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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 $T$?

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

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