von Neumann algebras
The spectral theorem

The theory of von Neumann algebras is a tool for exploring non-commutative situations, an ambition that goes far beyond the scope of this article. Here, we will focus on just a few elementary aspects, with the main aim of deepening our knowledge of the Borelian calculus and, in particular, identifying its image and continuity properties.

Among the topologies carried by $\mathfrak{L}\left(H\right)$ , apart from that of the norm, two are particularly important: the strong topology generated by the family of semi-norms $\mathrm{T}\to \Vert$