Given a (bounded) normal operator $N$ (satisfying $N^*N=NN^*$) in a von Neumann algebra $A$, does $A$ always contain a unitary operator $U$ having the same spectral (projection-valued) measure as $N$?
If $N$ has a purely discrete spectrum, then the answer is clearly yes, because then $N$ has the form $N = \sum_n z_n P_n$ where the $P_n$ are mutually orthogonal projection operators and the $z_n$ are distinct complex coefficients. Replacing the original coefficients with any list of distinct unit-magnitude coefficients gives a unitary operator $U$ with the desired property.
Is the answer always yes? If so, is there a simple proof (or a reference)? If not, then what is a counterexample?
I browsed other questions on math.stackexchange with similar keywords but didn't notice any that answered this particular question.