Let $H$ be a complex [Hilbert space](/page/Hilbert%20Space), and let $A \subseteq \mathcal{L}(H)$ be a complex subalgebra that is closed in the operator norm and satisfies $T^* \in A$ for every $T \in A$. Then $A$, equipped with the inherited algebra operations, the inherited operator norm, and the inherited involution $T \mapsto T^*$, is a possibly nonunital $C^*$-algebra.