How do i fully delete an old account so that it stops asking for updates? My ex-partner had an account on my Mac. When we broke up, we deleted his account. However, some of the applications that he has downloaded keep asking for updates. I cannot delete these apps, as it wants him to sign in but I do not have his log-in details. How can I delete these apps and stop the computer from continually asking for updates!?
By the general term "app", you ostensibly mean App Store? And if you did delete his Apple ID, then it wouldn't ask for updates.

You can either use a third-party program to remove those apps, or first delete them from the applications directory, then from Application Support located at ~/Library/Application\ Support/. If you cannot find it, then go to Terminal, located in Applications/Utilities  and execute:

open ~/Library/Application\ Support/


Then you can just delete the folder under the name of that app
Then, simply logout of his Apple ID if you failed in deleting it, and login your Apple ID