Von Neumann Algebras . Vaughan

The purpose of these notes is to provide a rapid introduction to von Neumann algebras which gets to the examples and active topics with a minimum of technical baggage. In this sense it is opposite in spirit from the treatises of philosophy is to lavish attention on a few key results and examples, and we prefer to make simplifying assumptions rather than go for the most general case. Thus we do not hesitate to give several proofs of a single result, or repeat an argument with dierent hypotheses. The notes are built around semester-long courses given at UC Berkeley though they contain more material than could be taught in a single semester. The notes are informal and the exercises are an integral part of the exposition. These exercises are vital and mostly intended to be easy. A Hilbert Space is a complex vector space H with inner product , : HxH → C which is linear in the rst variable, satises ξ, η = η, ξ, is positive denite, i.e. ξ, ξ > 0 for ξ = 0, and is complete for the norm dened by test ||ξ|| = ξ, ξ. Exercise 2.1.1. Prove the parallelogram identity : ||ξ − η|| 2 + ||ξ + η|| 2 = 2(||ξ|| 2 + ||η|| 2) and the Cauchy-Schwartz inequality: ||ξ, η| ≤ ||ξ|| ||η||. Theorem 2.1.2. If C is a closed convex subset of H and ξ is any vector in H, there is a unique η ∈ C which minimizes the distance from ξ to C, i.e. ||ξ − η || ≤ ||ξ − η|| ∀η ∈ C. Proof. This is basically a result in plane geometry. Uniqueness is clearif two vectors η and η in C minimized the distance to ξ, then ξ, η and η lie in a (real) plane so any vector on the line segment between η and η would be strictly closer to ξ. To prove existence, let d be the distance from C to ξ and choose a sequence η n ∈ C with ||η n − ξ|| < d + 1/2 n. For each n, the vectors ξ, η n and η n+1 dene a plane. Geometrically it is clear that, if η n and η n+1 were not close, some point on the line segment between them would be closer than d to ξ.