Course on von Neumann algebras, Vanderbilt 2015 Von Neumann algebras, lite.

### Von Neumann algebras, lite. Current version of notes. Cstar notes for lectures beginning Wed sept 23. Cstar chaper 1. Cstar chaper 2. Cstar chaper 3. Cstar chaper 4. Cstar chaper 5. Cstar chaper 6. Cstar chaper 7. Cstar chaper 8. Cstar chaper 9.

Homework number 1. Due Monday 7 September
• Exercises 2.1.1, 2.1.7,2.1.11,2.1.13,2.2.2,2.2.4,2.3.1,2.4.1
• Homework number 2. Due Wednesday 23 September

• Exercises 3.4.1,3.4.3,3.4.6,4.3.3,4.4.3

Homework number 3. Due Wednesday 7 October

• In the proof of the Kaplansky density theorem we were able to suppose that our *-subalgebra of B(H) was a C*-algebra. Explain why.
• Show that the Powers state \phi_0 gives an irreducible representation of the 2^\infty UHF algebra.
• Find a pure state of the irrational rotation C^*-algebra.
• Exercises 5.1.2,6.1.9,4.3.1.

• Homework number 4. Due Wednesday 21 October.
• Prove or disprove: In a II_1 factor there is a self adjoint operator whose spectrum conists of zero and {1/n| n is a positive integer}.
• Prove or disprove: A II_1 factor contains no compact operators other than 0.
• Do exercises 7.3.1-7.3.10

• Homework number 5. Due Wednesday 18 November.
• 1) Show that a II_1 factor is algebraically simple. (Try hard to do it first then if you can't, consult the notes.)
• 2) Let N a II_1 factor and G be a finite group acting outerly on N. Let M be the crossed product of N by G. SHow that any *-subalgebra of M containing N is the crossed produc to N by a subgroup of G. Deduce that any subalgebra of N containing the fixed point algebra N^G is of the form N^H for a subgroup of G.
• 3)Do exercise 11.3.4 11.3.5 11.3.6
• 4)Do exercise 11.3.2