Positive combinations and sums of projections in purely infinite simple 𝐶*-algebras and their multiplier algebras

Kaftal, Victor; Ng, P. W.; Zhang, Shuang

doi:10.1090/S0002-9939-2011-10995-X

Positive combinations and sums of projections in purely infinite simple $C^*$-algebras and their multiplier algebras
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Proc. Amer. Math. Soc. 139 (2011), 2735-2746 Request permission

Abstract:

Every positive element in a purely infinite simple $\sigma$-unital $C^*$-algebra $\mathscr {A}$ is a finite linear combination of projections with positive coefficients. Also, every positive $a$ in the multiplier algebra $\mathscr M(\mathscr {A})$ of a purely infinite simple $\sigma$-unital $C^*$-algebra $\mathscr {A}$ is a finite linear combination of projections with positive coefficients. Furthermore, if the essential norm $\|a\|_{ess} > 1$, then $a$ is a finite sum of projections in $\mathscr M(\mathscr {A})$. As a consequence, any positive element in the generalized Calkin Algebra $\mathscr M(\mathscr {A})/\mathscr {A}$ or in $\mathscr M(\mathscr {A})$ but not in $\mathscr {A}$ is a positive scalar multiple of a finite sum of projections.

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