Von Neumann algebra

A subalgebra $ A $ of the algebra $ {\mathcal B} ( H) $ of bounded linear operators on a Hilbert space $ H $ that is self-adjoint (that is, contains together with every operator $ T $ its adjoint operator $ T ^ {*} $) and that coincides with its bicommutant (that is, it contains all operators $ T \in {\mathcal B} ( H) $ that commute with every operator commuting with all operators in $ A $). These algebras were introduced by J. von Neumann . According to a theorem of von Neumann, a self-adjoint subalgebra $ A \subset {\mathcal B} ( H) $ is a von Neumann algebra if and only if $ A $( or its unit ball) is closed in the weak, strong, ultraweak, or ultrastrong operator topology (the uniform operator topology does not suffice). A given symmetric Banach algebra $ B $( cf. also Symmetric algebra) is isometrically isomorphic to some von Neumann algebra if and only if it is a $ C ^ {*} $- algebra isometric to some dual space; the Banach space $ E $ for which $ E ^ {*} = B $ is uniquely determined up to an isometric isomorphism and can be identified with the space of ultraweakly-continuous linear forms on the von Neumann algebra isometrically isometric to $ B $; this space is denoted by $ B _ {*} $ and is called the pre-dual of $ B $. Such symmetric Banach algebras are called $ W ^ {*} $- algebras. Let $ A $ be a von Neumann algebra on a Hilbert space $ H $, $ A ^ \prime $ its commutator, $ Z = A \cap A ^ \prime $ its centre, $ P $ a projection belonging to $ A $, and $ P ^ \prime $ a projection belonging to $ A ^ \prime $. The subspace $ P ^ \prime H $ is invariant under $ A $, and the family of operators from $ A $ restricted to $ P ^ \prime H $ forms a von Neumann algebra in $ P ^ \prime H $, which is denoted by $ A _ {P ^ \prime } $ and is called the induced algebra, while the mapping $ T \rightarrow T \mid _ {P ^ \prime H } $ is called the induced mapping of $ A $ onto $ A _ {P ^ \prime } $; the family of bounded operators of the form $ P T P $, $ T \in A $, on the subspace $ P H $ forms a von Neumann algebra $ A _ {P} $ in $ P H $, which is called reduced. If $ P = P ^ \prime \subset Z $, then the reduced and the induced von Neumann algebras are the same. An isometric isomorphism of a von Neumann algebra is said to be algebraic; a von Neumann algebra on a Hilbert space $ H $ is said to be spatially isomorphic to a von Neumann algebra $ B $ on a space $ K $ if there exists a unitary operator $ U $ mapping $ H $ onto $ K $ and such that $ B = U A U ^ {-1} $. The intersection of any family of von Neumann algebras on a given Hilbert space is a von Neumann algebra; the smallest von Neumann algebra containing a given set $ M $ is said to be the von Neumann algebra generated by the set $ M $. Let $ H _ {i} $, $ i \in I $, be Hilbert spaces, $ H = \sum ^ \oplus H _ {i} $ their direct sum, $ A _ {i} $ a von Neumann algebra on $ H _ {i} $, and $ A $ the von Neumann algebra on $ H $ generated by those operators $ T $ in $ {\mathcal B} ( H) $ for which every $ H _ {i} $ is invariant under $ T $ and the restriction of $ T $ to $ H _ {i} $ lies in $ A _ {i} $; this von Neumann algebra is called the direct product of the $ A _ {i} $ and is denoted by $ A = \times _ {i \in I } A _ {i} $. The operations of forming the tensor product, both finite and infinite, are also defined for von Neumann algebras. A von Neumann algebra is called a factor if its centre consists of multiples of the identity.

Let $ A $ be a von Neumann algebra and $ A ^ {+} $ the set of its positive operators. A weight on $ A $ is an additive mapping $ \phi $ from $ A ^ {+} $ into $ [ 0 , \infty ] $ that is homogeneous under multiplication by positive numbers. A weight $ \phi $ is called a trace if $ \phi ( U T U ^ {-1} ) = \phi ( T) $ for all $ T \in A ^ {+} $ and all unitary operators $ U $ in $ A $. A trace is said to be finite if $ \phi ( T) < \infty $ for all $ T \in A $; semi-finite if for any $ S \in A ^ {+} $ the quantity $ \phi ( S) $ is the least upper bound of the numbers of the form $ \phi ( T) $, where $ \phi ( T) < \infty $ and $ 0 \leq T \leq S $; exact if $ \phi ( T) = 0 $, $ T \in A ^ {+} $, implies $ T = 0 $; normal if for any increasing family $ T _ \alpha $ of elements in $ A ^ {+} $ with least upper bound $ T $ the quantity $ \phi ( T) $ is the least upper bound of the numbers $ \phi ( T _ \alpha ) $. A von Neumann algebra $ A $ is called finite if there is a family of normal finite traces on $ A $ separating the points of $ A $; properly infinite if there are no non-zero finite traces on $ A $; semi-finite if there is an exact normal semi-finite trace on $ A $; and purely infinite, or an algebra of type $ \textrm{ III } $, if there are no non-zero normal semi-finite traces on $ A $. A von Neumann algebra is called discrete, or of type $ \textrm{ I } $, if it is algebraically isomorphic to a von Neumann algebra with a commutative commutant; such an algebra is semi-finite. A von Neumann algebra is called continuous if for any non-zero central projection $ P $ the von Neumann algebra $ A _ {P} $ is not discrete. A continuous semi-finite algebra is said to be of type $ \textrm{ II } $. A finite algebra of type $ \textrm{ II } $ is said to be of type $ \textrm{ II } _ {1} $; a properly infinite algebra of type $ \textrm{ II } $ is said to be of type $ \textrm{ II } _ \infty $. Whether a von Neumann algebra belongs to a definite type is equivalent to the fact that its commutant belongs to the same type, but the commutant of a finite von Neumann algebra need not be a finite von Neumann algebra.

Let $ A $ be a von Neumann algebra, $ P $ and $ Q $ projections belonging to $ A $. Then $ P $ and $ Q $ are called equivalent, $ P \sim Q $, if there is an element $ U \in A $ such that $ P = U ^ {*} U $ and $ Q = U U ^ {*} $. One writes $ P \prec Q $ if there is a projection $ P _ {1} \in A $ such that $ P \sim P _ {1} $ and $ P _ {1} \leq Q $; the relation $ \prec $ is a partial order. A classification of von Neumann algebras according to type can be carried out in terms of this relation; in particular: A projection $ P \in A $ is called finite if $ P _ {1} \in A $, $ P _ {1} \sim P $, $ P _ {1} \leq P $ implies $ P _ {1} = P $; a von Neumann algebra is finite if and only if the identity projection is finite, and semi-finite if and only if the least upper bound of the family of finite projections is the identity projection.

A von Neumann algebra $ A $ is semi-finite if and only if it can be realized as the left von Neumann algebra of a certain Hilbert algebra; the elements of the latter are those $ x \in A $ for which $ \phi ( x ^ {*} x ) < \infty $, where $ \phi $ is an exact normal semi-finite trace on $ A $. For algebras of type $ \textrm{ III } $ the corresponding realization can be obtained by means of generalized Hilbert algebras and weights on von Neumann algebras.

Let $ H _ {p} $ be fixed Hilbert spaces of dimension $ p $, $ p = 1 \dots \aleph _ {0} $, let $ Z $ be a Borel space, let $ \mu $ be a positive measure on $ Z $, let $ Z = \cup_{p=1}^ {\aleph _ {0} } Z _ {p} $ be a partition of $ Z $ into disjoint measurable subsets, let $ L _ {2} ( Z _ {p} , \mu , H _ {p} ) $ be the Hilbert space of square-summable $ \mu $- measurable mappings of $ Z _ {p} $ into $ H _ {p} $, let

$$ H = \oplus_{p=1}^ \infty L _ {2} ( Z _ {p} , \mu , H _ {p} ) , $$

and let $ H ( \zeta ) = H _ {p} $ for $ \zeta \in Z _ {p} $. If $ f \in H $, then $ f = \sum _ {p} f _ {p} $, where $ f _ {p} \in L _ {2} ( Z _ {p} , \mu , H _ {p} ) $. Let $ f ( \zeta ) = f _ {p} ( \zeta ) $ for $ \zeta \in Z _ {p} $. A mapping $ \zeta \mapsto T ( \zeta ) $, where $ T ( \zeta ) $ is a continuous linear operator on the Hilbert space $ H ( \zeta ) $, is called a measurable field of operators if for any $ f \in H $ the function $ \zeta \mapsto T ( \zeta ) f ( \zeta ) $ is measurable on every set $ Z _ {p} $. If $ \zeta \mapsto T ( \zeta ) $ is a measurable field of operators and the function $ \zeta \mapsto \| T ( \zeta ) \| $ is essentially bounded on $ Z $, then for every $ f \in H $ there is a unit vector $ g \in H $ such that $ g ( \zeta ) = T ( \zeta ) f ( \zeta ) $ $ \mu $- almost everywhere. The mapping $ T : H \rightarrow H $ defined by $ T f = g $ for all $ f \in H $ is a bounded linear operator on $ H $, and

