On the uniform approximation property in Banach spaces

Abstract

We derive a criterion for a Banach space to fail the uniform approximation property (UAP). This criterion is applied to prove that q, spaces of matrices fail UAPifp>80. 0. A Banach space X has the uniform approximation property (abbreviated UAP) if for every integer k there exists an integer m(k) such that for every k-dimensional subspace E of X there exists an operator T : X ~ X such that ][T][~A, rkT<=m(k) and TpE=Id~ for some A < ~ which is independent of k. (TE denotes the restriction of T to E and IdE denotes the identity on E.)