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Almost unimodular groups

Aldo Garcia Guinto, Brent Nelson

TL;DR

This work characterizes almost unimodular locally compact groups via the Plancherel weight: ker Δ_G open if and only if φ_G is almost periodic, enabling a Murray–von Neumann dimension theory for L(G)-modules. It develops a dimension scaling formula for finite covolume subgroups, generalizing the Atiyah–Schmid formula to second countable almost unimodular groups. Irreducible and factorial square integrable representations of G are shown to be induced from ker Δ_G, with diagonalizable formal degree operators and almost periodic formal degrees. The paper also analyzes the group von Neumann algebra structure, showing the basic construction, factoriality criteria, and a Galois-type correspondence for intermediate algebras, revealing a rich interaction between modular theory, representation theory, and operator algebraic structures in almost unimodular groups.

Abstract

We show that a locally compact group has open unimodular part if and only if the Plancherel weight on its group von Neumann algebra is almost periodic. We call such groups almost unimodular. The almost periodicity of the Plancherel weight allows one to define a Murray-von Neumann dimension for certain Hilbert space modules over the group von Neumann algebra, and we show that for finite covolume subgroups this dimension scales according to the covolume. Using this we obtain a generalization of the Atiyah-Schmid formula in the setting of second countable almost unimodular groups with finite covolume subgroups. Additionally, for the class of almost unimodular groups we present many examples, establish a number of permanence properties, and show that the formal degrees of irreducible and factorial square integrable representations are well behaved.

Almost unimodular groups

TL;DR

This work characterizes almost unimodular locally compact groups via the Plancherel weight: ker Δ_G open if and only if φ_G is almost periodic, enabling a Murray–von Neumann dimension theory for L(G)-modules. It develops a dimension scaling formula for finite covolume subgroups, generalizing the Atiyah–Schmid formula to second countable almost unimodular groups. Irreducible and factorial square integrable representations of G are shown to be induced from ker Δ_G, with diagonalizable formal degree operators and almost periodic formal degrees. The paper also analyzes the group von Neumann algebra structure, showing the basic construction, factoriality criteria, and a Galois-type correspondence for intermediate algebras, revealing a rich interaction between modular theory, representation theory, and operator algebraic structures in almost unimodular groups.

Abstract

We show that a locally compact group has open unimodular part if and only if the Plancherel weight on its group von Neumann algebra is almost periodic. We call such groups almost unimodular. The almost periodicity of the Plancherel weight allows one to define a Murray-von Neumann dimension for certain Hilbert space modules over the group von Neumann algebra, and we show that for finite covolume subgroups this dimension scales according to the covolume. Using this we obtain a generalization of the Atiyah-Schmid formula in the setting of second countable almost unimodular groups with finite covolume subgroups. Additionally, for the class of almost unimodular groups we present many examples, establish a number of permanence properties, and show that the formal degrees of irreducible and factorial square integrable representations are well behaved.

Paper Structure

This paper contains 10 sections, 28 theorems, 159 equations.

Table of Contents

  1. Preliminaries
  2. Almost Unimodular Groups
  3. Permanence Properties
  4. Square Integrable Representations
  5. Irreducible square integrable representations
  6. Factorial square integrable representations
  7. Group von Neumann Algebra Properties
  8. The basic construction and factoriality
  9. Intermediate von Neumann algebras
  10. Finite covolume subgroups and Murray--von Neumann dimension

Key Result

Theorem A

For a locally compact group $G$, the following are equivalent: In this case, the point spectrum of $\Delta_G$ is $\Delta_G(G)$, and one has $L(\ker{\Delta_G})\cong L(G)^{\varphi_G}$ where the restriction $\varphi_G|_{L(\ker{\Delta_G})}$ is the Plancherel weight corresponding to the restriction of the left Haar measure $\mu_G$ to $\ker{\Delta_G}$.

Theorems & Definitions (60)

  • Theorem A: Theorem \ref{['thm:when_is_plancherel_weight_almost_periodic']}
  • Theorem B: Theorem \ref{['thm:sq_int_factor_reps']}
  • Theorem C: Theorem \ref{['thm:covolume_dimension_formula']}
  • Theorem D: Theorem \ref{['thm:formula']}
  • Lemma 1.1
  • proof
  • Lemma 1.2
  • proof
  • Remark 1.3
  • Theorem 2.1: Theorem \ref{['introthm:A']}
  • ...and 50 more