Unitarily invariant norms related to semi-finite factors
Let $\mathcal M$ be a semi-finite factor and let $\mathcal J(\mathcal M)$ be the set of operators $T$ in $\mathcal M$ such that $T=ETE$ for some finite projection $E$. We obtain a representation theorem for unitarily invariant norms on $\mathcal J(\mathcal M)$ in terms of Ky Fan norms. As an application, we prove that the class of unitarily invariant norms on $\mathcal J(\mathcal M)$ coincides with the class of symmetric gauge norms on a classical abelian algebra, which generalizes von Neumann's classical 1940 result on unitarily invariant norms on $M_n(\mathbb C)$. As another application, Ky Fan's dominance theorem of 1951 is obtained for semi-finite factors.