Spectral theorem for von Neumann algebras

Question

What is the motivation of studying spectral theorem? Is it just to know when two normal operators are unitary equivalent or not? Further, what is the viewpoint of seeing spectral theorem in terms of von Neumann algebras??

Answer 1

The spectral theorem basically says that if $T$ is a normal operator on a Hilbert space, then $T$ is unitarily diagonalizable. In linear algebra it is a standard fact that if $T$ is a hermitian operator on finite-dimensional spaces, then $T$ is unitarily diagonalizable, i.e. we can write $T=\sum_{\lambda}P_{\lambda}$ where the sum runs over the eigenvalues and $P_{\lambda}$ is the orthogonal projection on the corresponding eigenspace. The general spectral theorem says more or less the same but the finite sum is replaced with some kind of integral.

Anyways, the spectral theorem allows you to plug in $T$ in nice functions $f$, i.e. one can define $f(T)$ (this is the so called Borel functional calculus). If $T$ belongs to a von Neumann algebra, then so does $f(T)$. This is very useful when studying the structure of von Neumann algebras as you can construct elements with specific properties inside it.