Natural Operator Space Tensor Norms
摘要
Grothendieck’s Résumé contained the list of all natural tensor norms. These norms come from applying a finite number of natural operations to the projective and injective tensor norms. They are obtained by taking left/right projective and injective hulls in some order (see Sections 15 and 20 in Defant and Floret (Tensor Norms and Operator Ideals. North-Holland Mathematics Studies, vol. 176. North-Holland Publishing Co., Amsterdam, 1993)). Grothendieck proved that there were at most fourteen possible natural norms, but he did not know the exact dominations among them, or if there was a possible reduction on the table of natural norms (this was, in fact, one of the open problems posed in the Résumé). This was solved, several years later, thanks to very deep ideas of Gordon and Lewis. All these results are now classical and can be found for example in Defant and Floret (Tensor Norms and Operator Ideals. North-Holland Mathematics Studies, vol. 176. North-Holland Publishing Co., Amsterdam, 1993, Sec. 27) and Diestel et al. (The Metric Theory of Tensor Products: Grothendieck’s résumé Revisited. American Mathematical Society, Providence, 2008, 4.4.2).