Zone Alarm has been reporting that PostUpdater wanted access to the internet, but I decided to decline as I didn't know what it was. Scouring the Internet for clues led me to the fact that it is related to Shockwave, and is reputed to be dangerous. I uninstalled Shockwave, but Zone Alarm still reported the problem. I also discovered a new icon in the Notification area, labelled "Shockwave Updater". It doesn't seem to do anything, but I can't be rid of it. Ctrl+Alt+Del didn't show it up, though I did see a process with no name and no details and wondered if I should stop it, but it's not there now so that's a mystery! The unwanted icon remains, however. Several foolproof ideas elsewhere on the Internet turned out not to be foolproof - anyone got any ideas?

   It has to be starting somehow.  Either it is in one of Windows' startup lists or it is started when another program is run.  I would suspect Windows' startup.  Look in msconfig or Speed Startup to see what is in Windows' startup lists.

Thanks for your response, Ray. msconfig doesn't show it, and I seem to have forgotten where to find Speed Startup. Please could you remind me.

Speed Startup is at http://superwin.com/speedstartup.htm

Thanks again, Ray. Oddly enough, I think the problem may have resolved itself! I left my PC running for some hours, and on return I found it seemed to have restarted itself, but had been interrupted by Spybot-SD Resident, which reported that a registry entry for Shockwave Updater was being deleted. I decided to let it do so, and there was an immediate request for it to be replaced by what I assume to be a new version. This I declined. Next time I started the machine, the unwanted icon had vanished. From my Internet searches in other forums, it seems plenty of others have had the same thing happen, so maybe this is the way to fix it ....
I've downloaded the trial version of Speed Startup, and will try it out sometime.
Thanks again.