Reflexive operator algebra

In functional analysis, a reflexive operator algebra A is an operator algebra that has enough invariant subspaces to characterize it. Formally, A is reflexive if it is equal to the algebra of bounded operators which leave invariant each subspace left invariant by every operator in A. This should not be confused with a reflexive space.

## Contents

* 1 Examples
* 2 Hyper-reflexivity
* 3 Examples
* 4 See also
* 5 References

## Examples

Nest algebras 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 × 2 matrices

[math]\displaystyle{ \left\\{ \begin{pmatrix} a&b\\\ 0 & a \end{pmatrix} \ :\ a,b\in\mathbb{C}\right\\}. }[/math]

This algebra is smaller than the Nest algebra

[math]\displaystyle{ \left\\{ \begin{pmatrix} a&b\\\ 0 & c \end{pmatrix} \ :\ a,b,c\in\mathbb{C}\right\\} }[/math]

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

[math]\displaystyle{ \left\\{ \begin{pmatrix} a & b & 0\\\ 0 & a & 0\\\ 0 & 0 & a \end{pmatrix} \ :\ a,b\in\mathbb{C}\right\\} }[/math]

which is equal to the set of all polynomials in

[math]\displaystyle{ T=\begin{pmatrix} 0 & 1 & 0\\\ 0 & 0 & 0\\\ 0 & 0 & 0 \end{pmatrix} }[/math]

and the identity is reflexive.

## Hyper-reflexivity

Let [math]\displaystyle{ \mathcal{A} }[/math] 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

[math]\displaystyle{ \beta(T,\mathcal{A})=\sup \left\\{ \left\| P^\perp TP \right\| \ :\ P\mbox{ is a projection and } P^\perp \mathcal{A} P = (0) \right\\} . }[/math]

Observe that P is a projection involved in this supremum precisely if the range of P is an invariant subspace of [math]\displaystyle{ \mathcal{A} }[/math].

The algebra [math]\displaystyle{ \mathcal{A} }[/math] is reflexive if and only if for every T in B(H):

[math]\displaystyle{ \beta(T,\mathcal{A})=0 \mbox{ implies that } T \mbox{ is in } \mathcal{A} . }[/math]

We note that for any T in B(H) the following inequality is satisfied:

[math]\displaystyle{ \beta(T,\mathcal{A})\le \mbox{dist}(T,\mathcal{A}) . }[/math]

Here [math]\displaystyle{ \mbox{dist}(T,\mathcal{A}) }[/math] 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 [math]\displaystyle{ \mathcal{A} }[/math] hyperreflexive if there is a constant K such that for every operator T in B(H),

[math]\displaystyle{ \mbox{dist}(T,\mathcal{A})\le K \beta(T,\mathcal{A}) . }[/math]

The smallest such K is called the distance constant for [math]\displaystyle{ \mathcal{A} }[/math]. 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 algebras 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 subspace
* subspace lattice
* reflexive subspace lattice
* nest algebra

## References

* William Arveson, Ten lectures on operator algebras, ISBN:0-8218-0705-6
* H. Radjavi and P. Rosenthal, Invariant Subspaces, ISBN:0-486-42822-2

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