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In this chapter we return to the abstract theory of o.s. tensor norms. In particular, we study o.s. tensor norms which behave well with respect to complete injections or projections. We know that the minimal o.s. tensor norm respects complete injections (Effros and Ruan, Operator Spaces. London Mathematical Society Monographs. New Series, vol. 23. The Clarendon Press Oxford University Press, New York, 2000, Prop. 8.1.5) whereas the projective o.s. tensor norm respects complete projections (Effros and Ruan, Operator Spaces. London Mathematical Society Monographs. New Series, vol. 23. The Clarendon Press Oxford University Press, New York, 2000, Prop. 7.1.7). A well-known property of the Haagerup o.s. tensor norm is that it respects both complete projections and injections (Effros and Ruan, Operator Spaces. London Mathematical Society Monographs. New Series, vol. 23. The Clarendon Press Oxford University Press, New York, 2000, Prop. 9.2.5). It should be noted that this cannot happen in the Banach space/normed space framework as a consequence of Grothendieck’s inequality (Defant and Floret, Tensor Norms and Operator Ideals. North-Holland Mathematics Studies, vol. 176. North-Holland Publishing Co., Amsterdam, 1993, Prop. 20.20). Based on the classical definitions, we consider in this chapter two natural procedures on o.s. tensor norms: the completely injective hull and the completely projective hull. Similar constructions were considered in Blecher (Can. J. Math. 44(1):75–90, 1992).

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Completely Projective/Injective Operator Space Tensor Norms

  • 章节

  • 发表日期：

    2025年

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  • Javier Alejandro Chávez-Domínguez,
  • Verónica Dimant,
  • Daniel Galicer

摘要

In this chapter we return to the abstract theory of o.s. tensor norms. In particular, we study o.s. tensor norms which behave well with respect to complete injections or projections. We know that the minimal o.s. tensor norm respects complete injections (Effros and Ruan, Operator Spaces. London Mathematical Society Monographs. New Series, vol. 23. The Clarendon Press Oxford University Press, New York, 2000, Prop. 8.1.5) whereas the projective o.s. tensor norm respects complete projections (Effros and Ruan, Operator Spaces. London Mathematical Society Monographs. New Series, vol. 23. The Clarendon Press Oxford University Press, New York, 2000, Prop. 7.1.7). A well-known property of the Haagerup o.s. tensor norm is that it respects both complete projections and injections (Effros and Ruan, Operator Spaces. London Mathematical Society Monographs. New Series, vol. 23. The Clarendon Press Oxford University Press, New York, 2000, Prop. 9.2.5). It should be noted that this cannot happen in the Banach space/normed space framework as a consequence of Grothendieck’s inequality (Defant and Floret, Tensor Norms and Operator Ideals. North-Holland Mathematics Studies, vol. 176. North-Holland Publishing Co., Amsterdam, 1993, Prop. 20.20). Based on the classical definitions, we consider in this chapter two natural procedures on o.s. tensor norms: the completely injective hull and the completely projective hull. Similar constructions were considered in Blecher (Can. J. Math. 44(1):75–90, 1992).