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Reflexive operator algebra

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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.

Contents

This should not be confused with a reflexive space.

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 by 2 matrices

{ ( a b 0 a )   :   a , b ∈ C } .

This algebra is smaller than the Nest algebra

{ ( a b 0 c )   :   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

{ ( a b 0 0 a 0 0 0 a )   :   a , b ∈ C }

which is equal to the set of all polynomials in

T = ( 0 1 0 0 0 0 0 0 0 )

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 ⊥ T P ∥   :   P  is a projection and  P ⊥ A P = ( 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 ) ≤ K β ( 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.

    • References

    Reflexive operator algebra Wikipedia
    (Text) CC BY-SA


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