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- W2949757956 abstract "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 $NN(M_n(cc))$ ($ngeq 3$). For a type ${rm II}sb 1$ factor $M$, we prove that if $t$ ($0leq tleq 1$) is a rational number then the Ky Fan $t$-th norm is an extreme point of $NN(M)$." @default.
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- W2949757956 date "2007-07-28" @default.
- W2949757956 modified "2023-09-26" @default.
- W2949757956 title "Tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property" @default.
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