Von Neumann algebra conditional expectations with applications to generalized representing measures for noncommutative function algebras

Expectations in Von Neumann Algebras

Representation of faithful normal expectations in von Neumann algebras

Free Probability, Free Entropy and Applications to von Neumann Algebras

1. Non-commutative probability spaces In general, a non-commutative probability space is a pair (A, τ), where A is a unital algebra (over the field of complex numbers C) and τ a linear functional with τ(I) = 1, where I is the identity of A. Elements of A are called random variables. Since positivity is a key concept in (classical) probability theory, this can be captured by assuming that A is a...

von Neumann Algebras

For every selfadjoint operator T in the Hilbert space H, f(T) makes sense not only in the obvious case where / is a polynomial but also if / is just measurable, and if fn(x)-+f(x) for all x£R (with (/,) bounded) then fn(T)-+f(T) weakly, i.e. <fn(T)£9i)+<f(T)£9ti)V£9fiCH. Moreover the set {f(T)9 f measurable} is the set of all operators S in H invariant under all unitary transformations of H whi...

Von Neumann Algebras

The purpose of these notes is to provide a rapid introduction to von Neumann algebras which gets to the examples and active topics with a minimum of technical baggage. In this sense it is opposite in spirit from the treatises of Dixmier [], Takesaki[], Pedersen[], Kadison-Ringrose[], Stratila-Zsido[]. The philosophy is to lavish attention on a few key results and examples, and we prefer to make...