Throughout is a field, is a finite dimensional central simple -algebra and is a simple -subalgebra of We will use the notation for centralizers given in this post. The goal is to prove that This is called the double centralizer theorem for an obvious reason. We proved another double centralizer theorem in here. In there for some and did not have to be simple. So that double centralizer theorem has nothing to do with this one. We first show that the centralizer of a simple subalgebra of a finite dimensional central simple -algebra is simple.

**Lemma**. The -subalgebra is simple.

*Proof*. Since is central simple and and hence is simple, the algebra is also simple by the first part of the corollary in this post. Clearly is finite dimensional over because is so. So, as we mentioned before in Remark 1, has a unique simple -module and any -module is isomorphic to the direct sum of a finite number of copies of Thus, since has a structure of an -module, we must have

for some integer Let

Since is a simple -module, is a division ring by Schur’s lemma. On the other hand, as we proved in this post,

Now, (1), (2) and the remark in this post gives us

To be continued in part (2).