Introduction to von Neumann algebras, addendum to Lecture 1 (solution of Exercise B: the norm of a selfadjoint operator)

One of the challenges I had in preparing this course, was to find a quick route to the modern theory that is different from the standard modern route, in order to save time and be able to reach significant results and examples in the limited time of a one semester course. A main issue was to avoid the (beautiful, beautiful, beautiful) Gelfand theory of commutative Banach and C*-algebras, and base everything on the spectral theorem for a single selfadjoint operator (which is significantly simpler than the one for normal operators). In the previous lecture, I stated Exercise B, which gave some important properties of the spectrum of a selfadjoint operator. Since my whole treatment is based on this, I felt that for completeness I should give the details.

Spoiler alert: If you are a student in the course and you plan to submit the solution of this exercise, then you shouldn’t read the rest of this post.

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