On Certain Projections of $C^*$-Matrix Algebras

TL;DR

This paper studies projections in $C^*$-matrix algebras, showing their properties in finite and infinite cases, revising known proofs, and demonstrating factorization of unitaries into *-symmetries involving these projections.

Contribution

It extends the understanding of projections in $C^*$-matrix algebras, revises Leen's proof on *-symmetries, and shows how unitaries in certain algebras can be factorized using these projections.

Findings

In $ ext{M}_2( ext{C})$, the set of projections $P_{i,j}(a)$ covers all projections.

In infinite $C^*$-algebras with matrix units, $A ext{ is isomorphic to } ext{M}_n(A)$.

Unitaries in $ ext{O}_n$ can be expressed as products of *-symmetries involving $P_{i,j}( ext{unitaries})$.

Abstract

H. Dye defined the projections $P_{i,j}(a)$ of a $C^*$-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 $C^*$-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 $C^*$-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 $1-2P_{i,j}(\omega),\ \omega \in \mathcal{U}(A)$. 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. 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 $1-2P_{i,j}(\omega)$