If you're talking about the University of Ottawa, University is always uppercase, even if standing alone; the same applies when you shorten the name of other universities in a text:
The University of Calgary has a large campus. The University also boasts state-of-the-art chemistry laboratories.
Use lowercase "u" when university is simply a common noun:
The university setting can be intimidating for first-year students.
and when you pluralize it before a series of specific institution names:
The event attracted representatives from the universities of Calgary, Ottawa, Windsor and Toronto.