On the structure of isometries between noncommutative Lp spaces

TL;DR

This paper characterizes the structure of isometries between noncommutative Lp spaces associated with von Neumann algebras, extending previous results and introducing new techniques for understanding these mappings.

Contribution

It provides a canonical form for isometries under certain conditions and broadens the classification beyond existing theorems, using novel methods involving modular theory and interpolation.

Findings

Isometries can be expressed in a simple form under specific properties.

The classification extends Yeadon's theorem for a wider class of algebras.

New constructions of Lp isometries via modular projections and interpolation.

Abstract

We prove some structure results for isometries between noncommutative Lp spaces associated to von Neumann algebras. We find that an isometry T: Lp(M_1) to Lp(M_2) (1 le p < infty, p not 2) can be canonically expressed in a certain simple form whenever M_1 has variants of Watanabe's extension property. Conversely, this form always defines an isometry provided that M_1 is "approximately semifinite" (defined below). Although neither of these properties is fully understood, we show that they are enjoyed by all semifinite algebras and hyperfinite algebras (with no summand of type I_2), plus others. Thus the classification is stronger than Yeadon's theorem for semifinite algebras (and its recent improvement in [JRS]), and the proof uses independent techniques. Related to this, we examine the modular theory for positive projections from a von Neumann algebra onto a Jordan image of another von Neumann algebra, and use such projections to construct new Lp isometries by interpolation. Some complementary results and questions are also presented.