In this paper we set up a representation theorem for tracial gauge norms on
finite von Neumann algebras satisfying the weak Dixmier property in terms of Ky
Fan norms. Examples of tracial gauge norms on finite von Neumann algebras
satisfying the weak Dixmier property include unitarily invariant norms on
finite factors (type ${\rm II}\sb 1$ factors and $M_n(\cc)$) and symmetric
gauge norms on $L^\infty[0,1]$ and $\cc^n$. As the first application, we obtain
that the class of unitarily invariant norms on a type ${\rm II}\sb 1$ factor
coincides with the class of symmetric gauge norms on $L^\infty[0,1]$ and von
Neumann's classical result \cite{vN} on unitarily invariant norms on
$M_n(\cc)$. As the second application, Ky Fan's dominance theorem \cite{Fan} is
obtained for finite von Neumann algebras satisfying the weak Dixmier property.
As the third application, 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 obtained for
general unitarily invariant norms on finite factors. We also investigate the
extreme points of $\NN(\M)$, the convex compact set (in the pointwise weak
topology) of normalized unitarily invariant norms (the norm of the identity
operator is 1) on a finite factor $\M$. We obtain all extreme points of
$\NN(M_2(\cc))$ and many extreme points of $\NN(M_n(\cc))$ ($n\geq 3$). For a
type ${\rm II}\sb 1$ factor $\M$, we prove that if $t$ ($0\leq t\leq 1$) is a
rational number then the Ky Fan $t$-th norm is an extreme point of $\NN(\M)$.
