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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Automorphic images of commutative subspace lattices
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by K. J. Harrison and W. E. Longstaff PDF
Trans. Amer. Math. Soc. 296 (1986), 217-228 Request permission

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.
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Additional Information
  • © Copyright 1986 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 296 (1986), 217-228
  • MSC: Primary 46C10; Secondary 06B99, 47A15, 47D25
  • DOI: https://doi.org/10.1090/S0002-9947-1986-0837808-1
  • MathSciNet review: 837808