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Tree algebras, semidiscreteness and dilation theory.

Davidson, K. R. and Power, S. C. and Paulsen, V. I. (1994) Tree algebras, semidiscreteness and dilation theory. Proceedings of the London Mathematical Society, 68 (1). pp. 178-202. ISSN 1460-244X

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Abstract

We introduce a class of finite-dimensional algebras built from a partial order generated as a transitive relation from a finite tree. These algebras, known as tree algebras, have the property that every locally contractive representation has a *-dilation. Furthermore, they satisfy an appropriate analogue of the Sz. Nagy–Foia Commutant Lifting Theorem. Then we define the infinite-dimensional analogue of these algebras in the class of completely distributive CSL algebras. These algebras are shown to have the semidiscreteness and complete compact approximation properties with respect to the class of finite-dimensional tree algebras. Consequently, they also have the property that contractive weak-* continuous representations have *-dilations, and satisfy the Sz. Nagy–Foia Commutant Lifting Theorem.

Item Type: Article
Journal or Publication Title: Proceedings of the London Mathematical Society
Subjects: Q Science > QA Mathematics
Departments: Faculty of Science and Technology > Mathematics and Statistics
ID Code: 19605
Deposited By: ep_ss_importer
Deposited On: 13 Nov 2008 09:06
Refereed?: Yes
Published?: Published
Last Modified: 09 Apr 2014 20:23
Identification Number:
URI: http://eprints.lancs.ac.uk/id/eprint/19605

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