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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

{ ( a b 0 a )   :   a , b ∈ C } . {\displaystyle \left\{{\begin{pmatrix}a&b\\0&a\end{pmatrix}}\ :\ a,b\in \mathbb {C} \right\}.}

This algebra is smaller than the Nest algebra

{ ( a b 0 c )   :   a , b , c ∈ C } {\displaystyle \left\{{\begin{pmatrix}a&b\\0&c\end{pmatrix}}\ :\ a,b,c\in \mathbb {C} \right\}}

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

which is equal to the set of all polynomials in

T = ( 0 1 0 0 0 0 0 0 0 ) {\displaystyle T={\begin{pmatrix}0&1&0\\0&0&0\\0&0&0\end{pmatrix}}}

and the identity is reflexive.

Hyper-reflexivity

Let A {\displaystyle {\mathcal {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 ) } . {\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\}.}

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

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

β ( T , A ) = 0  implies that  T  is in  A . {\displaystyle \beta (T,{\mathcal {A}})=0{\mbox{ implies that }}T{\mbox{ is in }}{\mathcal {A}}.}

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

β ( T , A ) ≤ dist ( T , A ) . {\displaystyle \beta (T,{\mathcal {A}})\leq {\mbox{dist}}(T,{\mathcal {A}}).}

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

dist ( T , A ) ≤ K β ( T , A ) . {\displaystyle {\mbox{dist}}(T,{\mathcal {A}})\leq K\beta (T,{\mathcal {A}}).}

The smallest such K is called the distance constant for A {\displaystyle {\mathcal {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

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