A note on bundle gerbes and infinite-dimensionality

Let $(P, Y)$ be a bundle gerbe over a fibre bundle $Y to M$. We show that if $M$ is simply-connected and the fibres of $Y to M$ are connected and finite-dimensional then the Dixmier-Douady class of $(P, Y)$ is torsion. This corrects and extends an earlier result of the first author.