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Spectral Fredholm Theory in Von Neumann Algebras

Stefan Ivkovic

Abstract

In this paper, we extend Fredholm theory in von Neumann algebras established by Breuer in [5] and [6] to spectral Fredholm theory. We consider 2 by 2 upper triangular operator matrices with coefficients in a von Neumann algebra and give the relationship between the generalized essential spectra in the sense of Breuer of such matrices and of their diagonal entries. Next, we prove that if a generalized Fredholm operator in the sense of Breuer has 0 as an isolated point of its spectrum, then the corresponding spectral projection is finite. Finally, we define the generalized B-Fredholm operator in a von Neumann algebra as a generalization in the sense of Breuer of the classical B-Fredholm operators on Hilbert and Banach spaces. We provide sufficient conditions under which a sum of a generalized B-Fredholm operator and a finite operator in a von Neumann algebra is again a generalized B-Fredholm operator.

Spectral Fredholm Theory in Von Neumann Algebras

Abstract

In this paper, we extend Fredholm theory in von Neumann algebras established by Breuer in [5] and [6] to spectral Fredholm theory. We consider 2 by 2 upper triangular operator matrices with coefficients in a von Neumann algebra and give the relationship between the generalized essential spectra in the sense of Breuer of such matrices and of their diagonal entries. Next, we prove that if a generalized Fredholm operator in the sense of Breuer has 0 as an isolated point of its spectrum, then the corresponding spectral projection is finite. Finally, we define the generalized B-Fredholm operator in a von Neumann algebra as a generalization in the sense of Breuer of the classical B-Fredholm operators on Hilbert and Banach spaces. We provide sufficient conditions under which a sum of a generalized B-Fredholm operator and a finite operator in a von Neumann algebra is again a generalized B-Fredholm operator.
Paper Structure (5 sections, 31 theorems, 47 equations)

This paper contains 5 sections, 31 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. On some properties of 2 by 2 operator matrices in von Neumann algebras
  4. Spectral Fredholm theory in von Neumann algebras
  5. Semi-B-Fredholm theory in von Neumann algebras

Key Result

Lemma 2.5

BJMA Let $a \in \mathcal{A}$ and $p,q \in \mathcal{F} .$ Then $a$ is invertible up to pair $(p,q)$ if and only if $a^{*}$ is invertible up to pair $(q,p) .$

Theorems & Definitions (70)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Corollary 2.7
  • Corollary 2.8
  • proof
  • Definition 2.9
  • ...and 60 more