Analysis of the Plancherel weight and factoriality of the group von Neumann algebras of non-unimodular almost unimodular groups

2025/05/27 by Yuki Miyamoto, Miyamoto, Yuki · 1 citation
Mathematics · #Advanced Banach Space Theory #Advanced Operator Algebra Research #FOS: Mathematics #Holomorphic and Operator Theory #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.2505.21253

openalex publication_date 2025/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be a locally compact group, L(G) be its group von Neumann algebra equipped with the Plancherel weight φ_G. In this paper, we consider the following two questions. (1) When is the restriction of φ_G to the subalgebra generated by a closed subgroup H semifinite? If so, is it equal (up to a constant) to φ_H? (2) When is L(G) a factor? We give a complete answer to (1), and when G is second countable, G₁ := kerΔ_G is open in G (called almost unimodular) and admits a sufficiently large non-unimodular part, we provide an answer to (2). When L(G) is a factor, we also provide the formula of the S-invariant of L(G).

Citations

• Almost unimodular groups
• Crossed-products by locally compact groups: Intermediate subfactors
• Extensions of locally compact quantum groups and the bicrossed product construction
• A Radon-Nikodym theorem for von Neumann algebras

Cited by

• On ergodicity of linear actions on ℝⁿ and factoriality of group von Neumann algebras

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