p. 13 — every matrix ∗-algebra has a unique unit

Let $\mathcal A \subseteq \mathbb C^{n\times n}$ be a matrix ∗-algebra. There is a unique element $E \in \mathcal A$ such that

called the unit of $\mathcal A$.

The unit need not be the identity matrix: it is the orthogonal projection onto the part of $\mathbb C^n$ on which $\mathcal A$ acts, and for $\mathcal A = \{0\}$ it is $0$. It enters the definition of a simple matrix ∗-algebra. The paper states the claim on p. 13 and shows it on p. 14 ("(ii) shows that $I_1 + \dots + I_k$ is the unit of $\mathcal A$").

Formalization Note $\mathcal A$ is a NonUnitalStarSubalgebra; no nonzero or unital hypothesis is made.