Journal of Operator Theory

Volume 64, Issue 2, Fall 2010 pp. 321-347.

Simplicity of $C^*$-algebras using unique eigenstates

Authors: Lon H. Mitchell (1) and William L. Paschke (2)
Author institution: (1) Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, Virginia, 23284, U.S.A.
(2) Department of Mathematics, University of Kansas, Lawrence, Kansas, 66045, U.S.A.

Summary: We consider a one-parameter family of operators that are constructed from a pair of isometries on Hilbert space with orthogonal ranges. For special values of the parameter, the operator plays a role in the representation theory of free groups and in free probability theory. For each parameter value, we identify the irreducible $*$-representations of the pair of isometries in which the operator has an eigenvalue. This yields a new technique for showing that certain $C^*$-algebras, including the $C^*$-algebra generated by the operator, are simple. We establish several other fundamental properties of this $C^*$-algebra and its generator.

Keywords: $C^*$-algebra, Cuntz algebra, trace, simple, nonnuclear, free group, eigenstate, irreducible representation, spectrum

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