In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
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 defi ...
a direct integral is a generalization of the concept of
direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mor ...
. The theory is most developed for direct integrals of
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s and direct integrals of
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 ...
s. The concept was introduced in 1949 by
John von Neumann
John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest c ...
in one of the papers in the series ''On Rings of Operators''. One of von Neumann's goals in this paper was to reduce the classification of (what are now called) von Neumann algebras on separable Hilbert spaces to the classification of so-called factors. Factors are analogous to full matrix algebras over a field, and von Neumann wanted to prove a continuous analogue of the
Artin–Wedderburn theorem classifying semi-simple rings.
Results on direct integrals can be viewed as generalizations of results about finite-dimensional
C*-algebra
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuou ...
s of matrices; in this case the results are easy to prove directly. The infinite-dimensional case is complicated by measure-theoretic technicalities.
Direct integral theory was also used by
George Mackey
George Whitelaw Mackey (February 1, 1916 – March 15, 2006) was an American mathematician known for his contributions to quantum logic, representation theory, and noncommutative geometry.
Career
Mackey earned his bachelor of arts at Rice Un ...
in his analysis of
systems of imprimitivity and his general theory of
induced representation
In group theory, the induced representation is a representation of a group, , which is constructed using a known representation of a subgroup . Given a representation of '','' the induced representation is, in a sense, the "most general" represe ...
s of locally compact separable groups.
Direct integrals of Hilbert spaces
The simplest example of a direct integral are the ''L''
2 spaces associated to a (σ-finite) countably additive measure μ on a
measurable space
In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured.
Definition
Consider a set X and a σ-algebra \mathcal A on X. Then the ...
''X''. Somewhat more generally one can consider a separable Hilbert space ''H'' and the space of square-integrable ''H''-valued functions
:
Terminological note: The terminology adopted by the literature on the subject is followed here, according to which a measurable space ''X'' is referred to as a ''Borel space'' and the elements of the distinguished σ-algebra of ''X'' as Borel sets, regardless of whether or not the underlying σ-algebra comes from a topological space (in most examples it does). A Borel space is ''standard''
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bic ...
it is isomorphic to the underlying Borel space of a
Polish space
In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named be ...
; all Polish spaces of a given cardinality are isomorphic to each other (as Borel spaces). Given a countably additive measure μ on ''X'', a measurable set is one that differs from a Borel set by a
null set
In mathematical analysis, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length.
The notion of null ...
. The measure μ on ''X'' is a ''standard'' measure if and only if there is a null set ''E'' such that its complement ''X'' − ''E'' is a
standard Borel space
In mathematics, a standard Borel space is the Borel space associated to a Polish space. Discounting Borel spaces of discrete Polish spaces, there is, up to isomorphism of measurable spaces, only one standard Borel space.
Formal definition
...
. All measures considered here are σ-finite.
Definition. Let ''X'' be a Borel space equipped with a countably additive measure μ. A ''measurable family of Hilbert spaces'' on (''X'', μ) is a family
''x''∈ ''X'', which is locally equivalent to a trivial family in the following sense: There is a countable partition
:
by measurable subsets of ''X'' such that
:
where H
''n'' is the canonical ''n''-dimensional Hilbert space, that is
:
A ''cross-section'' of {''H''
''x''}
''x''∈ ''X'' is a family {''s''
''x''}
''x'' ∈ ''X'' such that ''s''
''x'' ∈ ''H''
''x'' for all ''x'' ∈ ''X''. A cross-section is measurable if and only if its restriction to each partition element ''X''
''n'' is measurable. We will identify measurable cross-sections ''s'', ''t'' that are equal
almost everywhere
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
. Given a measurable family of Hilbert spaces, the direct integral
:
consists of equivalence classes (with respect to almost everywhere equality) of measurable square integrable cross-sections of {''H''
''x''}
''x''∈ ''X''. This is a Hilbert space under the inner product
:
Given the local nature of our definition, many definitions applicable to single Hilbert spaces apply to measurable families of Hilbert spaces as well.
