## Automorphic images of commutative subspace lattices

HTML articles powered by AMS MathViewer

- by K. J. Harrison and W. E. Longstaff
- Trans. Amer. Math. Soc.
**296**(1986), 217-228 - DOI: https://doi.org/10.1090/S0002-9947-1986-0837808-1
- PDF | 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.## References

- William Arveson,
*Operator algebras and invariant subspaces*, Ann. of Math. (2)**100**(1974), 433–532. MR**365167**, DOI 10.2307/1970956 - William G. Bade,
*Weak and strong limits of spectral operators*, Pacific J. Math.**4**(1954), 393–413. MR**63567** - William G. Bade,
*On Boolean algebras of projections and algebras of operators*, Trans. Amer. Math. Soc.**80**(1955), 345–360. MR**73954**, DOI 10.1090/S0002-9947-1955-0073954-0 - Reinhold Baer,
*Linear algebra and projective geometry*, Academic Press, Inc., New York, N.Y., 1952. MR**0052795** - John B. Conway,
*A complete Boolean algebra of subspaces which is not reflexive*, Bull. Amer. Math. Soc.**79**(1973), 720–722. MR**320778**, DOI 10.1090/S0002-9904-1973-13279-3 - Kenneth R. Davidson,
*Commutative subspace lattices*, Indiana Univ. Math. J.**27**(1978), no. 3, 479–490. MR**482264**, DOI 10.1512/iumj.1978.27.27032 - Thomas Donnellan,
*Lattice theory*, Pergamon Press, Oxford-New York-Toronto, 1968. MR**0233738** - Nelson Dunford and Jacob T. Schwartz,
*Linear operators. Part I*, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR**1009162** - P. A. Fillmore and W. E. Longstaff,
*On isomorphisms of lattices of closed subspaces*, Canad. J. Math.**36**(1984), no. 5, 820–829. MR**762744**, DOI 10.4153/CJM-1984-048-x - P. R. Halmos,
*Reflexive lattices of subspaces*, J. London Math. Soc. (2)**4**(1971), 257–263. MR**288612**, DOI 10.1112/jlms/s2-4.2.257 - Alan Hopenwasser,
*The equation $Tx=y$ in a reflexive operator algebra*, Indiana Univ. Math. J.**29**(1980), no. 1, 121–126. MR**554821**, DOI 10.1512/iumj.1980.29.29009

## Bibliographic 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