[Paper Review] Mixing subalgebras of finite von Neumann algebras

Jan Cameron, Junsheng Fang+1 more|arXiv (Cornell University)|Dec 31, 2009

Advanced Operator Algebra Research39 references6 citations

TL;DR

This paper extends the concepts of mixing and weak mixing to general von Neumann subalgebras within finite von Neumann algebras, establishing a characterization of weak mixing via finite bimodules and proving that amalgamated free products and crossed products yield new examples of mixing subalgebras. The key contribution is a topological characterization: a subalgebra $ B \subseteq M $ is weakly mixing relative to $ A $ if and only if $ x \in M $ satisfying $ Ax \subset \sum x_i B $ implies $ x \in B $.

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 $B\subseteq M$ relative to $A$. We show that weak mixing of $B\subset M$ relative to $A$ is equivalent to the following property: if $x\in M$ and there exist a finite number of elements $x_1,...,x_n\in M$ 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.

Motivation & Objective

• To generalize the notions of mixing and weak mixing from abelian to general von Neumann subalgebras in finite von Neumann algebras.
• To establish a connection between mixing properties and normalizers of subalgebras in the ambient algebra.
• To provide a characterization of weak mixing subalgebras in terms of finite bimodules and operator norm convergence.
• To construct new examples of mixing subalgebras using amalgamated free products and crossed products.
• To recover and generalize classical results in ergodic theory, such as Halmos' theorem on automorphisms of compact abelian groups.

Proposed method

• Introduces a relative weak mixing condition: $ B \subseteq M $ is weakly mixing relative to $ A \subseteq B $ if $ Ax \subset \sum_{i=1}^n x_i B $ implies $ x \in B $ for all $ x \in M $.
• Uses the trace-preserving conditional expectation $ \mathbb{E}_B $ and defines mixing via the limit $ \lim_{n \to \infty} \| \mathbb{E}_B(x u_n y) - \mathbb{E}_B(x) u_n \mathbb{E}_B(y) \|_2 = 0 $ for unitaries $ u_n \to 0 $ weakly.
• Applies ultra-power techniques to show that $ B^\omega \ominus B $ is weakly closed span of unitaries in $ B^\omega \setminus B $, when $ B $ is diffuse.
• Employs orthogonality in free product structures: $ M_1 \ast_A M_2 $ has orthogonal components like $ (M_1 \setminus A) \otimes (M_2 \setminus A) $, which vanish under $ \mathbb{E}_{M_1} $.
• Uses the shift action on free products $ \ast_{g \in G} B_g \rtimes G $, with unitaries $ v_g $ implementing automorphisms, to show $ \mathbb{E}_B $-orthogonality of $ x_h v_g b_n v_g^* x_h^* $.
• Applies Lemma 6.4 to show $ \| \mathbb{E}_A(u_n) \|_2 \to 0 $ for $ u_n \to 0 $ weakly in $ M_1 $, enabling norm decay in conditional expectations.

Experimental results

Research questions

• RQ1 How can the notions of mixing and weak mixing be generalized from abelian to non-abelian von Neumann subalgebras in finite von Neumann algebras?
• RQ2 What is the precise relationship between weak mixing and the normalizer structure of a subalgebra in its ambient algebra?
• RQ3 Can weak mixing be characterized algebraically via finite bimodules and finite linear combinations?
• RQ4 Do amalgamated free products and crossed products naturally yield mixing subalgebras?
• RQ5 Can classical ergodic results, such as Halmos' theorem on automorphisms of compact abelian groups, be recovered or generalized in the operator algebraic setting?

Key findings

• A subalgebra $ B \subseteq M $ is weakly mixing relative to $ A \subseteq B $ if and only if $ Ax \subset \sum_{i=1}^n x_i B $ implies $ x \in B $ for all $ x \in M $, providing a purely algebraic characterization.
• If $ B \subseteq M $ is mixing, then for any bounded sequence $ \{b_n\} \subset B $ with $ b_n \to 0 $ weakly, $ \| \mathbb{E}_B(x b_n y) - \mathbb{E}_B(x) b_n \mathbb{E}_B(y) \|_2 \to 0 $ for all $ x,y \in M $.
• The subalgebra $ M_1 \subseteq M_1 \ast_A M_2 $ is mixing when $ A $ is atomic and $ M_1, M_2 $ are diffuse, as shown via orthogonality in free product components.
• The copy $ B \cong B_e $ inside the crossed product $ M = (*_{g \in G} B_g) \rtimes G $ is mixing in $ M $, due to orthogonality of $ \mathbb{E}_B(x_h v_g b_n v_g^* x_h^*) $ and decay of trace terms.
• The result generalizes Halmos' theorem: mixing subalgebras arising from group actions recover and extend classical ergodicity results in the operator algebra framework.
• Singular subalgebras need not be weakly mixing in the non-abelian case, showing that singularity and weak mixing are not equivalent beyond the abelian setting.