论文标题
关于矩阵空间的大致换算性
On approximate commutativity of spaces of matrices
论文作者
论文摘要
$ M_N（\ Mathbb {C}）$的交换子空间的最大维度是已知的。当达到最大维度时，这种子空间的结构也是如此。我们考虑这些结果的扩展，并提出以下自然问题：如果$ v $是$ m_n（\ mathbb {c}）$的子空间，而$ k $的整数小于$ n $，以至于每对$ a $ a $ a $ a $ a $ b $ $ v $的成员，$ ba $ ab ab $ a $ ba $，然后是$ k $，然后是$ k $，然后是dimension？如果达到最大值，我们可以确定$ v $的结构吗？我们回答第一个问题。我们还提出了关于第二个问题的猜想，尤其意味着这种子空间$ v $必须是代数，就像在已知的$ k = 0 $的情况下一样。如果已经假定为代数，我们将证明$ V $的拟议结构。
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.
