The isometry group of is SOT-contractible

We will show that if ( Ω , Σ , μ ) is an atomless positive measure space, X is a Banach space and 1 ⩽ p < ∞ , then the group of isometric automorphisms on the Bochner space L p ( μ , X ) is contractible in the strong operator topology. We do not require Σ or X above to be separable.