Addendum: I'm actually working with the Heisenberg group which is not compact.

Yes, I should have been more precise in my question. With unitary real/complex representation I mean exactly what @QiaochuYuan wrote in the comment: $<\rho(g)u|\rho(g)v>=<u,v>$.

Ok, I will try, thank you very much!

Yeah sorry, you are right, there doesn't exist any condition on the spaces which guarantee the additivity for $dim(V)>1$. I guess I woke up a little stupid this morning. . Anyhow in my question I was quite general, I wrote: "some conditions on the vector spaces or the map", what about some conditions on T?

Thank you very much for your answer Joonas, but actually I asked a little different question. You showed me that an homogeneous but non additive function does exist, and I get it, but I already have a map T and I would like to know if there is some theorem which guarantees its additivity under some suitable conditions.

Perfect!Thank you!