Automorphic images of commutative subspace lattices

Abstract:

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