 
Summary: OPERATOR HILBERT SPACES WITHOUT THE
OPERATOR APPROXIMATION PROPERTY
Alvaro Arias
Abstract. We use a technique of Szankowski S] to construct operator Hilbert
spaces that do not have the operator approximation property, including an example
in a noncommutative Lp space for p 6= 2.
1. Introduction and Preliminaries
A Banach space X has the approximation property, or AP, if the identity op
erator on X can be approximated uniformly on compact subsets of X by linear
operators of nite rank. In the 50's, Grothendieck G] investigated this property
and found several equivalent statements. For example, he proved that X has the
AP i the natural map J : X ^X ! X X is onetoone (^ is the projective
tensor product of Banach spaces and is the injective tensor product of Banach
spaces). However, it remained unknown if every Banach space had the AP until
En o E] constructed the rst counter example in the early 70's. In S], Szankowski
gave a very explicit example of a subspace of `p, 1 < p < 2, without the AP. He
considered X = (
P1
n=1 `n
2)p, which is isomorphic to `p, and de ned Z to be the
