Automorphic images of commutative subspace lattices

Abstract

Let C ( H ) C(H) denote the lattice of all (closed) subspaces of a complex, separable Hilbert space H H . Let ( AC) ({\text {AC)}} be the following condition that a subspace lattice F ⊆ C ( H ) \mathcal {F} \subseteq C(H) may or may not satisfy: (AC) \[ F = ϕ ( L ) for some lattice automorphism ϕ of C ( H ) and some commutative subspace lattice L ⊆ C ( H ) . \begin {array}{*{20}{c}} {\mathcal {F} = \phi (\mathcal {L})\;{\text {for}}\;{\text {some}}\;{\text {lattice}}\;{\text {automorphism}}\;\phi \;{\text {of}}\;C(H)} \\ {{\text {and}}\;{\text {some}}\;{\text {commutative}}\;{\text {subspace}}\;{\text {lattice}}\;\mathcal {L} \subseteq C(H).} \\ \end {array} \] Then F \mathcal {F} satisfies ( AC ) ({\text {AC}}) if and only if A ⊆ B \mathcal {A} \subseteq \mathcal {B} for some Boolean algebra subspace lattice B ⊆ C ( H ) \mathcal {B} \subseteq C(H) with the property that, for every K , L ∈ B K,L \in \mathcal {B} , the vector sum K + L K + L is closed. If F \mathcal {F} is finite, then F \mathcal {F} satisfies ( AC ) ({\text {AC}}) if and only if F \mathcal {F} is distributive and K + L K + L is closed for every K , L ∈ F K,L \in \mathcal {F} . In finite dimensions F \mathcal {F} satisfies ( AC ) ({\text {AC}}) if and only if F \mathcal {F} is distributive. Every F \mathcal {F} satisfying ( AC ) ({\text {AC}}) is reflexive. For such F \mathcal {F} , given vectors x , y ∈ H x,y \in H , the solvability of the equation T x = y Tx = y for T ∈ Alg F T \in \operatorname {Alg}\,\mathcal {F} is investigated.