• DOI:10.1007/BF02393206
• Corpus ID: 121325383

Counterexamples to a conjecture of Grothendieck

@article{Pisier1983CounterexamplesTA,
  title={Counterexamples to a conjecture of Grothendieck},
  author={Gilles Pisier},
  journal={Acta Mathematica},
  year={1983},
  volume={151},
  pages={181-208}
}

• G. Pisier
• Published 1 December 1983
• Mathematics
• Acta Mathematica

In his thesis ([7] II. p. 136) and in his fundamental paper ([6] p. 74), Grothendieck formulated the following conjecture: If two Banach spaces X and Y are such that their injective and projective tensor products X ~ Y and X ~ Y coincide, then either X or Y must be finite dimensional. The aim of this paper is to give a counterexample. We will exhibit a separable infinite dimensional Banach space X such that X~X=X~X, both algebraically and topologically. The space X is of cotype 2 as well as its dual. Moreover, the natural map from X*~X into X*~X is surjective, but it is not injective, since X fails the approximation property (in short the A.P.); equivalently, every operator on X which is a uniform limit of finite rank operators is nuclear. This implies that there are (roughly) "very few" operators on X of finite rank and of small norm. For instance, there is a number 6>0 such that, for any finite dimensional subspace E of X and for any projection P: X---~E, we have