Counterexamples to a conjecture of Grothendieck. (English) Zbl 0542.46038

Pisier, Gilles

Acta Math. 151, 181-208 (1983).

The paper decides to the negative the following conjecture of A. Grothendieck [Produits tensoriels topologiques et espaces nucléaires, Mém. Am. Math. Soc. 16 (1955; Zbl 0064.355)]: If two Banach spaces are such that their injective and projective products coincide, then one of the two Banach spaces must be finite dimensional.
The paper contains the construction of a separable infinite dimensional Banach space X such that the injective and the projective tensor product of this space with itself coincide, both algebraically and topologically. This space is of cotype 2 as well as its dual. It can contain uniformly complemented \(\ell^ n_{p'}s\) for no p such that 1\(\leq p\leq \infty\), thus answering also to the negative a question of J. Lindenstrauss [Actes Congr. Internat. Math., Nice 1970, Vol.2, 365-372 (1971; Zbl 0224.46032)]. Since X is not isomorphic to a Hilbert space, although X and \(X^*\) are both of cotype 2, a problem of B. Maurey stated in Proc. Int. Congr. Math., Vancouver 1974, Vol. 2, 75-79 (1975; Zbl 0343.47012) is also solved to the negative.

Reviewer: W.Strauss

MSC:

46M05	Tensor products in functional analysis
46B20	Geometry and structure of normed linear spaces

Keywords:

injective and projective products coincide; cotype 2

Citations:

Zbl 0064.355; Zbl 0224.46032; Zbl 0343.47012

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