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Projection decomposition in multiplier algebras

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Abstract

In this paper we present new structural information about the multiplier algebra \({\mathcal M (\mathcal A )}\) of a σ-unital purely infinite simple C*-algebra \({\mathcal {A}}\), by characterizing the positive elements \({A\in \mathcal M (\mathcal A )}\) that are strict sums of projections belonging to \({\mathcal A }\) . If \({A\not\in \mathcal {A}}\) and A itself is not a projection, then the necessary and sufficient condition for A to be a strict sum of projections belonging to \({\mathcal {A} }\) is that \({\|A\| >1 }\) and that the essential norm \({\|A\|_{ess} \geq 1}\). Based on a generalization of the Perera–Rordam weak divisibility of separable simple C*-algebras of real rank zero to all σ-unital simple C*-algebras of real rank zero, we show that every positive element of \({\mathcal {A}}\) with norm >1 can be approximated by finite sums of projections. Based on block tri-diagonal approximations, we decompose any positive element \({A\in \mathcal M (\mathcal {A} )}\) with \({\| A\| >1 }\) and \({\| A\|_{ess} \geq 1}\) into a strictly converging sum of positive elements in \({\mathcal A}\) with norm >1.

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Authors and Affiliations

  1. Department of Mathematics, University of Cincinnati, P. O. Box 210025, Cincinnati, OH, 45221-0025, USA

    Victor Kaftal & Shuang Zhang

  2. Department of Mathematics, University of Louisiana, 217 Maxim D. Doucet Hall, P. O. Box 41010, Lafayette, LA, 70504-1010, USA

    P. W. Ng

Authors
  1. Victor Kaftal
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  2. P. W. Ng
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  3. Shuang Zhang
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Correspondence to P. W. Ng.

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Kaftal, V., Ng, P.W. & Zhang, S. Projection decomposition in multiplier algebras. Math. Ann. 352, 543–566 (2012). https://doi.org/10.1007/s00208-011-0649-0

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  • DOI: https://doi.org/10.1007/s00208-011-0649-0

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