On certain projections of $C^*$-matrix algebras

Abstract

‎In 1955‎, ‎H‎. ‎Dye defined certain projections of a‎ ‎$C^*$-matrix algebra by‎ $P_{i,j}(a)= (1+aa^*)^{-1}\otimes E_{i,i}‎ + ‎(1+aa^*)^{-1}a \otimes‎ ‎E_{i,j}+ a^*(1+aa^*)^{-1} \otimes E_{j,i}‎ + ‎a^*(1+aa^*)^{-1}a\otimes E_{j,j}‎$, ‎which was used to show that in the case of factors not of type‎ ‎$I_{2n}$‎, ‎the unitary group determines the algebraic type of that‎ ‎factor‎. ‎We study these projections and we show that in‎ ‎$\mathbb{M}_2(\mathbb{C})$‎, ‎the set of such projections includes all‎ ‎the projections‎. ‎For infinite $C^*$-algebra $A$‎, ‎having a system of‎ ‎matrix units‎, ‎we have $A\simeq \mathbb{M}_n(A)$‎. ‎M‎. ‎Leen proved that‎ ‎in a simple‎, ‎purely infinite $C^*$-algebra $A$‎, ‎the $*$-symmetries‎ ‎generate $\mathcal{U}_0(A)$‎. ‎Assuming $K_1(A)$ is trivial‎, ‎we revise‎ ‎Leen's proof and we use the same construction to show that any‎ ‎unitary close to the unity can be written as a product of eleven‎ ‎$*$-symmetries‎, ‎eight of such are of the form $1-2P_{i,j}(\omega ),\‎ ‎\omega \in \mathcal{U}(A)$‎. ‎In simple‎, ‎unital purely infinite‎ ‎$C^*$-algebras having trivial $K_1$-group‎, ‎we prove that all‎ ‎$P_{i,j}(\omega )$ have trivial $K_0$-class‎. ‎Consequently‎, ‎we prove‎ ‎that every unitary of $\mathcal{O}_n$ can be written as a finite‎ ‎product of $*$-symmetries‎, ‎of which a multiple of eight are‎ ‎conjugate as group elements‎.