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Journal of Operator Theory

Volume 51, Issue 1, Winter 2004  pp. 201-219.

One-sided projections on $C^*$-algebras

Authors David P. Blecher, (1) Roger R. Smith (2) and Vrej Zarikian
Author institution: (1) Department of Mathematics, University of Houston, Houston, TX 77204--3008, USA
(2) Department of Mathematics, Texas A&M University, College Station, TX 77843--3368, USA
(3) Department of Mathematics, University of Austin, Austin, TX 78712--1082, USA

Summary:  We obtain several equivalent characterizations of linear maps on a $C^*$-algebra $\mathcal{A}$ which are given by left multiplication by a fixed orthogonal projection in (resp. fixed element in) $\mathcal{A}$ or its multiplier algebra. These results are connected to the `complete one-sided $M$-ideals' in operator spaces recently introduced by Blecher, Effros, and Zarikian. Part of the proof makes use of a technique to "solve" multi-linear equations in von Neumann algebras. This technique is also applied to show that preduals of von Neumann algebras have no nontrivial complete one-sided $M$-ideals. We also show that the intersection of two complete one-sided $M$-summands need not be a one-sided $M$-summand.

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