Junior Geometry and QFT Seminar

In his representation theory course, Dr. David Ben-Zvi mentioned a few examples of the relationship between algebra and geometry. One example in passing was the duality between commutative von Neumann algebras and measure spaces. To get a commutative von Neumann algebra from a measure space, take the space of L∞ functions on it. To get a measure space from a commutative von Neumann algebra we can use the Gelfand spectrum or more generally, the Stone spectrum. The theorem we will focus on is the following: Let A be a commutative von Neumann algebra on a separable Hilbert space. Then there exists a second-countable compact Hausdorff space X and a positive measure μ on X such that A is *-isomorphic to L∞(X, μ). A related result is that any commutative von Neumann algebra on a separable Hilbert space is *-isomorphic to exactly one of: ℓ∞({1,2,…,n}) for some n ≥ 1, ℓ∞(ℕ), L∞([0,1]), L∞([0,1]∪{1,2,…,n}) for some n ≥ 1, or L∞([0,1]∪ℕ).