III.97 Von Neumann Algebras

A unitary representation of a GROUP [I.3 §2.1] G is a HOMOMORPHISM [I.3 §4.1] that associates with each element g of G a UNITARY MAP [III.50 §3.1] U_g defined on some HILBERT SPACE [III.37] H. A von Neumann algebra is a special kind of C*-ALGEBRA [III.12], intimately connected with the theory of unitary representations. There are several equivalent ways of defining von Neumann algebras. One is as follows. It can be checked that, given any unitary representation, its commutant, defined to be the set of all OPERATORS [III.50] in B(H) that commute with every single unitary map U_g in the representation, forms a C*-algebra. Von Neumann algebras are algebras that arise in this way. They can also be defined abstractly as follows: a C*- algebra A is a von Neumann algebra if there is a BANACH SPACE [III.62] X such that the DUAL [III.19 §4] of X is A (when A is itself considered as a Banach space).

The basic building blocks of von Neumann algebras are special kinds of von Neumann algebras called factors. The classification of factors is a major topic of research, which includes some of the most celebrated theorems of the second half of the twentieth century. See OPERATOR ALGEBRAS [IV.15 §2] for more details.