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On New Approach to semi-Fredholm theory in unital C*-algebras

Stefan Ivkovic

Abstract

Axiomatic Fredholm theory in unital C*-algebras was established by Keckic and Lazovic in [15]. Following the purely algebraic approach by Keckic and Lazovic, in [14] we extended further this theory to axiomatic semi-Fredholm and semi-Weyl theory in unital C*-algebras. However, recently, in [11] we developed another approach to axiomatic Fredholm theory in unital C*-algebras which is based on the theory of Hilbert modules and which is equivalent to the algebraic approach by Keckic and Lazovic. In this paper, we extend further this new Hilbert-module approach from Fredholm theory to semi-Fredholm and semi-Weyl theory in unital C*-algebras. Hence, we provide a new proof of the results in [14].

On New Approach to semi-Fredholm theory in unital C*-algebras

Abstract

Axiomatic Fredholm theory in unital C*-algebras was established by Keckic and Lazovic in [15]. Following the purely algebraic approach by Keckic and Lazovic, in [14] we extended further this theory to axiomatic semi-Fredholm and semi-Weyl theory in unital C*-algebras. However, recently, in [11] we developed another approach to axiomatic Fredholm theory in unital C*-algebras which is based on the theory of Hilbert modules and which is equivalent to the algebraic approach by Keckic and Lazovic. In this paper, we extend further this new Hilbert-module approach from Fredholm theory to semi-Fredholm and semi-Weyl theory in unital C*-algebras. Hence, we provide a new proof of the results in [14].
Paper Structure (3 sections, 17 theorems, 51 equations)

This paper contains 3 sections, 17 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results

Key Result

Lemma 2.5

BJMA Let $a \in \mathcal{A}$ and $p,q,p^{\prime},q^{\prime}$ be projections in $\mathcal{A} .$ Suppose that $p,q,p^{\prime} \in \mathcal{F} .$ If $a$ is invertible up to pair $(p,q)$ and also invertible up to pair $(p^{\prime},q^{\prime}) ,$ then $q^{\prime} \in \mathcal{F} .$ Similarly, if instead

Theorems & Definitions (37)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • Definition 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Definition 2.9
  • Remark 2.10
  • ...and 27 more