Extension of Spectral Scales to Unbounded Operators

M D Wills

doi:10.1155/2010/713563

Abstract

We extend the notion of a spectral scale ton-tuples of unbounded operators affiliated with a finite von Neumann Algebra. We focus primarily on the single-variable case and show that many of the results from the bounded theory go through in the unbounded situation. We present the currently available material on the unbounded multivariable situation. Sufficient conditions for a set to be a spectral scale are established. The relationship between convergence of operators and the convergence of the corresponding spectral scales is investigated. We establish a connection between the Akemann et al. spectral scale (1999) and that of Petz (1985).

Highlights

Introduction and PreliminariesThe notion of the spectrum of a self-adjoint operator has proved to be of great interest and use in various branches of mathematics
Let M be a finite von Neumann algebra equipped with a normal, faithful tracial state, τ
It is easy to see that a spectral scale must be a prespectral scale: condition i is noted on page 3 of this paper, condition ii follows from Definition 2.1, condition iii follows from the fact that τ is a state, condition iv follows from Proposition 2.3, and condition v follows from the definition of the lower boundary Notation 2.14

Summary

Introduction and Preliminaries

The notion of the spectrum of a self-adjoint operator has proved to be of great interest and use in various branches of mathematics. Many of the results can be extended to n-tuples of self-adjoint operators in M. If M is closed in the weak operator topology, self-adjoint, and contains 1, M is a von Neumann algebra 2, page 308. Τ is a faithful, finite, normal trace on M 4, pages 504-5. Let τ be a faithful, finite, normal trace on M. And Nelson shows that elements of L1 M are closed, densely defined operators affiliated with M 8, Theorem 1, page 107, and Theorem 5, page 114 It follows that a bounded linear functional, g ∈ M∗ can be represented by a possibly unbounded linear operator b affiliated with M and we get the equality g a τ ba for every a ∈ M

Spectral Scale Theory for Unbounded Operators—the Single-Variable Case

The Geometry of Spectral Scales in Higher Dimensions

Invariance Properties of the Spectral Scale

Miscellaneous Results

Future Research