The SHAI property for the operators on L^p

TL;DR

This paper investigates the SHAI property in Banach spaces, providing a sufficient condition and demonstrating it for L^p spaces, symmetric basis spaces with finite cotype, and Schatten p-spaces.

Contribution

It introduces a sufficient condition for Banach spaces to have the SHAI property and applies it to several important classes of spaces.

Findings

L^p(0,1) spaces with 1<p<∞ have the SHAI property

Spaces with symmetric bases and finite cotype have the SHAI property

Schatten p-spaces for 1<p<∞ have the SHAI property

Abstract

A Banach space X has the SHAI (surjective homomorphisms are injective) property provided that for every Banach space Y, every continuous surjective algebra homomorphism from the bounded linear operators on X onto the bounded linear operators on Y is injective. The main result gives a sufficient condition for X to have the SHAI property. The condition is satisfied for L^p (0, 1) for 1 < p < \infty, spaces with symmetric bases that have finite cotype, and the Schatten p-spaces for 1 < p < \infty.