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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Applications of the Wold decomposition to the study of row contractions associated with directed graphs


Authors: Elias Katsoulis and David W. Kribs
Journal: Trans. Amer. Math. Soc. 357 (2005), 3739-3755
MSC (2000): Primary 47A63, 47L40, 47L80
Published electronically: March 31, 2005
MathSciNet review: 2146647
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Abstract: Based on a Wold decomposition for families of partial isometries and projections of Cuntz-Krieger-Toeplitz-type, we extend several fundamental theorems from the case of single vertex graphs to the general case of countable directed graphs with no sinks. We prove a Szego-type factorization theorem for CKT families, which leads to information on the structure of the unit ball in free semigroupoid algebras, and show that joint similarity implies joint unitary equivalence for such families. For each graph we prove a generalization of von Neumann's inequality which applies to row contractions of operators on Hilbert space which are related to the graph in a natural way. This yields a functional calculus determined by quiver algebras and free semigroupoid algebras. We establish a generalization of Coburn's theorem for the $\mathrm{C}^*$-algebra of a CKT family, and prove a universality theorem for $\mathrm{C}^*$-algebras generated by these families. In both cases, the $\mathrm{C}^*$-algebras generated by quiver algebras play the universal role.


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Additional Information

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

David W. Kribs
Affiliation: Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario, Canada N1G 2W1
Email: dkribs@uoguelph.ca

DOI: http://dx.doi.org/10.1090/S0002-9947-05-03692-5
PII: S 0002-9947(05)03692-5
Keywords: Directed graph, partial isometry, row contraction, Wold decomposition, von Neumann inequality, Cuntz-Krieger $\mathrm{C}^*$-algebra, quiver algebra, free semigroupoid algebra
Received by editor(s): November 11, 2003
Received by editor(s) in revised form: March 15, 2004
Published electronically: March 31, 2005
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.