Special factors of invertible elements in simple unital purely infinite $C^*$-algebras

Abstract

In simple unital purely infinite $C^*$-algebra $A$‎, ‎Leen proved that any element in the identity component of the invertible group is‎ ‎a finite product of symmetries of $A$‎. ‎Revising Leen's factorization‎, ‎we show that a multiple of eight of such factors are $*$-symmetries of the form $1-2P_{i,j}(u)$‎, ‎where $P_{i,j}(u)$ are certain projections of the $C^*$-matrix algebra‎, ‎defined by Dye as‎ ‎\begin{equation*}‎ ‎P_{i,j}(u) = \frac{1}{2}(e_{i,i}+e_{j,j}‎ ‎+e_{i,1}ue_{1,j}+e_{j,1}u^*e_{1,i}),‎ ‎\end{equation*}‎ ‎for a given system of matrix units $\{e_{i,j}\}_{i,j=1}^n$ of $A$ and a unitary $u\in \mathcal{U}(A)$‎. ‎