Math 261y: von Neumann Algebras (Lecture 1)
August 31, 2011
Let G be a compact group. The representation theory of G is completely reducible: every nitedimensional representation can be written as a d
Math 261y: von Neumann Algebras (Lecture 6)
September 15, 2011
Let us begin this lecture with a short review of the theory of trace-class operators. Let V be a Hilbert
space. For every pair of nonzero
Math 261y: von Neumann Algebras (Lecture 7)
September 15, 2011
Let V be a Hilbert space. In the last lecture, we dened the subspace B tc (V ) B (V ) of trace-class
operators: an operator is trace-clas
Math 261y: von Neumann Algebras (Lecture 5)
September 9, 2011
In this lecture, we will state and prove von Neumanns double commutant theorem. First, lets establish
a bit of notation. Let V be a Hilber
Math 261y: von Neumann Algebras (Lecture 3)
September 7, 2011
In the last lecture, we introduced the notion of a positive element of a C -algebra A. Our rst goal in
this lecture is to prove the follow
Math 261y: von Neumann Algebras (Lecture 2)
September 2, 2011
Let us begin by setting up some terminological conventions which will we will use in this course. We will
always work over the eld C of co
Math 261y: von Neumann Algebras (Lecture 3)
September 7, 2011
In this lecture, we continue our study of C -algebras. Recall that C -algebra is a Banach algebra equipped
with an anti-involution x x sat
Math 261y: von Neumann Algebras (Lecture 8)
September 20, 2011
Let C denote the category of C -algebras (in which the morphisms are homomorphisms of -algebras)
and let D denote the category of von Neu
Math 261y: von Neumann Algebras (Lecture 9)
September 20, 2011
Let us begin this lecture by continuing the analysis of ideals in von Neumann algebras. Let A be a von
Neumann algebra and let I A B (V )
Math 261y: von Neumann Algebras (Lecture 14)
October 3, 2011
In this lecture, we will continue our study of abelian von Neumann algebras. Let A be any -algebra.
We let P (A) denote the collection of p
Math 261y: von Neumann Algebras (Lecture 15)
October 3, 2011
In the last lecture, we saw that an abelian von Neumann algebra A is determined by the Boolean algebra
P (A) of projections of A. Our rst g
Math 261y: von Neumann Algebras (Lecture 13)
September 30, 2011
In this lecture, we will begin the study of abelian von Neumann algebras. We rst describe the prototypical
example of an abelian von Neu
Math 261y: von Neumann Algebras (Lecture 12)
September 27, 2011
In this lecture, we will complete our algebraic characterization of von Neumann algebra morphisms by
proving the following result:
Lemma
Math 261y: von Neumann Algebras (Lecture 10)
September 25, 2011
The following result provides an intrinsic characterization of von Neumann algebras:
Theorem 1. Let A be a C -algebra. Suppose there exi
Math 261y: von Neumann Algebras (Lecture 11)
September 26, 2011
In the last lecture, we promised a proof of the following assertion:
Theorem 1. Let : A B be a -algebra homomorphism between von Neumann