Norms Associated to Weights in von Neumann Algebras and Decompositions of Positive Operators
Catalin Dragan · OhioLink ETD Center (Ohio Library and Information Network) · 2016
Let M be an infinite, σ-finite von Neumann factor, A a positive operator in M and {B j } ∞ j=1 ⊆ M + .Various sufficient conditions are presented for the decomposition A = ∞ j=1 C j to hold when C j ∼ B j for all j (the equivalence C ∼ B means C = XX * and B = X * X for some X ∈ M) and when C j are unitarily equivalent to B j for all j.This extends a recent work of Bourin and Lee for the case of B j = B and M = B(H) and answers affirmatively their conjecture.For the case when B j = B for all j, necessary conditions are provided, which in the type III case are also sufficient.For selfadjoint operators A the condition (-1, 1) ⊆ W e (A) (W e (A) denotes the essential numerical range of A) is characterized in terms of compressions of A implemented by isometries.In the process, a "weak selfadjoint" pinching result is obtained, namely that under the above condition on the essential numerical range of A, given any sequence of selfadjoint operators {X j } ⊆ M with X j < 1 for all j, there is a sequence of isometries {V j } ⊆ M with mutually orthogonal ranges such that V * j AV j = X j for all j.This is in the same spirit as the so called "pinching conjecture" in factors posed by Bourin and Lee.Given a normal, semifinite weight φ on a von Neumann algebra M, a new norm associated to φ is defined on M, called the triple norm of φ and denoted by ||| • ||| φ .When the weight is a trace, the study of these norms was initiated by Popa and Rȃdulescu having as main motivation to characterize the ideal of compact operators in a semifinite von Neumann algebra.Using the notion of singular values, defined in algebras with faithful, normal, semifinite traces, a complete description for the triple norm of a weight is given in the case where the algebra is a semifinite factor or the weight is a trace.When the weight is a trace τ , its triple norm is shown to characterize the ideal of τ -compact elements.Furthermore, the triple norm of τ is extended to a class of unbounded elements affiliated to M, namely L 2 (M) + M, where L 2 (M) is the noncommutative L 2 -space associated with a semifinite von Neumann algebra, and the following duality results are shown:This generalizes to the von Neumann algebra setting the classical Hilbert space results: K(H) * = L 1 (B(H)), K(H) * * = B(H), where K(H) denotes the ideal of compact operators on H and L 1 (B(H)) the ideal of trace class operators on H.It took me some time to complete this dissertation and it would not have been possible without help.First, I would like to thank my advisor, Victor Kaftal, for giving me problems to solve, for all his comments and suggestions.This was really important.