# \ell^1-contractive maps on noncommutative L^p-spaces.

## Speaker:

Christian Le Merdy

## Affiliation:

Laboratoire de Mathématiques, Université de Franche-comté, Besançon

## Date:

Fri, 16/08/2019 - 12:00pm

## Venue:

RC-4082, The Red Centre, UNSW

## Abstract:

Let $T\colon L^p(\mathcal{M})\to L^p(\mathcal{N})$ be a bounded operator between two noncommutative $L^p$-spaces, $1< p<\infty$. We say that $T$ is $\ell^1$-bounded (resp. $\ell^1$-contractive) if $T\otimes I_{\ell^1}$ extends to a bounded (resp. contractive) map from $L^p(\mathcal{M};\ell^1)$ into $L^p(\mathcal{N};\ell^1)$.

In the first part of the talk I will explain the definition of the spaces $L^p(\mathcal{M};\ell^1)$. Then I will discuss the connection of $\ell^1$-boundedness with positivity, regularity and complete regularity. In the last part of the talk I will present the following result: Yeadon's factorization theorem for $L^p$-isometries, $1< p\not=2 <\infty$, applies to an isometry $T\colon L^2(\mathcal{M})\to L^2(\mathcal{N})$ if and only if $T$ is $\ell^1$-contractive.