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Approximating the group algebra of the lamplighter by infinite matrix products

Pere Ara, Joan Claramunt

Abstract

In this paper, we introduce a new technique in the study of the $*$-regular closure of some specific group algebras $KG$ inside $\mathcal{U}(G)$, the $*$-algebra of unbounded operators affiliated to the group von Neumann algebra $\mathcal{N}(G)$. The main tool we use for this study is a general approximation result for a class of crossed product algebras of the form $C_K(X) \rtimes_T \mathbb{Z}$, where $X$ is a totally disconnected compact metrizable space, $T$ is a homeomorphism of $X$, and $C_K(X)$ stands for the algebra of locally constant functions on $X$ with values on an arbitrary field $K$. The connection between this class of algebras and a suitable class of group algebras is provided by Fourier transform. Utilizing this machinery, we study an explicit approximation for the lamplighter group algebra. This is used in another paper by the authors to obtain a whole family of $\ell^2$-Betti numbers arising from the lamplighter group, most of them transcendental.

Approximating the group algebra of the lamplighter by infinite matrix products

Abstract

In this paper, we introduce a new technique in the study of the -regular closure of some specific group algebras inside , the -algebra of unbounded operators affiliated to the group von Neumann algebra . The main tool we use for this study is a general approximation result for a class of crossed product algebras of the form , where is a totally disconnected compact metrizable space, is a homeomorphism of , and stands for the algebra of locally constant functions on with values on an arbitrary field . The connection between this class of algebras and a suitable class of group algebras is provided by Fourier transform. Utilizing this machinery, we study an explicit approximation for the lamplighter group algebra. This is used in another paper by the authors to obtain a whole family of -Betti numbers arising from the lamplighter group, most of them transcendental.

Paper Structure

This paper contains 14 sections, 46 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Background and preliminaries
  3. Von Neumann regular rings, -regular rings and rank functions
  4. Division closure and rational closure
  5. -Betti numbers for group algebras and the Atiyah Conjecture
  6. A dynamical approximation for crossed product algebras
  7. The -regular closure and its ideal structure
  8. Difference between and assuming existence of a periodic point
  9. The -regular closure
  10. Group algebras arising as -crossed product algebras
  11. The lamplighter group algebra
  12. The approximating algebras for the lamplighter group algebra
  13. The algebra of special terms for the lamplighter group algebra
  14. Analysis of the algebra ${\pazocal A}_0$

Key Result

Theorem 1.1

Let ${\pazocal A} = C_K(X)\rtimes_T {\mathbb Z}$, $T$ and $\mu$ as stated above. Then the nested sequence of approximating $*$-subalgebras ${\pazocal A}_n$ of ${\pazocal A}$ satisfies the following properties:

Theorems & Definitions (98)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1
  • Proposition 2.2: Corollary 6.2 of Jaik
  • Lemma 2.3
  • Definition 2.4
  • Example 3.1: Odometer
  • Theorem 3.2
  • Proposition 4.1
  • proof
  • ...and 88 more