, Volume 352, Issue 3, pp 543-566
Date: 22 Feb 2011

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.