Journal of Operator Theory

Volume 43, Issue 2, Spring 2000 pp. 375-387.

Completely complemented subspace problem

Authors: Timur Oikhberg
Author institution: The University of Texas at Austin, Austin, TX 78712--1082, USA

Summary: We will prove that, if every finite dimensional subspace of an {\it infinite dimensional} operator space

$E$ is

$1$ -completely complemented in it,

$E$ is

$1$ -Hilbertian and

$1$ -homogeneous. However, this is not true for finite dimensional operator spaces: we give an example of an

$n$ -dimensional operator space

$E$ , such that all of its subspaces are

$1$ -completely complemented in

$E$ , but which is not

$1$ -homogeneous. Moreover, we will show that, if

$E$ is an operator space such that both

$E$ and

$E^*$ are

$c$ -exact and every subspace of

$E$ is

$\lambda$ -completely complemented in it, then

$E$ is

$f(c,\lambda)$ -completely isomorphic either to row or column operator space.

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