Let $(M,\tau)$ be a finite von Neumann algebra with faithful normal tracial state $\tau$. Suppose that $A \subset M$ is a finite-dimensional abelian unital subalgebra--say $A = W^*(p_1,\dots, p_n)$ where $p_1,\cdots, p_n \in M$ are mutually orthogonal projections with sum $1_M$--and $N \subset M$ is an arbitrary unital ($1_N = 1_M$) von Neumann subalgebra. Consider the relative commutants $A'\cap M$ and $N'\cap M$ and their respective $||\cdot||_2$-closures $\overline{A'\cap M}, \overline{N'\cap M} \subset L^2(M,\tau)$.
Under what conditions is the algebraic sum $\overline{A'\cap M} + \overline{N'\cap M}$ a closed linear subspace of $L^2(M,\tau)$?
Note: When $M$ is finite dimensional this sum is always closed. Also, recall that $A'\cap M = \sum_{i=1}^n p_iMp_i$.
Some references to consider are the following.
Deutsch, Frank. The angle between subspaces of a Hilbert space. (English summary) Approximation theory, wavelets and applications (Maratea, 1994), 107–130, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 454, Kluwer Acad. Publ., Dordrecht, 1995. MathSciNet: MR1340886
Luxemburg, W. A. J. A note on the sum of two closed linear subspaces. Nederl. Akad. Wetensch. Indag. Math. 47 (1985), no. 2, 235–242. MathSciNet: MR0799084
In [1], it is shown that two closed linear subspaces of a Hilbert space have a closed algebraic sum if and only if the cosine of the Friedrichs angle between them is strictly less than 1 (cf. Theorem 13). So an alternative approach to this question would be to compute the Friedrichs angle between these two relative commutant subspaces.
Reference [2] gives a number of functional analytic results on how to determine when the algebraic sum of two closed linear subspaces of a Banach space is closed.
Special case: A good start would be to resolve this question in the case where $N = uAu^*$ for a unitary $u \in M$.
