebra $A.$ We introduce a function $D_c(x, y)$ and show that in a unital simple AF-algebra there is a constant $1>C>0$ such that $$ Ccdot D_c(x, y)le {rm dist}({cal U}(x),{cal U}(y))le D_c(x,y), $$ where ${cal U}(x)$ and ${cal U}(y)$ are the closures of the unitary orbits of $x$ and of $y,$ respectively. We also generalize this to unital simple C*-algebras with real rank zero, stable rank one and weakly unperforated $K_0$-group. More complicated estimates are given in the presence of non-trivial $K_1$-information.

$C^*$-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.