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,...,x_n∈ M such that Ax⊂ ∑_i=1ⁿx_iB, 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.