The Fredholm Index for Elements of Toeplitz-Composition C*-Algebras

Abstract

We analyze the essential sectrum and index theory of elements of Toeplitz-composition C*-algebras (algebras generated by the Toeplitz algebra and a single linear-fractional composition operator, acting on the Hardy space of the unit disk). For automorphic composition operators we show that the quotient of the Toeplitz-composition algebra by the compacts is isomorphic to the crossed product C*-algebra for the action of the symbol on the boundary circle. Using this result we obtain sufficient conditions for polynomial elements of the algebra to be Fredholm, by analyzing the spectrum of elements of the crossed product. We also obtain an integral formula for the Fredholm index in terms of a generalized Chern character. Finally we prove an index formula for the case of the non-parabolic, non-automorphic linear fractional maps studied by Kriete, MacCluer and Moorhouse. Mathematics Subject Classification (2000). Primary 47B33; Secondary 47A53, 47L80.

Composition operators within singly generated composition C*-algebras

“…The linear-fractional maps ψ 1 , · · · , ψ m in Equation (13) are taken from the four lists in (6). The maps in each of these lists have a common angular derivative set (a singleton) and a single common first order data vector.…”

“…An appropriate choice of ǫ again yields the inequality (12), where p has the same meaning as there. Again we write A in the form (13), and thus have…”

“…The main result of [15] identifies C * (T z , C ϕ )/K with a certain C * -algebra of 2 × 2 matrix valued functions; see Theorem 4.12 of [15]. The case where ϕ is replaced by an automorphism of D, or even a discrete group of automorphisms, has been studied by Jury [12], [13], and has a rather different character.…”

Toeplitz-composition $C^{*}$-algebras for certain finite Blaschke products

“…If ϕ(z) = e −2πiθ z for some irrational number θ, then the Toeplitz-composition C * -algebra T C ϕ is an extension of the irrational rotation algebra A θ by K and studied by Park [25]. Jury [9,10] investigated the C * -algebra generated by a group of composition operators with the symbols belonging to a non-elementary Fuchsian group Γ to relate it with extensions of the crossed product C(T) Γ by K.…”

Compact Differences of Composition Operators in Several Variables

“…Many properties of composition operators have been studied over the past four decades; the monographs [4] and [14] give an overview of the work before the mid-1990s. