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On Segal entropy

Andrzej Łuczak

Abstract

The paper is devoted to the investigation of Segal's entropy in semifinite von Neumann algebras. The following questions are dealt with: semicontinuity, the 'ideal-like' structure of the linear span of the set of operators with finite entropy, and topological properties of the set of operators with finite as well as infinite entropy. In our analysis, full generality is aimed at, in particular, the operators for which the entropy is considered are not assumed to belong to the underlying von Neumann algebra, instead, they are arbitrary positive elements in the space $L^1$ over the algebra.

On Segal entropy

Abstract

The paper is devoted to the investigation of Segal's entropy in semifinite von Neumann algebras. The following questions are dealt with: semicontinuity, the 'ideal-like' structure of the linear span of the set of operators with finite entropy, and topological properties of the set of operators with finite as well as infinite entropy. In our analysis, full generality is aimed at, in particular, the operators for which the entropy is considered are not assumed to belong to the underlying von Neumann algebra, instead, they are arbitrary positive elements in the space over the algebra.
Paper Structure (4 sections, 20 theorems, 190 equations)

This paper contains 4 sections, 20 theorems, 190 equations.

Table of Contents

  1. Preliminaries and notation
  2. Semicontinuity of Segal entropy
  3. Ideal-like structure of operators with finite Segal entropy
  4. Topological properties of operators having Segal entropy

Key Result

Lemma 1

Let $h\in\mathfrak{E}$. Then

Theorems & Definitions (38)

  • Lemma 1
  • proof
  • Proposition 2
  • proof
  • Theorem 3
  • proof
  • Corollary 4
  • Theorem 5
  • proof
  • Corollary 6
  • ...and 28 more