On the Structure of Isometries between Noncommutative $L^p$ Spaces

Abstract

We prove some structure results for isometries between noncommutative $L^p$ spaces associated to von Neumann algebras. We find that an isometry $T:L^p({\mathcal{M}}_1)\to L^p({\mathcal{M}}_2)$ ($1\le p<\infty$, $p\ne 2$) can be canonically expressed in a certain simple form whenever ${\mathcal{M}}_1$ has variants of Watanabe’s extension property. Although these properties are not fully understood, we show that they are possessed by all “approximately semifinite” (AS) algebras with no summand of type ${\mathrm{I}}_2$ . Moreover, when ${\mathcal{M}}_1$ is AS, we demonstrate that the canonical form always defines an isometry, resulting in a complete parameterization of the isometries from $L^p({\mathcal{M}}_1)$ to $L^p({\mathcal{M}}_2)$. AS algebras include much more than semifinite algebras, so this classification is stronger than Yeadon’s theorem (and its recent improvement), 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 $L^p$ isometries by interpolation. Some complementary results and questions are also presented.