New York Journal of Mathematics
Volume 19 (2013) 343-366


Jan Cameron, Junsheng Fang, and Kunal Mukherjee

Mixing subalgebras of finite von Neumann algebras

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Published: June 14, 2013
Keywords: Finite von Neumann algebras, II1 factors, mixing subalgebras, normalizers
Subject: 46L

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 subalgebra A ⊂ B 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.


The second author gratefully acknowledges partial support by the Fundamental Research Funds for the Central Universities of China and NSFC (11071027).
The third author was supported in part by NSF grant DMS-0600814 during graduate studies at Texas A&M University.

Author information

Jan Cameron:
Department of Mathematics, Vassar College, Poughkeepsie, NY 12604, USA

Junsheng Fang:
School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China

Kunal Mukherjee:
Department of Mathematics, IIT Madras, Chennai 600036, India