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. Later Pedersen observed that as soon as B itself is a C∗–subalgebra one recovers B=A in complete generality. In particular one shows:

• If A has no type II₁–summand then any norm‐dense *–subalgebra B all of whose MASA’s are norm‐closed must equal A.

2. The von Neumann case: what one would like
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One now asks the direct analogue:

“Let A be a von Neumann algebra and B⊂A an ultra‐weakly dense *–subalgebra. Assume that every MASA of B is already ultra‐weakly closed in A. Must B=A?”

– By the Pedersen‐argument one still gets B=A as soon as B is itself a von Neumann (or even just a C∗) subalgebra.
– By adapting Behncke–Cuntz one shows B=A when A has no II₁–part.

Thus the only remaining (and genuinely new) case is when A has a type II₁–summand.

3. The hyperfinite II₁–factor
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All the available methods break down exactly in the II₁–case. No proof is known that an ultra–weakly dense *–subalgebra B⊂R (R the hyperfinite II₁–factor) whose every MASA is ultra–weakly closed must in fact be all of R, and no counter‐example is known either. In particular:

 • Open If R is the hyperfinite II₁–factor and B⊂R is ultra–weakly dense with every MASA of B ultra–weakly closed in R, is B=R?

4. A still‐stronger hypothesis
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One can try to force more local completeness by requiring closure not just on MASA’s but on every amenable (or every hyperfinite) subalgebra:

“Let A be a II₁–factor and B⊂A an ultra–weakly dense *–subalgebra such that for every hyperfinite subfactor R⊂A the intersection R∩B is ultra–weakly closed in A. Must B=A?”

Again, in the absence of a II₁–summand one recovers B=A by the old arguments, but in the II₁–setting even this strengthened hypothesis has not been shown to force B=A. No counter‐example is known, but all known techniques fail to pin down a proof.

5. Summary
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– If A has no II₁–part, the answer is yes (Behncke–Cuntz + Pedersen).
– As soon as B itself happens to be a C∗– or von Neumann subalgebra one also gets B=A.
– The case of a genuine II₁–factor (in particular the hyperfinite II₁–factor), even under the extra “intersection with every hyperfinite subalgebra” hypothesis, remains wide open.

References you may consult:

– H. Behncke and J. Cuntz, “Local completeness of operator algebras,” Proc. AMS 62 (1977), 95–100.
– G. K. Pedersen, “Subalgebras of C∗–algebras,” Math. Scand. 33 (1973), 264–285, especially the remarks on local completeness.
– For background on MASA’s in II₁–factors see e.g. S. Popa, “Maximal injective subalgebras in factors associated with free groups,” Adv. Math. 50 (1983), 27–48.

In short: outside the II₁–case the answer is yes; in the II₁–case (and a fortiori for the hyperfinite II₁–factor) it is still an open problem.