# 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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