This is an announcement for the paper "Tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property" by Junsheng Fang, Don Hadwin, Eric Nordgren, and Junhao Shen.

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 $\NN(M_n(\cc))$ ($n\geq 3$). 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 $\NN(\M)$.

Archive classification: math.OA math.FA

Mathematics Subject Classification: 46L10, 46L51

Remarks: 56 pages

The source file(s), tracial-gauge-norms.tex: 172272 bytes, is(are) stored in gzipped form as 0707.4239.gz with size 39kb. The corresponding postcript file has gzipped size 244kb.

Submitted from: jfang@cisunix.unh.edu

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/0707.4239

or

http://arXiv.org/abs/0707.4239

or by email in unzipped form by transmitting an empty message with subject line

uget 0707.4239

or in gzipped form by using subject line

get 0707.4239

to: math@arXiv.org.