# Math 209 Course Notes

Notes through 4/26/2017.

**Change Log:**

*Section 6.1.1: Dixmier's Property*added*Section 6.2: Characterizing the Commutant*added

Notes through 4/26/2017.

**Change Log:**

*Section 6.1.1: Dixmier's Property*added*Section 6.2: Characterizing the Commutant*added

**Group measure space von Neumann algebras**presented by Srivatsav Kunnawalkam Elayavalli on Monday, February 27th. Based on section 1.5 of these notes.**Spectral theory of normal unbounded operators and applications**presented by Simon Becker on Monday, Spril 3rd. Based on Chapters 5, 7, and 8 of this book.**Morita Equivalence**presented by Tim Drake on Friday, April 14th.**Topological structure of the spectrum of a von Neumann algebra**presented by Kai-chieh Chen on Friday, April 21st. Based on Chapter III.1 of this book.**Elliptic operators, discrete groups and von Neumann algebras**presented by Yingdi Qin on Monday, April 24th. Based on this paper of Atiyah.**Ultraproducts of von Neumann algebras**presented by Clark Lyons on Wednesday, April 26th.

- the functional calclulus for unbounded operators (Simon)
- $L(\Gamma)=R(\Gamma)'$ for a discrete group $\Gamma$
- Equivalent characterizations of amenability
- group-measure space construction (Srivatsav)
- topological structure of the spectrum of a von Neumann algebra (Kai-Chieh)
- Ultrapowers of von Neumann algebras
- Index of a subfactor
- Popa's deformation/rigidity: Haagerup property vs property (T)
- Construction of the interpolated free group factors
- Morita equivalence (Tim)

**Theory of Operator Algebras I**by Masamichi Takesaki.**A Course in Operator Theory**by John B. Conway. Chapters 2 and 7 are especially relevant to our course. Chapter 1 offers some review of the prerequisite material.**Von Neumann Algebras**by Vaughan F.R. Jones.**Operator Algebras: Theory of C*-Algebras and von Neumann Algebras**by Bruce Blackadar. Chapter III covers many of the topics we will discuss in this course. Chapters I and II give a comprehensive overview of the prerequisite material.**C*-algebras by Example**by Kenneth R. Davidson. A standard reference for the prerequisite material.**An Introduction to Operator Algebras**by Laurent W. Marcoux. Very clearly written notes about the prerequisite material. The last chapter also covers some of the current course material.**Notes on von Neumann algebras**by Jesse Peterson. Great reference for von Neumann algebras. In particular, we will be following Sections 2.7 and 2.8.**An Introduction to $\mathrm{II}_1$ Factors**by Cyril Houdayer. Another great reference for von Neumann algebras, specifically $\mathrm{II}_1$ factors.**Math 206 Notes**by Srivatsav Kunnawalkam Elayavalli. Notes from Math 206 taught by Marc Rieffel in Fall 2016. Generously denoted by Srivatsav Kunnawalkam Elayavalli.