2009/12/31 by Jan Cameron, Cameron, Jan, Junsheng Fang +3 · 1 citation
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Operator Algebras (math.OA) #math.OA
paper · pdf · doi:10.48550/arxiv.1001.0169
24 pages, minor typos corrected
openalex publication_date 2009/12/31 · arxiv created 2011/07/27 · arxiv updated 2011/07/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Jolissaint and Stalder introduced definitions of mixing and weak mixing for von Neumann subalgebras of finite von Neumann algebras. In this note, we study various algebraic and analytical properties of subalgebras with these mixing properties. We prove some basic results about mixing inclusions of von Neumann algebras and establish a connection between mixing properties and normalizers of von Neumann subalgebras. The special case of mixing subalgebras arising from inclusions of countable discrete groups finds applications to ergodic theory, in particular, a new generalization of a classical theorem of Halmos on the automorphisms of a compact abelian group. For a finite von Neumann algebra M and von Neumann subalgebras A, B of M, we introduce a notion of weak mixing of B⊆ M relative to A. We show that weak mixing of B⊂ M relative to A is equivalent to the following property: if x∈ M and there exist a finite number of elements x1,...,xn∈ M such that Ax⊂ ∑i=1nxiB, then x∈ B. We conclude the paper with an assortment of further examples of mixing subalgebras arising from the amalgamated free product and crossed product constructions.