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concerning chapter 4.4. in Murphy's book:

He proceeds by stating

(1)If $A$ is a von Neumann algebra acting non-degenerately on the Hilbert space $H$ and if $\zeta$ is separating for $A'$, then $\zeta$ is cyclic for $A$

(2)If $A$ is an abelian von Neumann algebra acting non-degenerately on a separable Hilbertspace $H$, then $A$ has a separating vector.

(3)If $A$ is maximal abelian von Neumann algebra on a separable Hilbert space then $A$ has a cyclic vector

He proves (3) by employing (2) to $M$ and since $M=M'$ we know that the separating vector found by (2) is cyclic for $M$.

My question is : Do maximal abelian von Neumann algebras always act non-degenerately on the Hilbert space (if so, why?) or, which would be a little horrifying, does (3) actually only hold for max. abelian algebras that act non-degenerately on the space. (That would be terrifying, because in my opinion then the proof, that every abelian von Neumann algebra that acts on a separable Hilbert space is isomorphic to $L^\infty(\Omega,\mu)$ ..., would only hold for algebras that act non-degenerately, too.

Thanks!

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If your abelian algebra $A$ is degenerate, that means that $I_A\ne I_H$. Then $A+\mathbb C\,(I_H-I_A)$ is an abelian algebra that contains $A$. So $A$ is not a masa.

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