On Certain Projections of $C^{\ast }$-Matrix Algebras

H. Dye defined the projections $P_{i,j}(a)$ of a $C^{\ast }$-matrix algebra by {eqnarray*} 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}, {eqnarray*} and he used it 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 $C^{\ast }$-algebra $A$, having a system of matrix units, including the Cuntz algebra ${\mathcal{O}}_n$, we have $A\simeq {\mathbb{M}}_n(A)$. M. Leen proved that in a simple, purely infinite $C^{\ast }$-algebra $A$, the *-symmetries generate ${\mathcal{U}}_0(A)$. We revise and modify Leen’s proof to show that part of such -isometry factors are of the form $1-2P_{i,j}(\omega),\ \omega \in $\mathcal{U}$(A)$\mathrm{.}Insimple,unitalpurelyinfinite$C^$-algebras having trivial $K_1$-group, we prove that all $P_{i,j}(\omega )$ have trivial $K_0$-class. In particular, if $u\in \mathcal{U}({\mathcal{O}}_n)$, then $u$ can be factorized as a product of *-symmetries, where eight of them are of the form $1-2P_{i,j}(\omega )$