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

{(ab0a) : a,b∈ℂ}.

This algebra is smaller than the Nest algebra

{(ab0c) : a,b,c∈ℂ}

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

{(ab00a000a) : a,b∈ℂ}

which is equal to the set of all polynomials in

T=(010000000)

and the identity is reflexive.

## Hyper-reflexivity

Let 𝒜 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

β(T,𝒜)=sup⁡{‖P⊥TP‖ : P is a projection and P⊥𝒜P=(0)}.

Observe that P is a projection involved in this supremum precisely if the range of P is an invariant subspace of 𝒜.

The algebra 𝒜 is reflexive if and only if for every T in B(H):

β(T,𝒜)=0 implies that T is in 𝒜.

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

β(T,𝒜)≤dist(T,𝒜).

Here dist(T,𝒜) 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 𝒜 hyperreflexive if there is a constant K such that for every operator T in B(H),

dist(T,𝒜)≤Kβ(T,𝒜).

The smallest such K is called the distance constant for 𝒜. 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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