As we can see for, self adjoint operator $x$ in discrete spectrum if eigenvalues repeat $m$ times then a copy of $M_{m}(\mathbb{C})$ sits inside commutant of the vN algebra generated by $x$. So my question is for general spectrum $x$ how is this gonna work? In addition, how the direct integral decomposition shows up the multiplicty by dividing the spectrum on disjoint sets with $\{t \in \sigma(x):m(t)=k\}$? Kindly elaborate the details. I am missing the loopholes. Thanks in Advance!!!
2026-03-27 16:53:58.1774630438
Relation of spectral multiplicity and commutant
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