It is known that the Elves built the White Gold Tower.

I want to know why? for who? for what purpose? why did they left it behind?

I'm just going to answer the obvious one: They didn't really leave it behind as most of the elven races are alive and well (for certain values of the term "well") in Tamriel. But when the Ayleids were driven from what is now the Imperial City, they couldn't exactly pack up a massive tower to take with them.