2012/05/25 by Quertermous, Katie S.
#47A53 #47L80 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #Primary 47B33 #Secondary 46L80
paper · doi:10.48550/arxiv.1205.5786
Let φ be a linear-fractional, non-automorphism self-map of \mathbbD that fixes ζ∈ \mathbbT and satisfies φ′(ζ) ≠ 1 and consider the composition operator Cφ acting on the Hardy space H2(\mathbbD). We determine which linear-fractionally-induced composition operators are contained in the unital C^*-algebra generated by Cφ and the ideal K of compact operators. We apply these results to show that C^*(Cφ, K) and C^*(Fζ), the unital C^*-algebra generated by all composition operators induced by linear-fractional, non-automorphism self-maps of \mathbbD that fix ζ, are each isomorphic, modulo the ideal of compact operators, to a unitization of a crossed product of C0([0,1]). We compute the K-theory of C^*(Cφ, K) and calculate the essential spectra of a class of operators in this C^*-algebra. We also obtain a full description of the structures, modulo the ideal of compact operators, of the C^*-algebras generated by the unilateral shift Tz and a single linear-fractionally-induced composition operator.