# Journal of Operator Theory

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

Simplicity of $\Cstar$-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 $\Cstar$-algebras, including the $\Cstar$-algebra generated by the operator, are simple. We establish several other fundamental properties of this $\Cstar$-algebra and its generator.

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

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