Journal of Operator Theory

Volume 60, Issue 2, Fall 2008 pp. 429-443.

$C^*$-algebras with multiple subnormal generators

Authors: Nathan S. Feldman (1) and Paul J. McGuire (2)
Author institution: (1) Mathematics Department , Washington and Lee University, Lexington, VA 24450, USA
(2) Mathematics Department, Bucknell University, Lewisburg, PA 17837, USA

Summary: If $A$ is an irreducible essentially normal operator, then we prove that the $C^*$-algebra generated by $A$ has a finite number of irreducible subnormal operators as generators if and only if the essential spectrum of $A$ is uncountable. It is shown that, in general, at most eight irreducible subnormal generators are required. Additionally, it is shown that frequently two irreducible subnormal operators will suffice and that, in many instances, the subnormal operators can be taken to be unilateral shifts of multiplicity one or unitarily equivalent to the dual of the Bergman shift.

Keywords: $C^*$-algebra, subnormal operator, generator, irreducible, essentially normal

Contents Full-Text PDF