Fixed Point Composition C* -algebras.

摘要

In this dissertation, we investigate C*-algebras generated by composition operators and compact operators on the Hardy space. We primarily consider composition operators Cϕ that are induced by linear-fractional, non-automorphism self-maps ϕ of the open unit disk that fix a point zeta on the unit circle and satisfy ϕ'(zeta) ≠ 1.;We begin by studying a C*-algebra generated by a semigroup of composition operators of this form. We show that this C*-algebra is isomorphic, modulo its commutator ideal, to the direct sum of the complex numbers and the algebra of almost periodic functions on the real line. As a key step in our argument, we calculate the joint approximate point spectra of finite collections of unitarily equivalent operators and relate these sets to the joint approximate point spectra of collections of almost periodic Toeplitz operators on the Hardy space of the real line. We also prove that the semigroup C*-algebra is irreducible and use the properties of the semigroup to calculate the Harte and Taylor spectra of n-tuples of related operators.;We next consider two types of C*-algebras: C*( Cϕ, K ), the C*-algebra generated by the compact operators and a single linear-fractionally-induced composition operator of the form described above, and C*( Fz ), the C*-algebra generated by the collection of all composition operators induced by linear-fractional, non-automorphisms that fix a given point zeta on the unit circle. We first investigate the role that ϕ'(zeta) plays in determining the structure of C*(C ϕ, K ). We then study C*-algebras generated by composition operators induced by parabolic, non-automorphisms and groups of automorphism-induced unitary operators and demonstrate that these C*-algebras are isomorphic, modulo the ideal of compact operators, to crossed product C*-algebras. We use these results to identify C*(Cϕ, K )/ K and C*( Fz )/ K with C*-subalgebras of these crossed product C*-algebras.;Finally, we apply known results for crossed products by the integers to determine the K-theory of C*(Cϕ , K ) and calculate the essential spectra of a class of operators in this C*-algebra.