Reflexive operator algebra
Encyclopedia
In functional analysis
, a reflexive operator algebra
A is an operator algebra that has enough invariant subspace
s to characterize it. Formally, A is reflexive if it is equal to the algebra of bounded operator
s which leave invariant
each subspace
left invariant by every operator in A.
This should not be confused with a reflexive space
.
s are examples of reflexive operator algebras. In finite dimensions, these are simply algebras of all matrices of a given size whose nonzero entries lie in an upper-triangular pattern.
In fact if we fix any pattern of entries in an n by n matrix containing the diagonal, then the set of all n by n matrices whose nonzero entries lie in this pattern forms a reflexive algebra.
An example of an algebra which is not reflexive is the set of 2 by 2 matrices
This algebra is smaller than the Nest algebra
but has the same invariant subspaces, so it is not reflexive.
If T is a fixed n by n matrix then the set of all polynomials in T and the identity operator forms a unital operator algebra. A theorem of Deddens and Fillmore states that this algebra is reflexive if and only if the largest two blocks in the Jordan normal form
of T differ in size by at most one. For example, the algebra
which is equal to the set of all polynomials in
and the identity is reflexive.
be a weak*-closed operator algebra contained in B(H), the set of all bounded operators on a Hilbert space
H and for T any operator in B(H), let
.
Observe that P is a projection involved in this supremum precisely if the range of P is an invariant subspace of
.
The algebra
is reflexive if and only if for every T in B(H):
.
We note that for any T in B(H) the following inequality is satisfied:
.
Here
is the distance of T from the algebra, namely the smallest norm of an operator T-A where A runs over the algebra. We call 
hyperreflexive if there is a constant K such that for every operator T in B(H),
.
The smallest such K is called the distance constant for
. A hyper-reflexive operator algebra is automatically reflexive.
In the case of a reflexive algebra of matrices with nonzero entries specified by a given pattern, the problem of finding the distance constant can be rephrased as a matrix-filling problem: if we fill the entries in the complement of the pattern with arbitrary entries, what choice of entries in the pattern gives the smallest operator norm?
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 and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, a reflexive operator algebra
Operator algebra
In functional analysis, an operator algebra is an algebra of continuous linear operators on a topological vector space with the multiplication given by the composition of mappings...
A is an operator algebra that has enough invariant subspace
Invariant subspace
In mathematics, an invariant subspace of a linear mappingfrom some vector space V to itself is a subspace W of V such that T is contained in W...
s to characterize it. Formally, A is reflexive if it is equal to the algebra of bounded operator
Bounded operator
In functional analysis, a branch of mathematics, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L to that of v is bounded by the same number, over all non-zero vectors v in X...
s which leave invariant
Invariant (mathematics)
In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...
each subspace
Subspace
-In mathematics:* Euclidean subspace, in linear algebra, a set of vectors in n-dimensional Euclidean space that is closed under addition and scalar multiplication...
left invariant by every operator in A.
This should not be confused with a reflexive space
Reflexive space
In functional analysis, a Banach space is called reflexive if it coincides with the dual of its dual space in the topological and algebraic senses. Reflexive Banach spaces are often characterized by their geometric properties.- Normed spaces :Suppose X is a normed vector space over R or C...
.
Examples
Nest algebraNest algebra
In functional analysis, a branch of mathematics, nest algebras are a class of operator algebras that generalise the upper-triangular matrix algebras to a Hilbert space context. They were introduced by John Ringrose in the mid-1960s and have many interesting properties...
s are examples of reflexive operator algebras. In finite dimensions, these are simply algebras of all matrices of a given size whose nonzero entries lie in an upper-triangular pattern.
In fact if we fix any pattern of entries in an n by n matrix containing the diagonal, then the set of all n by n matrices whose nonzero entries lie in this pattern forms a reflexive algebra.
An example of an algebra which is not reflexive is the set of 2 by 2 matrices
This algebra is smaller than the Nest algebra
but has the same invariant subspaces, so it is not reflexive.
If T is a fixed n by n matrix then the set of all polynomials in T and the identity operator forms a unital operator algebra. A theorem of Deddens and Fillmore states that this algebra is reflexive if and only if the largest two blocks in the Jordan normal form
Jordan normal form
In linear algebra, a Jordan normal form of a linear operator on a finite-dimensional vector space is an upper triangular matrix of a particular form called Jordan matrix, representing the operator on some basis...
of T differ in size by at most one. For example, the algebra
which is equal to the set of all polynomials in
and the identity is reflexive.
Hyper-reflexivity
Let be a weak*-closed operator algebra contained in B(H), the set of all bounded operators on a Hilbert spaceHilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
H and for T any operator in B(H), let
.Observe that P is a projection involved in this supremum precisely if the range of P is an invariant subspace of
.The algebra
is reflexive if and only if for every T in B(H):.We note that for any T in B(H) the following inequality is satisfied:
.Here
is the distance of T from the algebra, namely the smallest norm of an operator T-A where A runs over the algebra. We call hyperreflexive if there is a constant K such that for every operator T in B(H),.The smallest such K is called the distance constant for
. A hyper-reflexive operator algebra is automatically reflexive.In the case of a reflexive algebra of matrices with nonzero entries specified by a given pattern, the problem of finding the distance constant can be rephrased as a matrix-filling problem: if we fill the entries in the complement of the pattern with arbitrary entries, what choice of entries in the pattern gives the smallest operator norm?
Examples
- Every finite-dimensional reflexive algebra is hyper-reflexive. However, there are examples of infinite-dimensional reflexive operator algebras which are not hyper-reflexive.
- The distance constant for a one dimensional algebra is 1.
- Nest algebras are hyper-reflexive with distance constant 1.
- Many von Neumann algebraVon Neumann algebraIn 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, motivated by his study of single operators, group...
s are hyper-reflexive, but it is not known if they all are.
- A type I von Neumann algebra is hyper-reflexive with distance constant at most 2.
See also
- Invariant subspaceInvariant subspaceIn mathematics, an invariant subspace of a linear mappingfrom some vector space V to itself is a subspace W of V such that T is contained in W...
- subspace lattice
- reflexive subspace lattice
- nest algebraNest algebraIn functional analysis, a branch of mathematics, nest algebras are a class of operator algebras that generalise the upper-triangular matrix algebras to a Hilbert space context. They were introduced by John Ringrose in the mid-1960s and have many interesting properties...