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Compact operators and nest representations of limit algebras


Authors: Elias Katsoulis and Justin R. Peters
Journal: Trans. Amer. Math. Soc. 359 (2007), 2721-2739
MSC (2000): Primary 47L80
DOI: https://doi.org/10.1090/S0002-9947-07-04071-8
Published electronically: January 4, 2007
MathSciNet review: 2286053
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Abstract: In this paper we study the nest representations $ \rho: \mathcal{A} \longrightarrow \operatorname{Alg} \mathcal{N}$ of a strongly maximal TAF algebra $ \mathcal{A}$, whose ranges contain non-zero compact operators. We introduce a particular class of such representations, the essential nest representations, and we show that their kernels coincide with the completely meet irreducible ideals. From this we deduce that there exist enough contractive nest representations, with non-zero compact operators in their range, to separate the points in $ \mathcal{A}$. Using nest representation theory, we also give a coordinate-free description of the fundamental groupoid for strongly maximal TAF algebras.

For an arbitrary nest representation $ \rho: \mathcal{A} \longrightarrow \operatorname{Alg} \mathcal{N}$, we show that the presence of non-zero compact operators in the range of $ \rho$ implies that $ \mathcal{N}$ is similar to a completely atomic nest. If, in addition, $ \rho (\mathcal{A} )$ is closed, then every compact operator in $ \rho (\mathcal{A} )$ can be approximated by sums of rank one operators $ \rho (\mathcal{A} )$. In the case of $ \mathbb{N}$-ordered nest representations, we show that $ \rho ( \mathcal{A})$ contains finite rank operators iff $ \ker \rho $ fails to be a prime ideal.


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Additional Information

Elias Katsoulis
Affiliation: Department of Mathematics, East Carolina University, Greenville, North Carolina 27858
Email: katsoulise@ecu.edu

Justin R. Peters
Affiliation: Department of Mathematics, Iowa State University, Ames, Iowa 50011
Email: peters@iastate.edu

DOI: https://doi.org/10.1090/S0002-9947-07-04071-8
Received by editor(s): April 15, 2004
Received by editor(s) in revised form: March 27, 2005
Published electronically: January 4, 2007
Additional Notes: The first author’s research was partially supported by a grant from ECU
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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