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Transactions of the American Mathematical Society

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$C^*$-algebras generated by a subnormal operator

Authors: Kit C. Chan and Zeljko Cuckovic
Journal: Trans. Amer. Math. Soc. 351 (1999), 1445-1460
MSC (1991): Primary 47B20, 32A37, 46L05; Secondary 46E20, 47A13, 47B37
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Abstract: Using the functional calculus for a normal operator, we provide a result for generalized Toeplitz operators, analogous to the theorem of Axler and Shields on harmonic extensions of the disc algebra. Besides that result, we prove that if $T$ is an injective subnormal weighted shift, then any two nontrivial subspaces invariant under $T$ cannot be orthogonal to each other. Then we show that the $C^*$-algebra generated by $T$ and the identity operator contains all the compact operators as its commutator ideal, and we give a characterization of that $C^*$-algebra in terms of generalized Toeplitz operators. Motivated by these results, we further obtain their several-variable analogues, which generalize and unify Coburn's theorems for the Hardy space and the Bergman space of the unit ball.

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Additional Information

Kit C. Chan
Affiliation: Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, Ohio 43403-0221

Zeljko Cuckovic
Affiliation: Department of Mathematics, University of Toledo, Toledo, Ohio 43606-3390

Keywords: $C^*$-algebras, subnormal operators, weighted shifts, functional calculus, analytic functions, joint spectrum, Toeplitz operators
Received by editor(s): December 4, 1995
Received by editor(s) in revised form: March 4, 1998
Article copyright: © Copyright 1999 American Mathematical Society