On approximate commutativity of spaces of matrices

TL;DR

This paper investigates the maximal dimension and structure of subspaces of matrices where the commutator's rank is bounded, extending known results on commutative subspaces to approximate commutativity.

Contribution

It extends classical results by characterizing subspaces with bounded commutator rank and proposes a conjecture on their algebraic structure.

Findings

Maximal dimension of subspaces with bounded commutator rank is determined.

Such subspaces are conjectured to be algebras, generalizing the commutative case.

Proven structure if the subspace is already an algebra.

Abstract

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.