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Reflexive operator algebra
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<h2 id="examples">Examples</h2> <p><a href="/facts/Nest_algebra/2jP8yYBA">Nest algebras</a> 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. </p><p>In fact if we fix any pattern of entries in an <i>n</i> by <i>n</i> matrix containing the diagonal, then the set of all <i>n</i> by <i>n</i> matrices whose nonzero entries lie in this pattern forms a reflexive algebra. </p><p>An example of an algebra which is <i>not</i> reflexive is the set of 2 × 2 matrices </p> { ( a b 0 a ) : a , b ∈ C } . {\displaystyle \left\{{\begin{pmatrix}a&b\\0&a\end{pmatrix}}\ :\ a,b\in \mathbb {C} \right\}.} <p>This algebra is smaller than the Nest algebra </p> { ( 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\}} <p>but has the same invariant subspaces, so it is not reflexive. </p><p>If <i>T</i> is a fixed <i>n</i> by <i>n</i> matrix then the set of all polynomials in <i>T</i> 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 <a href="/facts/Jordan_normal_form/M7ezYbnl">Jordan normal form</a> of <i>T</i> differ in size by at most one. For example, the algebra </p> { ( 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\}} <p>which is equal to the set of all polynomials in </p> 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}}} <p>and the identity is reflexive. </p> <h2 id="hyper-reflexivity">Hyper-reflexivity</h2> <p>Let A {\displaystyle {\mathcal {A}}} be a weak*-closed operator algebra contained in <i>B</i>(<i>H</i>), the set of all bounded operators on a <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> <i>H</i> and for <i>T</i> any operator in <i>B</i>(<i>H</i>), let </p> 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\}.} <p>Observe that <i>P</i> is a projection involved in this supremum precisely if the range of <i>P</i> is an invariant subspace of A {\displaystyle {\mathcal {A}}} . </p><p>The algebra A {\displaystyle {\mathcal {A}}} is reflexive if and only if for every <i>T</i> in <i>B</i>(<i>H</i>): </p> 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}}.} <p>We note that for any <i>T</i> in <i>B(H)</i> the following inequality is satisfied: </p> β ( T , A ) ≤ dist ( T , A ) . {\displaystyle \beta (T,{\mathcal {A}})\leq {\mbox{dist}}(T,{\mathcal {A}}).} <p>Here dist ( T , A ) {\displaystyle {\mbox{dist}}(T,{\mathcal {A}})} is the distance of <i>T</i> from the algebra, namely the smallest norm of an operator <i>T-A</i> where A runs over the algebra. We call A {\displaystyle {\mathcal {A}}} hyperreflexive if there is a constant <i>K</i> such that for every operator <i>T</i> in <i>B</i>(<i>H</i>), </p> dist ( T , A ) ≤ K β ( T , A ) . {\displaystyle {\mbox{dist}}(T,{\mathcal {A}})\leq K\beta (T,{\mathcal {A}}).} <p>The smallest such <i>K</i> is called the distance constant for A {\displaystyle {\mathcal {A}}} . A hyper-reflexive operator algebra is automatically reflexive. </p><p>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? </p> <h2 id="examples">Examples</h2> <ul><li>Every finite-dimensional reflexive algebra is hyper-reflexive. However, there are examples of infinite-dimensional reflexive operator algebras which are not hyper-reflexive.</li> <li>The distance constant for a one-dimensional algebra is 1.</li> <li>Nest algebras are hyper-reflexive with distance constant 1.</li> <li>Many <a href="/facts/Von_Neumann_algebra/8NyWUIRH">von Neumann algebras</a> are hyper-reflexive, but it is not known if they all are.</li> <li>A <a href="/facts/Type_I_von_Neumann_algebra/8NyWUIRH">type I von Neumann algebra</a> is hyper-reflexive with distance constant at most 2.</li></ul> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Invariant_subspace/51Vm1W2X">Invariant subspace</a></li> <li>subspace lattice</li> <li>reflexive subspace lattice</li> <li><a href="/facts/Nest_algebra/2jP8yYBA">nest algebra</a></li></ul> <ul><li>William Arveson, <i>Ten lectures on operator algebras</i>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8218-0705-6</li> <li>H. Radjavi and P. Rosenthal, <i>Invariant Subspaces</i>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-42822-2</li></ul>