The isometry group of L^{p}(\mu,\X) is SOT-contractible

TL;DR

This paper proves that the group of isometric automorphisms of Bochner spaces L^{p}(9,9,X) over atomless measure spaces is contractible in the strong operator topology, regardless of separability.

Contribution

It establishes the SOT-contractibility of the isometry group of L^{p} spaces for general measure spaces and Banach spaces, without separability assumptions.

Findings

The isometry group of L^{p}(9,9,X) is SOT-contractible.

Contractibility holds for all 1  9<9,9> with atomless measure spaces.

No separability assumptions are needed on 9 or 9.

Abstract

We will show that if (\Omega,\Sigma,\mu) is an atomless positive measure space, X is a Banach space and 1\leq p<\infty, then the group of isometric automorphisms on the Bochner space L^{p}(\mu,X) is contractible in the strong operator topology. We do not require \Sigma or X above to be separable.