Autocomplete nested within AJAX partial

I have the scriptaculous autocomplete working perfectly for normal
page access. But having problems when combining with AJAX.

The page contains a div that is updated dynamically via AJAX.
Also the autocomplete field lives within this dynamic region.

On first load (normal page access) the autocomplete works fine. However
it stops to respond once AJAX updates the region.

It seems I am missing something. I must admit I don’t know if JS can
work in this way.

Please help, anyone?

Many thanks,