Let $A$ be a simple and finite dimensional C*-algebra. We first note that $aAb\neq 0$ for every nonzero $a,b\in A$. Let $B$ be a maximal abelian self-adjoint subalgebra of $A$. Being finite dimensional, the spectrum of $B$ is finite, say $\{x_1,...,x_n\}$. For each $i$, $1\leq i\leq n$, let $e_i$ denote the projection of $B$ corresponding to the characteristic function of the one point set $\{x_i\}$. we have then the $\{e_i\}$ are orthogonal and $\sum_{i=1}^n e_i = 1$, and $B$ is isomorphic to $\Bbb C e_1 \oplus ... \oplus\Bbb C e_n$. It follows then that $e_iAe_i, 1\leq i\leq n$, commutes with every $e_j$, $1\leq j\leq n$. Since $B$ is maximal abelian, $e_iAe_i$ is contained in $B$ because the $\{e_j\}$ generate $B$, which means that $e_iAe_i = \Bbb C e_i$.
I could not see $e_iAe_i$ is abelian and it's containd in $B$. Please give me a hint. Thanks in advance.
Hint: Since $B$ is isomorphic to $\Bbb C e_1 \oplus \cdots \oplus\Bbb C e_n$ and $B$ is commutative all projections $e_j$ has to be of rank $1$. Hence $e_j A e_j$ is onedimensional.