  The maximal dimension of commutative subspaces of $M_n(\mathbb{C})$ is known.
So is the structure of such a subspace when the maximal dimension is achieved.
We consider extensions of these results and ask the following natural
questions: If $V$ is a subspace of $M_n(\mathbb{C})$ and $k$ is an integer less
than $n$, such that for every pair $A$ and $B$ of members of $V$, the rank of
the commutator $AB - BA$ is at most $k$, then how large can the dimension of
$V$ be? If this maximum is achieved, can we determine the structure of $V$? We
answer the first question. We also propose a conjecture on the second question
which implies, in particular, that such a subspace $V$ has to be an algebra,
just as in the known case of $k = 0$. We prove the proposed structure of $V$ if
it is already assumed to be an algebra.
