# Does the von Neumann algebra generated by a normal operator contain all commuting projections?

Let $$H$$ be a Hilbert space and $$T\in B(H)$$ a bounded normal operator. Let $$\mathscr{A}$$ be the von Neumann algebra generated by $$T$$. Is it true that $$\mathscr{A}$$ contains every orthogonal projection which commutes with $$T$$?

No. For instance, if $$T$$ is the identity operator, then $$\mathscr{A}$$ is just the span of $$T$$, but every orthogonal projection commutes with $$T$$.