Remark. This definition is apparently more restrictive than the one given by von Neumann and discussed in Dixmier's classic treatise on von Neumann algebras. In the more general definition, the Hilbert space ''fibers'' ''H''
''x'' are allowed to vary from point to point without having a local triviality requirement (local in a measure-theoretic sense). One of the main theorems of the von Neumann theory is to show that in fact the more general definition is equivalent to the simpler one given here.
Note that the direct integral of a measurable family of Hilbert spaces depends only on the measure class of the measure μ; more precisely:
Theorem. Suppose μ, ν are σ-finite countably additive measures on ''X'' that have the same sets of measure 0. Then the mapping
:
is a unitary operator
:
Example
Technically the simplest examples are when ''X'' is a countable set and μ is a discrete measure. Throughout the article, we will consider the following running example in which ''X'' = N and μ is counting measure on N. In this case any sequence {''H''
''k''} of separable Hilbert spaces can be considered as a measurable family. Moreover,
:
Decomposable operators
In our running example, any bounded linear operator ''T'' on
:
is given by an infinite matrix
:
Consider operators that are ''block diagonal'', that is all entries off the diagonal are zero. We call these operators ''decomposable''. These operators can be characterized as those that commute with diagonal matrices:
:
We now proceed to the general definition: A family of bounded operators {''T''
''x''}
''x''∈ ''X'' with ''T''
''x'' ∈ L(''H''
''x'') is said to be ''strongly measurable'' if and only if its restriction to each ''X''
''n'' is strongly measurable. This makes sense because ''H''
''x'' is constant on ''X''
''n''.
Measurable families of operators with an essentially bounded norm, that is
:
define bounded linear operators
:
acting in a pointwise fashion, that is
:
Such operators are said to be ''decomposable''.
Examples of decomposable operators are those defined by scalar-valued (i.e. C-valued) measurable functions λ on ''X''. In fact,
Theorem. The mapping
:
given by
:
is an involutive algebraic isomorphism onto its image.
For this reason we will identify ''L''
∞μ(''X'') with the image of φ.
Theorem
[, Chapter IV, Theorem 7.10, p. 259] Decomposable operators are precisely those that are in the operator commutant of the abelian algebra ''L''
∞μ(''X'').
Decomposition of Abelian von Neumann algebras
The spectral theorem has many variants. A particularly powerful version is as follows:
Theorem. For any Abelian von Neumann algebra A on a separable Hilbert space ''H'', there is a standard Borel space ''X'' and a measure μ on ''X'' such that it is unitarily equivalent as an operator algebra to ''L''
∞μ(''X'') acting on a direct integral of Hilbert spaces
:
To assert A is unitarily equivalent to ''L''
∞μ(''X'') as an operator algebra means that there is a unitary
:
such that ''U'' A ''U''* is the algebra of diagonal operators ''L''
∞μ(''X''). Note that this asserts more than just the algebraic equivalence of A with the algebra of diagonal operators.
This version however does not explicitly state how the underlying standard Borel space ''X'' is obtained. There is a uniqueness result for the above decomposition.
Theorem. If the Abelian von Neumann algebra A is unitarily equivalent to both ''L''
∞μ(''X'') and ''L''
∞ν(''Y'') acting on the direct integral spaces
:
and μ, ν are standard measures, then there is a Borel isomorphism
:
where ''E'', ''F'' are null sets such that
:
φ is a measure class isomorphism, that is φ and its inverse preserve sets of measure 0.
This previous two theorems provide the complete classification of Abelian von Neumann algebras on separable Hilbert spaces. Note that this classification actually takes into account the realization of the von Neumann algebra as an algebra of operators. If we only consider the underlying von Neumann algebra independently of its realization as a von Neumann algebra, then its structure is determined by very simple measure-theoretic invariants.
Direct integrals of von Neumann algebras
Let {''H''
''x''}
''x'' ∈ ''X'' be a measurable family of Hilbert spaces. A family of von Neumann algebras {''A''
''x''}
''x'' ∈ ''X''
with
:
is measurable
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bic ...
there is a countable set ''D'' of measurable operator families that pointwise generate {''A''
''x''}
''x'' ∈ ''X'' as a von Neumann algebra in the following sense: For almost all ''x'' ∈ ''X'',
:
where W*(''S'') denotes the von Neumann algebra generated by the set ''S''. If {''A''
''x''}
''x'' ∈ ''X'' is a measurable family of von Neumann algebras, the direct integral of von Neumann algebras
:
consists of all operators of the form
:
for ''T''
''x'' ∈ ''A''
''x''.
