This course is an introduction to the theory of operator algebras, specifically the two branches of the subject that can be described as the noncommutative counterparts to topology (C*-algebras) and measure theory (von Neumann algebras). Topics will include spectral theory, the Gelfand-Naimark theorem, the functional calculus, the GNS construction, projections, positive elements, unitaries, completely positive maps, the bicommutant theorem, strong and weak operator topologies, group C*-algebras, group von Neumann algebras, hyperfiniteness, and AF algebras. The course forms the first part of a specialization module in Operator Algebras and Noncommutative Geometry. A second course, Operator Algebras II, will be offered in the summer 2023 semester.

Main text:

• G. J. Murphy, C*-Algebras and Operator Theory

Additional resources:

• G. K. Petersen, Analysis Now
• R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, I and II
• M. Takesaki, Theory of Operator Algebras I
• K. Davidson, C*-Algebras by Example

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