Let us work over $\mathbb{C}$ to make life easier.

I've came across to the following definition. Let $F$ be a totally real number field of degree $g$, with ring of integers $\mathcal{O}_F$. An abelian variety $A$ of dimension $g$ has real multiplication by $\mathcal{O}_F$ if there is a ring embedding $\mathcal{O}_F \to {\rm End}(A)$.

**Question**: Is it always the case that $F$ maps into the center of ${\rm End}^0(A)$? What if $A$ is simple?

In general, can the structure of ${\rm End}^0(A)$ be completely understood?