Type I Von Neumann algebra
Type I Von Neumann algebras are special algebras considered in the mathematical theory of Von Neumann algebras. It is the first of three types of type classification of Von Neumann algebras . Type I Von Neumann algebras are also called discrete .
Definitions
A projection in a Von Neumann algebra is a self-adjoint idempotent element , that is, it holds . Such a projection is called Abelian if the algebra is commutative. A Von Neumann algebra is called type I (read: type one) if it has an Abelian projection , so that the unit element is the smallest projection from the center of the algebra, the product of which is equal to . It is called more precisely of the type I n if the one element is the sum of pairwise orthogonal, equivalent Abelian projections. Two projections are called orthogonal if , and they are called equivalent if there is an element with . In the case of infinite, the sum is to be understood in the sense of the strong operator topology .
Examples
- Abelian Von Neumann algebras are of type I, because in this case the unit element itself is an Abelian projection.
- The algebra of the continuous linear operators over a Hilbert space is of the type I n , where is the dimension of the Hilbert space. If namely is an orthogonal basis and is the projection onto the one-dimensional subspace , then they are Abelian, mutually equivalent and it is .
properties
We only consider Von Neumann algebras on a separable Hilbert space . Then one only has to consider the cases for type I n algebras ; otherwise one would have to differentiate between thicknesses for the infinite case.
Every Von Neumann algebra of type I breaks down into a direct sum
- ,
in which
- each is a projection from the center of (possibly 0)
- they are orthogonal in pairs
- in terms of the strong operator topology.
- is a Von Neumann algebra of type I n on the Hilbert space , if .
Every Von Neumann algebra of type I n is isomorphic to the tensor product , where is an n-dimensional Hilbert space and is the center of .
Since the centers are Abelian Von Neumann algebras and these are known, the structure of the Type I Von Neumann algebras is thus revealed; they are direct sums of tensor products of algebras with Abelian Von Neumann algebras. It follows easily from this that every finite-dimensional Von Neumann algebra is of type I and isomorphic to a finite direct sum of matrix algebras .
A Von Neumann algebra is of type I if and only if it is isomorphic to a Von Neumann algebra with an Abelian commutant .
See also
Individual evidence
- ↑ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0-1254-9450-5 , Chapter 5.5: Von Neumann Algebras of Type I.
- ^ RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras II , 1983, ISBN 0-1239-3302-1 , Theorem 6.6.5
- ^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras II , 1983, ISBN 0-1239-3302-1 , Theorem 6.6.6
- ↑ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0-1254-9450-5 , Theorem 5.5.11