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.
Citation
A. Al-Rawashdeh. "On certain projections of $C^*$-matrix algebras." Ann. Funct. Anal. 3 (2) 144 - 154, 2012. https://doi.org/10.15352/afa/1399899939
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