Abelian von Neumann algebra


Abelian von Neumann algebra

In functional analysis, an Abelian von Neumann algebra is a von Neumann algebra of operators on a Hilbert space in which all elements commute.

The prototypical example of an abelian von Neumann algebra is the algebra L^infty(X,mu) for μ a σ-finite measure on "X" realized as an algebra of operators on the Hilbert space L^2(X,mu) as follows: Each fin L^infty(X,mu) is identified with the multiplication operator

: psi mapsto f psi.

Of particular importance are the abelian von Neumann algebras on separable Hilbert spaces, particularly since they are completely classifiable by simple invariants.

Though there is a theory for von Neumann algebras on non-separable Hilbert spaces (and indeed much of the general theory still holds in that case) the theory is considerably simpler for algebras on separable spaces and most applications to other areas of mathematics or physics only use separable Hilbert spaces. Note that if the measure spaces ("X", μ) is a standard measure space (that is "X" − "N" is a standard Borel space for some null set "N" and μ is a σ-finite measure) then "L"2μ("X") is separable.

Classification

The relationship between commutative von Neumann algebras and measure spaces is analogous to that between commutative C*-algebras and locally compact Hausdorff spaces. Every commutative von Neumann algebra on a separable Hilbert space is isomorphic to "L"("X") for some standard measure space ("X", μ) and conversely, for every standard measure space "X", "L"("X") is a von Neumann algebra. This isomorphism as stated is an algebraic isomorphism. In fact we can state this more precisely as follows:

Theorem. Any abelian von Neumann algebra on a separable Hilbert space is *-isomorphic to exactly one of the following
*ell^infty({1,2, ldots, n}), quad n geq 1
* ell^infty(mathbf{N})
* L^infty( [0,1] )
* L^infty( [0,1] cup {1,2, ldots, n}), quad n geq 1
* L^infty( [0,1] cup mathbf{N}).

The isomorphism can be chosen to preserve the weak operator topology.

In the above list, the interval [0,1] has Lebesgue measure and the sets {1, 2, ..., "n"} and N have counting measure. The unions are disjoint unions. This classification is essentially a variant of Maharam's classification theorem for separable measure algebras. The version of Maharam's classification theorem that is most useful involves a point realization of the equivalence, and is somewhat of a folk theorem.

: Let μ an ν be non-atomic probability measures on standard Borel spaces "X" and "Y" respectively. Then there is a μ null subset "N" of "X", a ν null subset "M" of "Y" and a Borel isomorphism

:: phi: X setminus N ightarrow Y setminus M, quad

:which carries μ into ν.

Notice that in the above result, it is necessary to clip away sets of measure zero to make the result work.

In the above theorem, the isomorphism is required to preserve the weak operator topology. As it turns out (and follows easily from the definitions), for algebras "L"μ("X"), the following topologies are equivalent:

# The weak operator topology on "L"μ("X");
# The ultraweak operator topology on "L"μ("X");
# The topology of weak* convergence on "L"μ("X") considered as the dual space of "L"1μ("X").

However, for an abelian von Neumann algebra "A" the realization of "A" as an algebra of operators on a separable Hilbert space is highly non-unique. The complete classification of the operator algebra realizations of "A" is given by spectral multiplicity theory and requires the use of direct integrals.

Spatial isomorphism

Using direct integral theory, it can be shown that the Abelian von Neumann algebras of the form "L"μ("X") acting as operators on "L"2μ("X") are all maximal Abelian. This means that any they cannot extended to properly larger Abelian algebras. They are also referred to as "Maximal Abelian self-adjoint algebras" (or M.A.S.A.s). Another phrase used to describe them is Abelian von Neumann algebras of "uniform multiplicity 1"; this description makes sense only in relation to multiplicity theory described below.

Von Neumann algebras "A" on "H", "B" on "K" are "spatially isomorphic" (or "unitarily isomorphic") if and only if there is a unitary operator "U": "H" → "K" such that

: U A U^* = B.

In particular spatially isomorphic von Neumann algebras are algebraically isomorphic.

To describe the most general Abelian von Neumann algebra on a separable Hilbert space "H" up to spatial isomorphism, we need to refer the direct integral decomposition of "H". The details of this decomposition are discussed in decomposition of Abelian von Neumann algebras. In particular:

Theorem Any Abelian von Neumann algebra on a separable Hilbert space "H" is spatially isomorphic to "L"μ("X") acting on

: int_X^oplus H(x) , d mu(x)

for some measurable family of Hilbert spaces {"H""x"}"x" ∈ "X".

