Factors
A von Neumann algebra N whose center consists only of multiples of the identity operator is called a factor. von Neumann (1949) showed that every von Neumann algebra on a separable Hilbert space is isomorphic to a direct integral of factors. This decomposition is essentially unique. Thus, the problem of classifying isomorphism classes of von Neumann algebras on separable Hilbert spaces can be reduced to that of classifying isomorphism classes of factors.
Murray & von Neumann (1936) showed that every factor has one of 3 types as described below. The type classification can be extended to von Neumann algebras that are not factors, and a von Neumann algebra is of type X if it can be decomposed as a direct integral of type X factors; for example, every commutative von Neumann algebra has type I1. Every von Neumann algebra can be written uniquely as a sum of von Neumann algebras of types I, II, and III.
There are several other ways to divide factors into classes that are sometimes used:
- A factor is called discrete (or occasionally tame) if it has type I, and continuous (or occasionally wild) if it has type II or III.
- A factor is called semifinite if it has type I or II, and purely infinite if it has type III.
- A factor is called finite if the projection 1 is finite and properly infinite otherwise. Factors of types I and II may be either finite or properly infinite, but factors of type III are always properly infinite.
Read more about this topic: Von Neumann Algebra
Famous quotes containing the word factors:
“Language makes it possible for a child to incorporate his parents’ verbal prohibitions, to make them part of himself....We don’t speak of a conscience yet in the child who is just acquiring language, but we can see very clearly how language plays an indispensable role in the formation of conscience. In fact, the moral achievement of man, the whole complex of factors that go into the organization of conscience is very largely based upon language.”
—Selma H. Fraiberg (20th century)
“I always knew I wanted to be somebody. I think that’s where it begins. People decide, “I want to be somebody. I want to make a contribution. I want to leave my mark here.” Then different factors contribute to how you will do that.”
—Faith Ringgold (b. 1934)
“The economic dependence of woman and her apparently indestructible illusion that marriage will release her from loneliness and work and worry are potent factors in immunizing her from common sense in dealing with men at work.”
—Mary Barnett Gilson (1877–?)