Von Neumann Algebra

A Von Neumann algebra or W * algebra is a mathematical structure in functional analysis . Historically, the theory of Von Neumann algebras begins with the fundamental works by Francis J. Murray and John von Neumann On rings of operators , published from 1936 to 1943 . The name Von Neumann Algebra for such algebras goes back to a suggestion by Jean Dieudonné .

definition

A Von Neumann algebra (named after John von Neumann ) or (now obsolete) a ring of operators is a * -sub algebra with one of the algebra of the bounded linear operators of a Hilbert space , which is one (and thus all) of the three following equivalents Conditions met: ${\displaystyle A}$ ${\displaystyle L\left(H\right)}$ ${\displaystyle H}$

• ${\displaystyle A=A''}$.
• ${\displaystyle A}$is completed in the strong operator topology .
• ${\displaystyle A}$is closed in the weak operator topology .

Here is the commutant of and accordingly the commutant of . ${\displaystyle A':={\bigl \{}x\in L(H)\,|\,\forall a\in A:\,xa=ax{\bigr \}}}$ ${\displaystyle A}$ ${\displaystyle A''}$ ${\displaystyle A'}$

The equivalence of the three above statements is called the von Neumann double commutant theorem or bicommutant theorem . This statement can be tightened as follows:

• If a * subalgebra is one, then is the closure of both the weak and the strong operator topology. ${\displaystyle A\subset L(H)}$ ${\displaystyle A''}$ ${\displaystyle A}$

This formulation, which establishes an equivalence between the purely algebraic commutant formation and the purely topological density relationship or termination formation, is also referred to as the bicommutant theorem. The bicommutant set thus proves to be a tightness set. Together with Kaplansky's other theorem of density , it represents the starting point of the theory of Von Neumann algebras.

A Von Neumann algebra can also be defined abstractly without an underlying Hilbert space according to a theorem of Shōichirō Sakai :

• A Von Neumann algebra is a C * algebra that is the topological dual space of a Banach space . ${\displaystyle A}$ ${\displaystyle A_{\star }}$

Factors

The Von Neumann algebra is called a factor if it fulfills one of the following two equivalent conditions: ${\displaystyle A}$

• ${\displaystyle A\cap A'=\mathbb {C} \cdot 1_{H}}$.
• ${\displaystyle A\cup A'}$generated . ${\displaystyle L\left(H\right)}$

Since the set of operators off that commute with all operators off is the center of . Factors are therefore the Von Neumann algebras with the smallest possible center. Von Neumann algebras can be represented as a direct integral (a generalization of the direct sum) of factors, that is, Von Neumann algebras are composed of factors in this sense. ${\displaystyle A\cap A'}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle A\cap A'}$ ${\displaystyle A}$

${\displaystyle L\left(H\right)}$and are examples of factors. Mit is also a factor; apparently applies and . ${\displaystyle \mathbb {C} \cdot 1_{H}}$ ${\displaystyle A}$ ${\displaystyle A'}$ ${\displaystyle L\left(H\right)'=\mathbb {C} \cdot 1_{H}}$ ${\displaystyle (\mathbb {C} \cdot 1_{H})'=L\left(H\right)}$

There are 3 types of factors , called Type I , Type II and Type III .

Commutative Von Neumann algebras

→ Main article : Abelian Von Neumann algebra

Be a -finite measure space . Then L ^{2 is} a Hilbert space, and every essentially restricted function defines an operator via multiplication . The mapping is an * - isomorphism from to a commutative Von Neumann algebra , one can even show that the algebra agrees with its commutant. No real upper algebra can therefore be commutative, so it is a maximal commutative Von Neumann algebra. ${\displaystyle (X,{\mathfrak {X}},\mu )}$ ${\displaystyle \sigma }$ ${\displaystyle H=}$ ${\displaystyle (X,{\mathfrak {X}},\mu )}$ ${\displaystyle f\in L^{\infty }(X,{\mathfrak {X}},\mu )}$ ${\displaystyle M_{f}\in L(H),M_{f}(g):=f\cdot g}$ ${\displaystyle f\to M_{f}}$ ${\displaystyle f\in L^{\infty }(X,{\mathfrak {X}},\mu )}$ ${\displaystyle {\mathcal {M}}\subset L(H)}$ ${\displaystyle {\mathcal {M}}'={\mathcal {M}}}$ ${\displaystyle {\mathcal {M}}}$ ${\displaystyle {\mathcal {M}}}$

If one looks specifically at the measurement space (unit interval with the Lebesgue measure ), one can show that the bicommutant of coincides with . The transition from the topological construct to the measure theoretical construct corresponds to the transition from C * algebras to Von Neumann algebras. While in C * algebras one speaks of non-commutative topology because of the Gelfand-Neumark theorem , the consideration given here gives reason to regard a Von Neumann algebra as a non-commutative measure space, which is why one also speaks of non-commutative Measure theory . ${\displaystyle ([0,1],{\mathcal {B}},\lambda )}$ ${\displaystyle \{M_{f};\,f\in C([0,1])\}}$ ${\displaystyle {\mathcal {M}}\cong L^{\infty }([0,1])}$ ${\displaystyle C([0,1])}$ ${\displaystyle L^{\infty }([0,1])}$

properties

Every Von Neumann algebra is a C * algebra and therefore also a Banach algebra .

As can be seen from the restricted Borel functional calculus , Von Neumann algebras contain a great number of orthogonal projections ; every operator is in the norm topology Limes of linear combinations of orthogonal projections. This is an essential difference to the C * -algebras, which, as the example C ([0,1]) shows, need not contain any other projections besides 0 and 1. One can construct a lattice from the set of projections ; the structure of this association is used for the type classification of the Von Neumann algebras.

See also

• Type I Von Neumann algebra
• Type II Von Neumann algebra
• Type III Von Neumann algebra
• Tomita Takesaki Theory

literature

• Jacques Dixmier : Von Neumann algebras , North-Holland Publishing, Amsterdam et al. 1981 (North-Holland Mathematical Library, Volume 27), ISBN 0-444-86308-7 .
• RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras , Volumes I and II, Academic Press 1983, ISBN 0-123-93301-3 and 1986, ISBN 0-123-93302-1, respectively
• Shôichirô Sakai : C * -Algebras and W * -Algebras , Springer, Berlin et al. 1971 (Results of Mathematics and their Border Areas, Volume 60) ISBN 3-540-05347-6 (Reprint. Ibid 1998, ISBN 3-540-63633 -1 ).
• Jacob T. Schwartz : W * -Algebras. Gordon & Breach, New York NY et al. 1967.

Individual evidence

1. ^ FJ Murray, J. von Neumann: On rings of operators. Ann. of Math. (2), Volume 37, 1936, pages 116-229.
2. ^ FJ Murray, J. von Neumann: On rings of operators II. Trans. Amer. Math. Soc., Volume 41, 1937, pages 208-248
3. ^ FJ Murray, J. von Neumann: On rings of operators IV. Ann. of Math. (2), Vol. 44, 1943, pages 716-808.
4. ↑ Newsletter of the EMS, June 2009, interview with Jacques Dixmier, page 36