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 $u\in \mathcal{U}(A)$.
Citation
Ahmed Al-Rawashdeh. "Special factors of invertible elements in simple unital purely infinite $C^*$-algebras." Adv. Oper. Theory 4 (3) 641 - 650, Summer 2019. https://doi.org/10.15352/aot.1810-1432
Information