von Neumann Algebra

Ideas

Relation with $C^\ast$-algebra

A $C^\ast$-algebra $A$ is a von Neumann algebra iff it is dual to some Banach space, or equivalently $A = A’’$ where $A’$ is the commutant of $A$.

Decomposition of representation spaces

Given a locally compact group $G$, consider a unitary representation $\alpha: G\to U(\mathcal{H})$. One can form the group von Neumann algebra $V\alpha(G)$. The von Neumann algebra $V\alpha(G)$ while generally loses a great deal of information of the group $G$, retains the information about the structure of all $G$-equivariant direct sum decompositions of $\mathcal{H}$.

Superselection rules

By changing terminology from pure mathematics to physics, one can see that the information of the superselection sectors of a quantum system is encoded in its von Neumann algebra.

Measure theory

Commutative von Neumann algebras can be seen as Isbell duals of measurable spaces:

  1. Any commutative von Neumann algebra $A\subset B(\mathcal{H})$ is isomorphic to $L^\infty(X)$, the space of essentially bounded functions on a measurable space $X$.
  2. For a measurable space $X$, there is an embedding $L^\infty(X)\to B(L^2(X))$, whose image is a von Neumann algebra in $B(L^2(X))$.

Therefore a noncommutatie von Neumann algebra can be thought of as the Isbell dual of a generalized measurable space.

Hilbert lattices

Each von Neumann algebra $M$ defines a quantum logic via its subset of projections $P(M)$.

Structure

On a separable Hilbert space, a von Neumann algebra is always a direct sum or a direct integral of factors.