Complete isometries between subspaces of noncommutative Lp-spaces

TL;DR

This paper extends classical isometry theorems to noncommutative Lp-spaces, showing that under certain conditions, isometries are induced by *-isomorphisms of von Neumann algebras, with applications to noncommutative H^p spaces.

Contribution

It establishes noncommutative analogues of Plotkin and Rudin's theorem, characterizing isometries in noncommutative Lp-spaces via *-isomorphisms.

Findings

Unital completely isometric maps arise from *-isomorphisms of von Neumann algebras.

Results apply to noncommutative H^p spaces.

Provides a framework for understanding isometries in noncommutative Lp-spaces.

Abstract

We prove some noncommutative analogues of a theorem by Plotkin and Rudin about isometries between subspaces of Lp-spaces. Let 0<p<\infty, p not an even integer. The main result of this paper states that in the category of unital subspaces of noncommutative probability Lp-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 non commutative H^p spaces.