In
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...
, an abelian von Neumann algebra is a
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. It is a special type of C*-algebra.
Von Neumann algebra ...
of operators on a
Hilbert space in which all elements
commute
Commute, commutation or commutative may refer to:
* Commuting, the process of travelling between a place of residence and a place of work
Mathematics
* Commutative property, a property of a mathematical operation whose result is insensitive to th ...
.
The prototypical example of an abelian von Neumann algebra is the algebra ''L''
∞(''X'', μ) for μ a σ-finite measure on ''X'' realized as an algebra of operators on the Hilbert space ''L''
2(''X'', μ) as follows: Each ''f'' ∈ ''L''
∞(''X'', μ) is identified with the multiplication operator
:
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''
2(''X'', μ) is separable.
Classification
The relationship between
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
von Neumann algebras and
measure spaces is analogous to that between
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
C*-algebras and
locally compact Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the m ...
s. 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 of operators on a separable Hilbert space is *-isomorphic to exactly one of the following
*
*
*
*
*
The isomorphism can be chosen to preserve the
weak operator topology
In functional analysis, the weak operator topology, often abbreviated WOT, is the weakest topology on the set of bounded operators on a Hilbert space H, such that the functional sending an operator T to the complex number \langle Tx, y\rangle is c ...
.
In the above list, the interval
,1has Lebesgue measure and the sets and N have counting measure. The unions are disjoint unions. This classification is essentially a variant of
Maharam's classification theorem In mathematics, Maharam's theorem is a deep result about the decomposability of measure spaces, which plays an important role in the theory of Banach spaces. In brief, it states that every complete measure space is decomposable into "non-atomic ...
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.
Although every standard measure space is isomorphic to one of the above and the list is exhaustive in this sense, there is a more canonical choice for the measure space in the case of abelian von Neumann algebras ''A'': The set of all projectors is a
-complete Boolean algebra, that is a pointfree
-algebra. In the special case
one recovers the abstract
-algebra
. This pointfree approach can be turned into a duality theorem analogue to Gelfand-duality between the category of abelian von Neumann algebras and the category of abstract
-algebras.
: Let μ and 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
::
: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''
∞(''X'', μ), the following topologies agree on norm bounded sets:
# The weak operator topology on ''L''
∞(''X'', μ);
# The ultraweak operator topology on ''L''
∞(''X'', μ);
# The topology of weak* convergence on ''L''
∞(''X'', μ) considered as the dual space of ''L''
1(''X'', μ).
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 In abstract algebra, multiplicity theory concerns the multiplicity of a module ''M'' at an ideal ''I'' (often a maximal ideal)
:\mathbf_I(M).
The notion of the multiplicity of a module is a generalization of the degree of a projective variety. By S ...
and requires the use of
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 introduced i ...
s.
Spatial isomorphism
Using direct integral theory, it can be shown that the abelian von Neumann algebras of the form ''L''
∞(''X'', μ) acting as operators on ''L''
2(''X'', μ) are all maximal abelian. This means that they cannot be 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
:
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''
∞(''X'', μ) acting on
:
for some measurable family of Hilbert spaces
''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 on norm bounded sets 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
:
which is weak*-
bicontinuous
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorph ...
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
:
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
::
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.
[, Chapter IV, Theorem 8.23, p. 277] Suppose μ, ν are standard measures on ''X'', ''Y'' and
:
for measurable families of Hilbert spaces
''x'' ∈ ''X'',
''y'' ∈ ''Y''. If ''U'' : ''H'' → ''K'' is a unitary such that
:
then there is an almost everywhere defined Borel point transformation η : ''X'' → ''Y'' as in the previous theorem and a measurable family
''x'' ∈ ''X'' of unitary operators
:
such that
:
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 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 introduced i ...
s.
Notes
References
* J. Dixmier, ''Les algèbres d'opérateurs dans l'espace Hilbertien'', Gauthier-Villars, 1969. See chapter I, section 6.
*
Masamichi Takesaki
Masamichi Takesaki (竹崎 正道; born July 18, 1933 in Sendai) is a Japanese mathematician working in the theory of operator algebras.
Takesaki studied at Tohoku University, earning a bachelor's degree in 1956, a master's degree in 1958 and a ...
''Theory of Operator Algebras I,II,III", encyclopedia of mathematical sciences, Springer-Verlag, 2001–2003 (the first volume was published 1979 in 1. Edition) {{ISBN, 3-540-42248-X
Von Neumann algebras