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Transactions of the American Mathematical Society

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Automorphism Groups and
Invariant Subspace Lattices


Authors: Paul S. Muhly and Baruch Solel
Journal: Trans. Amer. Math. Soc. 349 (1997), 311-330
MSC (1991): Primary 46K50, 47D25, 47D99, 46L40; Secondary 46L50, 46L55, 46L99
DOI: https://doi.org/10.1090/S0002-9947-97-01755-8
MathSciNet review: 1376551
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Abstract: Let $(B,\mathbf {R},\alpha )$ be a $C^{*}$- dynamical system and let $% A=B^\alpha ([0,\infty ))$ be the analytic subalgebra of $B$. We extend the work of Loebl and the first author that relates the invariant subspace structure of $\pi (A),$ for a $C^{*}$-representation $\pi $ on a Hilbert space $\mathcal {H}_\pi $, to the possibility of implementing $\alpha $ on $% \mathcal {H}_\pi .$ We show that if $\pi $ is irreducible and if lat $\pi (A)$ is trivial, then $\pi (A)$ is ultraweakly dense in $\mathcal {L(H}_\pi ).$ We show, too, that if $A $ satisfies what we call the strong Dirichlet condition, then the ultraweak closure of $\pi (A)$ is a nest algebra for each irreducible representation $\pi .$ Our methods give a new proof of a ``density'' theorem of Kaftal, Larson, and Weiss and they sharpen earlier results of ours on the representation theory of certain subalgebras of groupoid $C^{*}$-algebras.


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Additional Information

Paul S. Muhly
Affiliation: Department of Mathematics The University of Iowa Iowa City, Iowa 52242
Email: muhly@math.uiowa.edu

Baruch Solel
Affiliation: Department of Mathematics Technion - Israel Institute of Technology Haifa 32000 Israel
Email: mabaruch@techunix.technion.ac.il

DOI: https://doi.org/10.1090/S0002-9947-97-01755-8
Received by editor(s): July 11, 1994
Received by editor(s) in revised form: October 8, 1995
Additional Notes: Supported in part by grants from the U. S. National Science Foundation and the U. S. - Israel Binational Science Foundation.
Supported in part by the U. S. - Israel Binational Science Foundation and the Fund for the Promotion of Research at the Technion.
Article copyright: © Copyright 1997 American Mathematical Society

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