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Strictly cyclic operator algebras


Author: John Froelich
Journal: Trans. Amer. Math. Soc. 325 (1991), 73-86
MSC: Primary 47D25; Secondary 47A15
DOI: https://doi.org/10.1090/S0002-9947-1991-0989575-0
MathSciNet review: 989575
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Abstract: We prove several results about the lattice of invariant subspaces of general strictly cyclic and strongly strictly cyclic operator algebras. A reflexive operator algebra $ A$ with a commutative subspace lattice is strictly cyclic iff $ \operatorname{Lat}{(A)^ \bot }$ contains a finite number of atoms and each nonzero element of $ \operatorname{Lat}{(A)^ \bot }$ contains an atom. This leads to a characterization of the $ n$-strictly cyclic reflexive algebras with a commutative subspace lattice as well as an extensive generalization of D. A. Herrero's result that there are no triangular strictly cyclic operators. A reflexive operator algebra $ A$ with a commutative subspace lattice is strongly strictly cyclic iff $ \operatorname{Lat}(A)$ satisfies A.C.C.

The distributive lattices which are attainable by strongly strictly cyclic reflexive algebras are the complete sublattices of $ \{ 0,1] \times \{ 0,1\} \times \cdots $ which satisfy A.C.C.

We also show that if $ \operatorname{Alg}(\mathcal{L})$ is strictly cyclic and $ \mathcal{L} \subseteq $ atomic m.a.s.a. then $ \operatorname{Alg}(\mathcal{L})$ contains a strictly cyclic operator.


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DOI: https://doi.org/10.1090/S0002-9947-1991-0989575-0
Article copyright: © Copyright 1991 American Mathematical Society

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