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Tree Algebras, Semidiscreteness, and Dilation Theory
Author(s) -
Davidson K. R.,
Paulsen V. I.,
Power S. C.
Publication year - 1994
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-68.1.178
Subject(s) - mathematics , tree (set theory) , distributive property , class (philosophy) , dilation (metric space) , pure mathematics , transitive relation , representation theorem , nest algebra , centralizer and normalizer , discrete mathematics , algebra over a field , non associative algebra , combinatorics , algebra representation , artificial intelligence , computer science
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–Foiaş 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–Foiaş Commutant Lifting Theorem.