Von Neumann group algebras and type analysis

Abstract

Any locally compact group G admits a left (resp. right) translation invariant measure, called a left (resp. right) Haar measure. These measures give access to a sensible theory of integration, as well as the Hilbert space L2(G). The unitary operators defined by (λ(g)ξ)(k) = ξ(g−1k) for all g ∈ G and ξ ∈ L2(G), generate a von Neumann algebra, called a group von Neumann algebra, denoted VNl(G). This study investigates group von Neumann algebras, with special attention paid to type analysis. We start by investigating locally compact groups, their semidirect products and modular functions, and the Haar measure. Special attention is paid to the existence and uniqueness of the Haar measure. Secondly, we briefly introduce von Neumann algebra theory and the associated direct integral theory. We also investigate the crossed products of von Neumann algebras, especially an interesting connection (unique to a group von Neumann algebra context) they have with semidirect products. We also develop what is known as the Plancherel weight and the canonical modular automorphism group associated to a group von Neumann algebra. To this end, we investigate some Tomita-Takesaki theory. Next, we assume G is separable. In this context, we investigate structure theorems derived by Colin E. Sutherland ([Sut78]), that provides both necessary and sufficient conditions for a group von Neumann algebra to have a central summand of pure type IIIλ (λ ∈ [0, 1]).