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Remarks on complemented subspaces of von-Neumann algebras

Gilles Pisier
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Abstract

In this note we include two remarks about bounded ($\underline{not}$ necessarily contractive) linear projections on a von Neumann-algebra. We show that if $M$ is a von Neumann-subalgebra of $B(H)$ which is complemented in B(H) and isomorphic to $M \otimes M$ then $M$ is injective (or equivalently $M$ is contractively complemented). We do not know how to get rid of the second assumption on $M$. In the second part,we show that any complemented reflexive subspace of a $C^*$- algebra is necessarily linearly isomorphic to a Hilbert space.

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Gilles Pisier. 1991-05-31. Remarks on complemented subspaces of von-Neumann algebras. https://arxiv.org/abs/math/9201227

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