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

Document Type: Original Article

Author

United Arab Emirates University, ‎UAE

Abstract

In simple unital purely infinite $C^*$-algebra $A$‎, ‎M‎. ‎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 H‎. ‎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 $u\in \mathcal{U}(A)$.

Keywords

Main Subjects

### History

• Receive Date: 28 October 2018
• Revise Date: 09 January 2019
• Accept Date: 09 January 2019