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 direct sum of irreducible representations. Suppose we wa
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 elements v, w V , we introduce a formal symbol ev,w .
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-class if it has the form
u
(u, wi )vi
where |vi |2 < and |w
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 Hilbert space, and let B (V ) denote the Hilbert space of bou
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 following:
Proposition 1. Let A be a C -algebra. Then an elem
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 complex numbers. By an algebra we will mean an associativ
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 satisfying
|x|2 = |x x|.
Notation 1. Let A be a -algebra.
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 Neumann algebras (in which the morphisms are ultraweakly c
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 ) be an ultraweakly closed -ideal. We saw in the last le
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 projections of A: that is, the collection of elements e
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 goal in this lecture is to describe those Boolean algebr
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 Neumann algebra. Let (X, ) be a measure space: that is, X
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 1. Let A be a von Neumann algebra and let : A C be a l
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 exists a Banach space E and a Banach space isomorphism
A E
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 algebras. If is completely
additive, then it is ultraw