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.

Keywords:

$C^*$ -algebra, von Neumann algebra,

$M$ -ideal,

$M$ -summand, one-sided

$M$ -ideal, multiplier algebra

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