Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On Certain Projections of $C^*$-Matrix Algebras   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
$1-2P_{i,j}(\omega)$

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The categories of this paper are: nucl-ex
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The categories of this paper are: physics
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The categories of this paper are: dg-ga
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The categories of this paper are: cmp-lg
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The categories of this paper are: funct-an
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The categories of this paper are: cs
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The categories of this paper are: comp-gas
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: stat
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The categories of this paper are: atom-ph
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The categories of this paper are: supr-con
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The categories of this paper are: mtrl-th
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The categories of this paper are: chem-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: bayes-an