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 . ${\displaystyle A}$ ${\displaystyle e}$ ${\displaystyle e=e^{*}=e^{2}}$ ${\displaystyle eAe\subset A}$ ${\displaystyle e}$ ${\displaystyle e}$ ${\displaystyle e}$ ${\displaystyle n}$ ${\displaystyle e_{1},e_{2}}$ ${\displaystyle e_{1}e_{2}=0}$ ${\displaystyle v\in A}$ ${\displaystyle e_{1}=v^{*}v,e_{2}=vv^{*}}$ ${\displaystyle n}$

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 . ${\displaystyle A=L(H)}$ ${\displaystyle H}$ ${\displaystyle n}$ ${\displaystyle (\xi _{i})_{i\in I}}$ ${\displaystyle e_{i}}$ ${\displaystyle \mathbb {C} \xi _{i}}$ ${\displaystyle e_{i}}$ ${\displaystyle \textstyle \sum _{i=I}e_{i}=1}$

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. ${\displaystyle n=1,2,3,\ldots ,\infty }$

Every Von Neumann algebra of type I breaks down into a direct sum ${\displaystyle A}$

${\displaystyle A=Ap_{1}\oplus Ap_{2}\oplus Ap_{3}\oplus \ldots \oplus Ap_{\infty }}$,

in which

• each is a projection from the center of (possibly 0) ${\displaystyle p_{n}}$ ${\displaystyle A}$
• they are orthogonal in pairs ${\displaystyle p_{n}}$
• ${\displaystyle 1=p_{\infty }+p_{1}+p_{2}+p_{3}+\ldots }$ in terms of the strong operator topology.
• ${\displaystyle Ap_{n}=p_{n}Ap_{n}}$is a Von Neumann algebra of type I _n on the Hilbert space , if . ${\displaystyle p_{n}(H)}$ ${\displaystyle p_{n}\not =0}$

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 . ${\displaystyle A}$ ${\displaystyle L(H){\overline {\otimes }}\,\mathrm {cen} (A)}$ ${\displaystyle H}$ ${\displaystyle \mathrm {cen} (A)}$ ${\displaystyle A}$

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 . ${\displaystyle L(H)}$ ${\displaystyle L(\mathbb {C} ^{n})}$

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

• Type IC * algebra
• Type II Von Neumann algebra
• Type III Von Neumann Algebra

Individual evidence

1. ↑ 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.
2. ^ RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras II , 1983, ISBN 0-1239-3302-1 , Theorem 6.6.5
3. ^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras II , 1983, ISBN 0-1239-3302-1 , Theorem 6.6.6
4. ↑ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0-1254-9450-5 , Theorem 5.5.11