# Analytic and algebraic aspects of noncommutative $L^p$ spaces

Abstract: In the first part of the talk we consider the complete isometries between $L^p$ spaces ($1 \le p < infty$, $p \ne 2$). We give a complete (of course) classification, exhibiting a simple canonical form which extends work of many authors, including Banach. In particular, the classical theorem which says that the isometric image of one $L^p$ (function) space inside another must be contractively complemented remains true; we view this as a rigidity property for embeddings of noncommutative measure spaces".

The second part of the talk develops the $L^p$ representation theory for a fixed von Neumann algebra $\mathcal{M}$. The desired modules should be columns of $L^p(\mathcal{M})$". We show that this is equivalent to the existence of an $L^{p/2}(\mathcal{M})$-valued inner product. When $p=2$, these are normal Hilbert space representations, and when $p=\infty$, these are C*-modules. The theory for general $p$ shares properties with both, but the bimodule theory is basically trivial.

Some of this is joint work with Marius Junge and Zhong-Jin Ruan.