Proof of Theorem 2.7, p. 14 — $i\sim j \iff I_i\mathcal A I_j\neq\{0\}$ is an equivalence relation

Let $\mathcal A \subseteq \mathbb C^{n\times n}$ be a matrix ∗-algebra and let $\mathcal B$ be an inclusionwise maximal commutative ∗-subalgebra of $\mathcal A$ in diagonal form $\mathcal B = \mathbb C I_1 + \dots + \mathbb C I_k$ for a partition $[n] = S_0 \cup S_1 \cup \dots \cup S_k$ with $S_1, \dots, S_k$ nonempty. Define the relation $\sim$ on $\{1, \dots, k\}$ by

i.e. there is $X \in \mathcal A$ with $I_i X I_j \neq 0$. Then $\sim$ is an equivalence relation.

Its classes group the diagonal blocks into the simple components of $\mathcal A$: summing $I_j$ over a class gives the minimal idempotents $E_1, \dots, E_d$ of the center.

Formalization Note Indices are 0-based (Fin k for $\{1,\dots,k\}$). The setting is the same as for observations (i)–(iii).