PhD, University of Cincinnati, 2016, Arts and Sciences: Mathematical Sciences
Let M be an infinite, ς-finite von Neumann factor, A a positive operator in M and {Bj} a sequence in M+. Various sufficient conditions are presented for the decomposition A = ∑j = 1∞ Cj to hold when Cj ∼ Bj for all j (the equivalence C ∼ B means C = XX* and B = X*X for some X in M) and when Cj are unitarily equivalent to Bj for all j. This extends a recent work of Bourin and Lee for the case of Bj = B and M = B(H) and answers affirmatively their conjecture. For the case when Bj = 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) ⊆ We(A) (We(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 {Xj} in M with ||Xj|| < 1 for all j, there is a sequence of isometries {Vj} in M with mutually orthogonal ranges such that Vj * A Vj = Xj 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 Radulescu 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 affiliat (open full item for complete abstract)
Committee: Victor Kaftal Ph.D. (Committee Chair); Herbert Halpern Ph.D. (Committee Member); Costel Peligrad Ph.D. (Committee Member); Gary Weiss Ph.D. (Committee Member); Shuang Zhang Ph.D. (Committee Member)
Subjects: Mathematics