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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Automorphic images of commutative subspace lattices


Authors: K. J. Harrison and W. E. Longstaff
Journal: Trans. Amer. Math. Soc. 296 (1986), 217-228
MSC: Primary 46C10; Secondary 06B99, 47A15, 47D25
MathSciNet review: 837808
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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{displaymath}\begin{array}{*{20}{c}} {\mathcal{F} = \phi (\mathcal{L})\;{\... ...;{\text{lattice}}\;\mathcal{L} \subseteq C(H).} \\ \end{array} \end{displaymath}

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.

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DOI: https://doi.org/10.1090/S0002-9947-1986-0837808-1
Article copyright: © Copyright 1986 American Mathematical Society