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The algebraic entropies of the Leavitt path algebra and the graph algebras agree

Wolfgang Bock, Cristóbal Gil Canto, Dolores Martín Barquero, Cándido Martín González, Iván Ruiz Campos, Alfilgen Sebandal

Abstract

In this note we prove that the algebras $L_K(E)$ and $KE$ have the same entropy. Entropy is always referred to the standard filtrations in the corresponding kind of algebra. The main argument leans on (1) the holomorphic functional calculus; (2) the relation of entropy with suitable norm of the adjacency matrix; and (3) the Cohn path algebras which yield suitable bounds for the algebraic entropies.

The algebraic entropies of the Leavitt path algebra and the graph algebras agree

Abstract

In this note we prove that the algebras and have the same entropy. Entropy is always referred to the standard filtrations in the corresponding kind of algebra. The main argument leans on (1) the holomorphic functional calculus; (2) the relation of entropy with suitable norm of the adjacency matrix; and (3) the Cohn path algebras which yield suitable bounds for the algebraic entropies.
Paper Structure (3 sections, 4 theorems, 17 equations)

This paper contains 3 sections, 4 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Agreement of the algebraic entropies of $L_K(E)$ and $KE$

Key Result

Proposition 3.1

Let $E$ be a finite directed graph. Then

Theorems & Definitions (11)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • Theorem 3.5
  • ...and 1 more