Note that for Abelian von Neumann algebras acting on such direct integral spaces, the equivalence of the weak operator topology, the ultraweak topology and the weak* topology still hold.

Point and spatial realization of automorphisms

Many problems in ergodic theory reduce to problems about automorphisms of Abelian von Neumann algebras. In that regard, the following results are useful:

Theorem. Suppose μ, ν are standard measures on "X", "Y" respectively. Then any involutive isomorphism

: Phi: L^infty_mu(X) ightarrow L^infty_ u(Y)

which is weak*-bicontinuous corresponds to a point transformation in the following sense: There are Borel null subsets "M" of "X" and "N" of "Y" and a Borel isomorphism

: eta: X setminus M ightarrow Y setminus N

such that
# η carries the measure μ into a measure μ' on "Y" which is equivalent to ν in the sense that μ' and ν have the same sets of measure zero;
# η realizes the transformation Φ, that is

:: Phi (f) = f circ eta^{-1}.

Note that in general we cannot expect η to carry μ into ν.

The next result concerns unitary transformations which induce a weak*-bicontinuous isomorphism between abelian von Neumann algebras.

Theorem. Suppose μ, ν are standard measures on "X", "Y" and

: H = int_X^oplus H_x d mu(x), quad K = int_Y^oplus K_y d u(y)

for measurable families of Hilbert spaces {"H""x"}"x" ∈ "X", {"K""y"}"y" ∈ "Y". If "U":"H" → "K" is a unitary such that

: U , L^infty_mu(X) , U^* = L^infty_ u(Y)

then there is an almost everywhere defined Borel point transformation η "X" → "Y" as in the previous theorem and a measurable family {"U""x"}"x" ∈ "X" of unitary operators

: U_x: H_x ightarrow K_{eta(x)}

such that

: U igg(int_X^oplus psi_x d mu(x) igg)= int_Y^oplus sqrt{ frac{d (mu circ eta^{-1})}{d u}(y)} U_{eta^{-1}(y)} igg(psi_{eta^{-1}(y)}igg) d u(y),

where the expression in square root sign is the Radon-Nikodym derivative of μ η -1 with respect to ν. The statement follows combining the theorem on point realization of automorphisms stated above with the theorem characterizing the algebra of diagonalizable operators stated in the article on direct integrals.

References

J. Dixmier, "Les algèbres d'opérateurs dans l'espace Hilbertien", Gauthier-Villars, 1969. See chapter I, section 6.


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Von Neumann algebra — In mathematics, a von Neumann algebra or W* algebra is a * algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. They were originally introduced by John von Neumann,… …   Wikipedia

  • John von Neumann — Von Neumann redirects here. For other uses, see Von Neumann (disambiguation). The native form of this personal name is Neumann János. This article uses the Western name order. John von Neumann …   Wikipedia

  • Abelian — Abelian, in mathematics, is used in many different definitions, named after Norwegian mathematician Niels Henrik Abel:In group theory:*Abelian group, a group in which the binary operation is commutative **Category of abelian groups Ab has abelian …   Wikipedia

  • Approximately finite dimensional C*-algebra — In C* algebras, an approximately finite dimensional, or AF, C* algebra is one that is the inductive limit of a sequence of finite dimensional C* algebras. Approximate finite dimensionality was first defined and described combinatorially by… …   Wikipedia

  • C*-algebra — C* algebras (pronounced C star ) are an important area of research in functional analysis, a branch of mathematics. The prototypical example of a C* algebra is a complex algebra A of linear operators on a complex Hilbert space with two additional …   Wikipedia

  • Spectrum of a C*-algebra — The spectrum of a C* algebra or dual of a C* algebra A, denoted Â, is the set of unitary equivalence classes of irreducible * representations of A. A * representation π of A on a Hilbert space H is irreducible if, and only if, there is no closed… …   Wikipedia

  • List of abstract algebra topics — Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. The phrase abstract algebra was coined at the turn of the 20th century to distinguish this …   Wikipedia

  • Equations defining abelian varieties — In mathematics, the concept of abelian variety is the higher dimensional generalization of the elliptic curve. The equations defining abelian varieties are a topic of study because every abelian variety is a projective variety. In dimension d ge; …   Wikipedia

  • Trace (linear algebra) — In linear algebra, the trace of an n by n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i.e., where aii represents the entry on the ith row and ith column …   Wikipedia

  • Direct integral — In mathematics and functional analysis a direct integral is a generalization of the concept of direct sum. The theory is most developed for direct integrals of Hilbert spaces and direct integrals of von Neumann algebras. The concept was… …   Wikipedia


Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.