Semi-Endless Von Neumann Algebra

Semi-infinite Von Neumann algebras are examined in the mathematical sub-area of functional analysis. They are Von Neumann algebras without a type III component.

definition

Every Von Neumann algebra contains a largest orthogonal projection in its center , so that a Von Neumann algebra is of type III. means semi-ending if . ${\displaystyle A}$ ${\displaystyle p_{\infty }}$ ${\displaystyle p_{\infty }A}$ ${\displaystyle A}$ ${\displaystyle p_{\infty }=0}$

Examples

• Finite Von Neumann algebras are semi-finite.
• Abelian Von Neumann algebras are semi-infinite.
• Type I and Type II Von Neumann algebras are semi-infinite.
• The Von Neumann algebra of all continuous, linear operators on a Hilbert space is semi-infinite.

properties

traces

Semi- infinite Von Neumann algebras are characterized by the fact that they have a semi-infinite, normal, faithful trace weight, that is, there is a mapping on the set of positive elements of with the following properties: ${\displaystyle A}$ ${\displaystyle \phi \colon A^{+}\rightarrow [0,\infty ]}$ ${\displaystyle A}$

• ${\displaystyle \phi (a+\lambda b)=\phi (a)+\lambda \phi (b)}$for everyone and with the usual conventions for calculating with infinity. ${\displaystyle a,b\in A^{+}}$ ${\displaystyle \lambda \in \mathbb {R} ^{+}}$
• ${\displaystyle \phi (u^{*}au)=\phi (a)}$for all and all unitary elements . ${\displaystyle a\in A^{+}}$ ${\displaystyle u\in A}$
• For each is the upper bound of with , and (semi finiteness of the track). ${\displaystyle a\in A^{+}}$ ${\displaystyle \phi (a)}$ ${\displaystyle \phi (b)}$ ${\displaystyle b\in A^{+}}$ ${\displaystyle b\leq a}$ ${\displaystyle \phi (b)<\infty }$
• For every ascending net in with supremum applies (normality of the track). ${\displaystyle \textstyle (a_{i})_{i\in I}}$ ${\displaystyle A^{+}}$ ${\displaystyle a\in A^{+}}$ ${\displaystyle \textstyle \phi (a)=\sup _{i\in I}\phi (a_{i})}$
• For each it follows from (loyalty to the trail). ${\displaystyle a\in A^{+}}$ ${\displaystyle a=0}$ ${\displaystyle \phi (a)=0}$

In the textbook Von Neumann Algebras by Jacques Dixmier given below , this is the definition of the semi-terminus Von Neumann algebras.

Inheritance properties

The commutant of a semi-terminating Von Neumann algebra is again semi-terminating. A Von Neumann algebra is semi-finite if and only if it is isomorphic to a Von Neumann algebra whose commutant is a finite Von Neumann algebra .

Tensor products of a finite number of semi-infinite Von Neumann algebras are again semi-infinite. Any direct products of semi-terminating Von Neumann algebras are semi-terminating again.

Since the algebra of all bounded, linear operators on a Hilbert space is semi-finite, there can be no inheritance of this property to subalgebras, because every Von Neumann algebra is a subalgebra of such an algebra . ${\displaystyle L(H)}$ ${\displaystyle L(H)}$

Hilbert algebras

→ Main article : Hilbert algebra

The semininite Von Neumann algebras are precisely those von Neumann algebras that are isomorphic to the left-associated Von Neumann algebra of a Hilbert algebra .

Tomita Takesaki Theory

In the Tomita-Takesaki theory one shows that a Von Neumann algebra is semi-finite if and only if its modular group is inner. More precisely, the following applies: If there is a true, normal state on a Von Neumann algebra and the associated modular group, then it is semi-finite if and only if there is a generally unbounded , positive and injective operator with ${\displaystyle \varphi }$ ${\displaystyle A}$ ${\displaystyle t\mapsto \sigma _{t}^{\varphi }}$ ${\displaystyle A}$ ${\displaystyle h}$

1. ${\displaystyle h=uhu^{*}}$ for all unitary operators ${\displaystyle u\in A^{'}}$
2. ${\displaystyle \sigma _{t}^{\varphi }(a)=h^{it}ah^{-it}}$for everyone and . ${\displaystyle t\in \mathbb {R} }$ ${\displaystyle a\in A}$

If restricted, then this operator would commutate with every unitary operator from the commutant according to the first condition , and therefore with every operator , and it would therefore be an element according to the bicommutant theorem . In this sense "belongs" to the unlimited operator to . With the unbounded Borel functional calculus it follows that the operators are unitary operators from , that is, they are inner automorphisms. ${\displaystyle h}$ ${\displaystyle A^{'}}$ ${\displaystyle A^{'}}$ ${\displaystyle A}$ ${\displaystyle h}$ ${\displaystyle A}$ ${\displaystyle h^{it}}$ ${\displaystyle A}$ ${\displaystyle \sigma _{t}^{\varphi }}$

Individual evidence

1. ^ RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras II , Academic Press (1983), ISBN 0-1239-3302-1 , end of section 6.5.2
2. ^ Jacques Dixmier: Von Neumann algebras , North-Holland Publishing, Amsterdam et al. 1981 (North-Holland Mathematical Library, Vol. 27), ISBN 0-444-86308-7 , Part I, Chap. 6, paragraph 7, definition 5 and sentence 9
3. ^ RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras II , Academic Press (1983), ISBN 0-1239-3302-1 , Corollary 9.1.4
4. ^ Jacques Dixmier: Von Neumann algebras , North-Holland Publishing, Amsterdam et al. 1981 (North-Holland Mathematical Library, Volume 27), ISBN 0-444-86308-7 , Part III, Chap. 2, para. 4, corollary 3
5. ^ Jacques Dixmier: Von Neumann algebras , North-Holland Publishing, Amsterdam et al. 1981 (North-Holland Mathematical Library, Vol. 27), ISBN 0-444-86308-7 , Part I, Chap. 6, para. 8, sentence 12
6. ^ Jacques Dixmier: Von Neumann algebras , North-Holland Publishing, Amsterdam et al. 1981 (North-Holland Mathematical Library, Vol. 27), ISBN 0-444-86308-7 , Part I, Chap. 6, paragraph 7, sentence 7
7. ^ RV Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras II , Academic Press (1983), ISBN 0-1239-3302-1 , Theorem 9.2.21