Consider a finite-dimensional Hilbert space $H$ and a von Neumann algebra $M$ on $H$, i.e., an algebra of operators on $H$ which is closed under Hermitian conjugation and contains the identity. Consider also a subspace $S\subseteq H$, and let $\iota:S\to H$ be the inclusion map. I am interested in the set $\iota^\dagger M\iota$, which is a set of operators on $S$. In terms of the decomposition $H=S\oplus S^\perp$, the matrices of $\iota^\dagger M\iota$ are the upper-left blocks of the matrices of $M$.
My question is: when (i.e., for which subspaces $S$) is $\iota^\dagger M\iota$ a von Neumann algebra?
The set $\iota^\dagger M\iota$ is always a vector space, contains the identity and is closed under Hermitian conjugation, but in general it is not a von Neumann algebra because it is not closed under multiplication.
I know that $\iota^\dagger M\iota$ is a von Neumann algebra if, for example, the orthogonal projection onto $S$ lies in $M$ or its commutant. But these are only sufficient conditions; I would like to find a necessary and sufficient condition.