# 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.