One of the main theorems of von Neumann and Murray in their original series of papers is a proof of the decomposition theorem: Any von Neumann algebra is a direct integral of factors. We state this precisely below.
Theorem. If {''A''
''x''}
''x'' ∈ ''X'' is a measurable family of von Neumann algebras and μ is standard, then the family of operator commutants is also measurable and
:
Central decomposition
Suppose ''A'' is a von Neumann algebra. let Z(''A'') be the
center
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentrici ...
of ''A'', that is the set of operators in ''A'' that commute with all operators ''A'', that is
:
Z(''A'') is an Abelian von Neumann algebra.
Example. The center of L(''H'') is 1-dimensional. In general, if ''A'' is a von Neumann algebra, if the center is 1-dimensional we say ''A'' is a factor.
Now suppose ''A'' is a von Neumann algebra whose center contains a sequence of minimal pairwise orthogonal non-zero projections {''E''
''i''}
''i'' ∈ N such that
:
Then ''A'' ''E''
''i'' is a von Neumann algebra on the range ''H''
''i'' of ''E''
''i''. It is easy to see ''A'' ''E''
''i'' is a factor. Thus in this special case
:
represents ''A'' as a direct sum of factors. This is a special case of the central decomposition theorem of von Neumann.
In general, we can apply the structure theorem of Abelian von Neumann algebras that represents Z(A) as an algebra of scalar diagonal operators. In any such representation, all the operators in A are decomposable operators. In fact, we can use this to prove the basic result of von Neumann that any von Neumann algebra admits a decomposition into factors.
Theorem. Suppose
:
is a direct integral decomposition of ''H'' and A is a von Neumann algebra on ''H'' so that Z(A) is represented by the algebra of scalar diagonal operators ''L''
∞μ(''X'') where ''X'' is a standard Borel space. Then
:
where for almost all ''x'' ∈ ''X'', ''A''
''x'' is a von Neumann algebra that is a ''factor''.
Measurable families of representations
If ''A'' is a separable C*-algebra, we can consider measurable families of non-degenerate *-representations of ''A''; recall that in case ''A'' has a unit, non-degeneracy is equivalent to unit-preserving. By the general correspondence that exists between strongly continuous unitary representations of a locally compact group ''G'' and non-degenerate *-representations of the groups C*-algebra C*(''G''), the theory for C*-algebras immediately provides a decomposition theory for representations of separable locally compact groups.
Theorem. Let ''A'' be a separable C*-algebra and π a non-degenerate involutive representation of ''A'' on a separable Hilbert space ''H''. Let W*(π) be the von Neumann algebra generated by the operators π(''a'') for ''a'' ∈ ''A''. Then corresponding to any central decomposition of W*(π) over a standard measure space (''X'', μ) (which as stated is unique in a measure theoretic sense), there is a measurable family of factor representations
:
of ''A'' such that
:
Moreover, there is a subset ''N'' of ''X'' with μ measure zero, such that π
''x'', π
''y'' are disjoint whenever ''x'', ''y'' ∈ ''X'' − ''N'', where representations are said to be ''disjoint'' if and only if there are no
intertwining operator
In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry grou ...
s between them.
One can show that the direct integral can be indexed on the so-called ''quasi-spectrum'' ''Q'' of ''A'', consisting of quasi-equivalence classes of factor representations of ''A''.
Thus there is a standard measure μ on ''Q'' and a measurable family of factor representations indexed on ''Q'' such that π
''x'' belongs to the class of ''x''. This decomposition is essentially unique. This result is fundamental in the theory of group representations.
References
*
J. Dixmier, ''Von Neumann algebras'',
* J. Dixmier, ''C* algebras''
*
G. W. Mackey, ''The Theory of Unitary Group Representations'', The University of Chicago Press, 1976.
*
J. von NeumannOn Rings of Operators. Reduction TheoryThe Annals of Mathematics 2nd Ser., Vol. 50, No. 2 (Apr., 1949), pp. 401–485.
*
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)
{{Spectral theory
Functional analysis
Measure theory
Von Neumann algebras