# On the Structure of Isometries between Noncommutative <em>L<sup>p</sup></em> Spaces

### David Sherman

University of California, Santa Barbara, USA

## 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) → *Lp*(_M_2) (1 ≤ *p* < ∞, *p* ≠ 2) can be canonically expressed in a certain simple form whenever _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 I2 . Moreover, when _M_1 is AS, we demonstrate that the canonical form always deﬁnes an isometry, resulting in a complete parameterization of the isometries from *Lp*(_M_1) to *Lp*(_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 *Lp* isometries by interpolation. Some complementary results and questions are also presented.

## Cite this article

David Sherman, On the Structure of Isometries between Noncommutative <em>L<sup>p</sup></em> Spaces. Publ. Res. Inst. Math. Sci. 42 (2006), no. 1, pp. 45–82

DOI 10.2977/PRIMS/1166642058