Logarithmic residues, Rouché’s theorem, and spectral regularity: The C∗C∗-algebra case

Using families of irreducible Hilbert space representations as a tool, the theory of analytic Fredholm operator valued function is extended to a C∗C∗-algebra setting. This includes a C∗C∗-algebra version of Rouché’s Theorem known from complex function theory. Also, criteria for spectral regularity of C∗C∗-algebras are developed. One of those, involving the (generalized) Calkin algebra, is applied to C∗C∗-algebras generated by a non-unitary isometry.