I just finished reading a proof of the spectral theorem in Conway where he defines the multiplicity function.
As far as I understood it this functions is well defined. And the multiplicity of an arbitrary point in the spectrum is just the value of the function $m_{N}$.
My friend however, claim that this is not a well defined concept due to the fact that you can change functions on zero sets w.o changing the function.
I was really convinced that this concept is working just as in finite dimensions but one just need more sophisticated tools i.e measures and borel functions instead of some finite sets of eigenvalues.
Q1 Have I misunderstood this or is he right? I.e whats wrong with letting the multipcility of a point be the borel set it belongs to in the definition of $m_{N}$ below.
He referred to some multiplication operator with absolutly contiuous spectral measure, and said that one could not define any multiplicity function for it.
The defintion of the function is very long and complicated but Ill try to define it fast, we start as follows,
Let $e_{1}$ be a seperareting vector for the Von Neumann algebra $W^{*}(N)$. Take the restriction of $N$ to $cl(W^{*}(N)e_{1})$. This is *-cyclic and this unitarly equivallent to a shift w.r.t the measure $\mu_{1}(\Delta)=\mid E(\Delta)e_1\mid^2$. Iterating we get a sequence of abs. cts measures such that \begin{equation} N=\oplus N_{\mu_{i}} \end{equation} As these measure are abs. cts. there are decreasing sequence of Borels sets such that they are mutually absolutely continuous. Hence \begin{equation} N=\oplus N_{\mu_{\mid \Delta{i}}} \end{equation} Now define $\mu_{\mid \Delta{i}}$ as the sum of pairwise singular measures $v_{n}=\mu\mid\Delta_{n} \setminus \Delta_{n+1} $
Doing this we get a sum \begin{equation} N=\oplus N_{v_{n}}=\oplus(\oplus N_{\mu\mid\Delta_{n} \setminus \Delta_{n+1}}) \end{equation} The multiplictity function is now defined as $m_{N}=\infty\chi_{\Delta_{\infty}}+1\chi_{\Delta_{1}}+2\chi_{\Delta_{2}}+...$