Tracial gauge norms on finite von Neumann algebras statisfying the weak Dixmier property
Junshen Fang
I will talk about a representation theorem for tracial gauge norems on finite von Neumann Algebras
satisfying the weak Dixmier property and a number of its consequences for unitarily invariant norms on
type II1 factors.
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A Folner Invariant for type II1 factors
Mohan Ravichandran
Given a group representation, the communtant algebra of the representation is a Von Neumann algebra.
This construction with an amenable group and any representation of the group gives us one of the so called hyperfinite
Von Neuman algebras. As in the case of groups and amenability, it is of importance to provide isomorphism invariants that
measure how far a given Von Neumann algebra(not necessarily coming from a group representation) is from being hyperfinite.
Measures of non-amenability are common in group theory. One instance of this in the case of groups is the Folner type
invariant Fol(G) definde by Arzhantseva, et al in Adv. Math. 197 (2005). These Folner conditions have analogues in Von Neumann
algebras and were used implicitly in Alain Connes' celebrated 1976 paer in which he classified all injective Von Neumann algebras [Ann. of Math., 2 104].
Motivated by his work, we introduce an numerical invariant for Von Neumann algebras and compute it in certain important case [Bannon, Ravichandran, Expo. Math. 25 (2007)].
We then suggest an approach to the problem of determining if Richard Thompson's group F is amenable or not.
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An elementary proof of a theorem of Douglas and Foias on invariant operator ranges
Donald Hadwin
All of the main structure theorems for nXn complex matrices (e.g., upper triangular form, Jordan form) can be expressed soley
in terms of invariant spaces. For bounded operators on an infinite-dimensional space it is not known whether every
operator even has a nontrivial closed invariant subspace. However, C. Foias showed that if closed linear subspaces are replaced
with operator ranges (i.e., ranges of operators), most of te finite-dimensional results (e.g., Burnside's theorem) are true
in infinite dimensions. Douglas and Foias proved a striking theorem: If T is a non-algebraic operator and S is an operator the
leaves invariant every T-invariant operator range, then there is an entire function f(z), such that S=f(T). We give a proof
that mostly uses very elementary linear algebra.
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