 

Jan Cameron, Junsheng Fang, and Kunal Mukherjee
Mixing subalgebras of finite von Neumann algebras view print


Published: 
June 14, 2013 
Keywords: 
Finite von Neumann algebras, II_{1} factors, mixing subalgebras, normalizers 
Subject: 
46L 


Abstract
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 x_{1},...,x_{n}∈ M such that Ax⊂ ∑_{i=1}^{n}x_{i}B,
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.


Acknowledgements
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 DMS0600814 during graduate studies at Texas A&M University.


Author information
Jan Cameron:
Department of Mathematics, Vassar College, Poughkeepsie, NY 12604, USA
jacameron@vassar.edu
Junsheng Fang:
School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China
junshengfang@gmail.com
Kunal Mukherjee:
Department of Mathematics, IIT Madras, Chennai 600036, India
kunal@iitm.ac.in

