The question asks whether an ultra-weakly dense *-subalgebra ( B \subset A ) of a von Neumann algebra ( A ), where every maximal abelian self-adjoint subalgebra (MASA) of ( B ) is ultra-weakly closed in ( A ), must equal ( A ).

Key Reasoning:

1. MASAs in ( B ) and ( A ): For any MASA ( D ) of ( A ), the intersection ( D \cap B ) is an abelian *-subalgebra of ( B ). ...

Short answer: this is known to be true in all types except possibly in the II1 case, and the II1 case (even for the hyperfinite II1 factor) is, as far as currently known, still open. Your strengthened “hyperfinite intersections are ultraweakly closed” hypothesis reduces the problem to exactly the same unresolved step in type II1.

Here is what one can say precisely.

• From your hypothesis (every ...

This is a very interesting question about the structure of von Neumann algebras and their subalgebras. Let me analyze the different aspects of your questions.

Background and Context

The original Behncke-Cuntz result deals with norm-dense -subalgebras of C-algebras where all maximal abelian subalgebras (MASAs) are norm-closed. Your questions translate this to the von Neumann algebra setting ...

The problem is still open.
What is known at the moment can be summarised as follows.

1. Reduction to the type II₁ case.
   Let A be a von-Neumann algebra, let B⊂A be a *-subalgebra which is ultra-weakly dense in A and assume that

   (⋆) every maximal abelian *-subalgebra (MASA) of B is ultra-weakly closed in A.
   

   Because the hypotheses are inherited by central summands, one may sep...

Here is what is known and what remains open.

1. The C∗‐case (Behncke–Cuntz and Pedersen)
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– In 1977 Behncke–Cuntz showed that if A is a C∗–algebra and B⊂A is a *–subalgebra which is dense in the norm–topology and such that every maximal abelian *–subalgebra (MASA) of B is already norm–closed in A, then in various “type I” and “purely infinite” situations one can force B=A. La...

This is an excellent and deep question that probes the structure of von Neumann algebras, particularly of type II₁. The answer distinguishes between the general case and more specific, yet still unresolved, cases.

The key to this problem lies in a famous answer by Sorin Popa on the MathOverflow post you linked. He provided a definitive counterexample for the first and most general question.

...

Yes – the hypothesis already forces the sub-algebra to be ultra-weakly closed, hence it coincides with the ambient von Neumann algebra. No counter–example can exist, neither in the hyperfinite (II_{1})-factor nor in any other von Neumann algebra.

Proposition.
Let (M) be a von Neumann algebra and (B\subset M) a σ-weakly (ultra-weakly) dense *-sub-algebra.
Assume that every maximal abeli...

<think> We are considering several questions about dense subalgebras of von Neumann algebras with certain maximal abelian subalgebras (MASAs) being closed. The original context is from a paper by Behncke and Cuntz, and we are now in the von Neumann algebra setting.

The main question is: Let ( A ) be a von Neumann algebra and ( B \subset A ) be an ultra-weakly dense *-subalgebra such that ever...

Subalgebras of von Neumann algebras

This question addresses an interesting structural problem about von Neumann algebras that builds on earlier work by Cuntz, Behncke, and Pedersen.

Background Context

The question stems from a 1977 paper by Behncke and Cuntz concerning when a dense -subalgebra B of a C-algebra A must equal A, given that every maximal abelian subalgebra (MASA) of B is nor...