Unitarily invariant norms related to factors

Let $\M$ be a semi-finite factor and let $\J(\M)$ be the set of operators $T$ in $\M$ such that $T=ETE$ for some finite projection $E$. In this paper we obtain a representation theorem for unitarily invariant norms on $\J(\M)$ in terms of Ky Fan norms. As an application, we prove that the class of unitarily invariant norms on $\J(\M)$ coincides with the class of symmetric gauge norms on a classical abelian algebra, which generalizes von Neumann's classical result \cite{vN} on unitarily invariant norms on $M_n(\cc)$. As another application, Ky Fan's dominance theorem \cite{Fan} is obtained for semi-finite factors. Some classical results in non-commutative $L^p$-theory (e.g., non-commutative H$\ddot{\text{o}}$lder's inequality, duality and reflexivity of non-commutative $L^p$-spaces) are extended to general unitarily invariant norms related to semi-finite factors. We also prove that up to a scale the operator norm is the unique unitarily invariant norm associated to a type ${\rm III}$ factor.