# Journal of Operator Theory

Volume 64, Issue 2, Fall 2010 pp. 265-298.

Complete isometries between subspaces of noncommutative $L_p$-spaces**Authors**: Mikael de la Salle

**Author institution:**Equipe d'Analyse Fonctionnelle, Institut de Mathematiques de Jussieu, Universite Paris 6 and DMA, Ecole Normale Superieure, 45 rue d'Ulm, 75230 Paris Cedex 05, France

**Summary:**We prove some noncommutative analogues of a theorem proved by Plotkin and Rudin about isometries between subspaces of $L_p$-spaces. Let $p$ be a finite positive number, $p$ not an even integer. The main result of this paper states that in the category of unital subspaces of noncommutative probability $L_p$-spaces, under some boundedness condition, the unital completely isometric maps come from $*$-isomorphisms of the underlying von Neumann algebras. Some applications are given, including to noncommutative $H^p$-spaces.

**Keywords:**Non commutative probability, complete isometries between non commutative $L_p$ spaces, von Neumann algebra

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