Tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property

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 II 1 factors and M n ( C ) ) and symmetric gauge norms on L ∞ [ 0 , 1 ] and C n . As the first application, we obtain that the class of unitarily invariant norms on a type II 1 factor coincides with the class of symmetric gauge norms on L ∞ [ 0 , 1 ] and von Neumann's classical result [J. von Neumann, Some matrix-inequalities and metrization of matrix-space, Tomsk. Univ. Rev. 1 (1937) 286–300] on unitarily invariant norms on M n ( C ) . As the second application, Ky Fan's dominance theorem [Ky Fan, Maximum properties and inequalities for the eigenvalues of completely continuous operators, Proc. Natl. Acad. Sci. USA 37 (1951) 760–766] 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 Hölder'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 N ( 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 N ( M 2 ( C ) ) and some extreme points of N ( M n ( C ) ) ( n ⩾ 3 ). For a type II 1 factor M , we prove that if t ( 0 ⩽ t ⩽ 1 ) is a rational number then the Ky Fan tth norm is an extreme point of N ( M ) .