Mixing subalgebras of finite von Neumann algebras

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 $B\subseteq M$ relative to $A$. We show that weak mixing of $B\subset M$relativeto$A$isequivalenttothefollowingproperty:if$x\in M$ and there exist a finite number of elements $x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n\in M$ such that $Ax\subset \sum_{i=1}^nx_iB$,then$x\in B$. We conclude the paper with an assortment of further examples of mixing subalgebras arising from the amalgamated free product and crossed product constructions.