Extensions
The spectral theorem

This presentation should not be seen as an end in itself, but rather as a starting point that can be extended in a variety of ways.

From a theoretical point of view, a finer analysis of the spectrum leads to the elaboration of a theory of multiplicity generalizing that of finite dimension. In addition, non-commutative algebras and their representations, commonly referred to as non-commutative geometry, are the subject of numerous and difficult theoretical studies within the framework of von Neumann algebras. An elementary application lies in identifying the image by $\tilde{\Phi }$ of the set of bounded Borelian functions on the spectrum in the case of a simple operator, as the von Neumann algebra generated...