One-sided Projections on C∗-algebras

In [4], the notion of a complete one-sided M -ideal for an operator space X was introduced as a generalization of Alfsen and Effros’ notion of an M -ideal for a Banach space [1]. In particular, various equivalent formulations of complete one-sided M -projections were given. In this paper, some sharper equivalent formulations are given in the special situation that X = A, a C∗-algebra (in which case the complete left M -projections are simply left multiplication on A by a fixed orthogonal projection in A or its multiplier algebra). The proof of the first equivalence makes use of a technique which is of interest in its own right—a way of “solving” 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. In addition, we show that in a C∗-algebra, the intersection of finitely many complete one-sided M -summands need not be a complete one-sided M -summand, unlike the classical situation.