Exercise 5, § 1

(a) Let $E = L^2([0,1], \mu)$ where $\mu$ is the Lebesgue measure. The image A of the algebra $L^{\infty}([0,1], \mu)$ under the mapping which associates with $f$ the endomorphism of multiplication by $f$ is a maximal commutative subalgebra of $\mathscr{L}(E)$ which is not of the form $\mathscr{D}_B(E)$ for an orthonormal basis B of E. (Verify that the algebra A does not contain a minimal projector.)

(b) Let E be an infinite-dimensional Hilbert space. There exists a maximal commutative subalgebra of $\mathscr{L}(E)$ which is not of the form $\mathscr{D}_B(E)$ for B an orthonormal basis of E.