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Vaughan Jones

Von Neumann algebras, Vanderbilt 2015

Course on von Neumann algebras, Vanderbilt 2015

Von Neumann algebras, lite.

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 3.4.1,3.4.3,3.4.6,4.3.3,4.4.3Homework 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