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Metric dimension and product entropy of group $C^{\ast}$-algebras

Arnab Chattopadhyay, Soumalya Joardar

Abstract

We consider reduced group $C^{\ast}$-algebras of finitely generated discrete groups metrized by seminorms obtained from word length functions. We study the metric dimensions of such $C^{\ast}$-algebras as defined by David Kerr. We also study the entropy of the automorphisms of group $C^{\ast}$-algebras induced by the automorphisms of the underlying groups. Both the metric dimension and entropy are related to the growth of the groups. We exhibit a class of examples of virtually abelian finitely generated discrete groups $Γ$ and automorphisms $ψ$ of $Γ$ such that $ψ$ has non-zero finite product entropy in the sense of David Kerr.

Metric dimension and product entropy of group $C^{\ast}$-algebras

Abstract

We consider reduced group -algebras of finitely generated discrete groups metrized by seminorms obtained from word length functions. We study the metric dimensions of such -algebras as defined by David Kerr. We also study the entropy of the automorphisms of group -algebras induced by the automorphisms of the underlying groups. Both the metric dimension and entropy are related to the growth of the groups. We exhibit a class of examples of virtually abelian finitely generated discrete groups and automorphisms of such that has non-zero finite product entropy in the sense of David Kerr.
Paper Structure (6 sections, 29 theorems, 89 equations)

This paper contains 6 sections, 29 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Compact quantum metric spaces
  4. Metric dimension and the product entropy
  5. Main results
  6. Higher order Leibnitz property

Key Result

Lemma 2.3

For all $k, n \in \mathbb N$ we have where $\dbinom {k} {r_{1}\ r_{2}\ \cdots\ r_{n}} = \dfrac{k!}{r_{1}!\ r_{2}!\ \cdots\ r_{n}!}$ and $f_{i} \in C_{c}(\Gamma)$ for $i = 1, 2, \cdots, n.$

Theorems & Definitions (56)

  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • Corollary 2.4
  • proof
  • Theorem 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • Theorem 2.8
  • ...and 46 more