$$ \| T \| = { \mathop{\rm ess} \sup } _ {\zeta \in Z } \ \| T ( \zeta ) \| . $$

Such an operator $ T $ on $ H $ is called decomposable. Suppose that for any $ \zeta \in Z $ a von Neumann algebra $ A ( \zeta ) $ is defined on $ H ( \zeta ) $; the mapping $ \zeta \mapsto A ( \zeta ) $ is called a measurable field of von Neumann algebras if there exists a sequence $ \{ \zeta \rightarrow T _ {n} ( \zeta ) \} $ of measurable fields of operators such that for any $ \zeta \in Z $ the von Neumann algebra $ A ( \zeta ) $ is generated by the operators $ T _ {n} ( \zeta ) $. The set of all decomposable operators $ T $ on $ H $ such that $ T ( \zeta ) \in A ( \zeta ) $ for every $ \zeta \in Z $ is a von Neumann algebra in $ H $. It is denoted by

$$ A = \int\limits ^ \oplus A ( \zeta ) d \mu ( \zeta ) , $$

and is called the direct integral of the von Neumann algebras $ A ( \zeta ) $ over $ \mu $. Every von Neumann algebra on a separable Hilbert space is isomorphic to a direct integral of factors. An arbitrary von Neumann algebra has an algebraic decomposition, and this is why the theory of factors is of interest for the general theory of von Neumann algebras.

Von Neumann algebras arise naturally in problems connected with operators on a Hilbert space and have numerous applications in operator theory itself and in the representation theory of groups and algebras, as well as in the theory of dynamical systems, statistical physics and quantum field theory.

References[edit]

Comments[edit]

In terms of the majorization relation $ \prec $, defined above, the types of von Neumann algebras are as follows. A von Neumann algebra $ A $ is of type $ \textrm{ I } $ if every non-zero central projection in $ A $ majorizes a non-zero Abelian projection. (An Abelian projection is a projection $ e $ such that $ e A e $ is Abelian.) If there are no non-zero finite projections in $ A $, i.e. $ A $ is purely infinite, then it is of type $ \textrm{ III } $. If $ A $ has no non-zero Abelian projections and if every non-zero central projection in $ A $ majorizes a non-zero finite projection of $ A $, then $ A $ is of type $ \textrm{ II } $. If $ A $ is finite and of type $ \textrm{ II } $, then it is of type $ \textrm{ II } _ {1} $. If $ A $ is of type $ \textrm{ II } $ and has no non-zero central finite projections, then $ A $ is of type $ \textrm{ II } _ \infty $. Every von Neumann algebra is uniquely decomposable into the direct sum of von Neumann algebras of types $ \textrm{ I } $, $ \textrm{ II } _ {1} $, $ \textrm{ II } _ \infty $, $ \textrm{ III } $, and a factor is hence of one of these types.

A factor of type $ \textrm{ I } $ is isomorphic to $ {\mathcal B} ( H) $ for some Hilbert space $ H $. A factor of type $ \textrm{ I } _ {n} $ is the algebra of $ ( n \times n) $- matrices over $ \mathbf C $.

A von Neumann algebra or factor is hyperfinite if it is generated by an ascending sequence of finite factors, i.e. matrix algebras. There is just one hyperfinite type $ \textrm{ II } _ {1} $ factor and one hyperfinite $ \textrm{ II } _ \infty $ factor (up to isomorphism), [a7]. For more details on types and factors, e.g. types $ \textrm{ III } _ {0} $, $ \textrm{ III } _ {1} $, $ \textrm{ III } _ \lambda $ and finer classification results, cf. [a1], [a4], [a5], [a7]–[a9].

Part of a recent breakthrough in knot theory came from work on the classification of subfactors; cf. (the editorial comments to) Knot theory and references given there, as well as [a3], [a6].

References[edit]