Remark

The Banach algebra $\mathscr{D}_B(E)$ is a maximal commutative closed subalgebra of $\mathscr{L}(E)$. Indeed, let $u$ be an endomorphism of E which commutes with $\mathscr{D}_B(E)$. Let $i\in I$. Since the orthogonal projector $p_i$ is diagonal in the basis B, one has $p_i(u(e_i)) =u(p_i(e_i)) =u(e_i)$, which implies that $u(e_i)$ is proportional to $e_i$. Thus $u$ is diagonal in the basis B.

If E is of infinite dimension, there exist maximal commutative involutive subalgebras of $\mathscr{L}(E)$ which are not isomorphic to $\mathscr{D}_B(E)$ (exercise 5 of IV, p. 314).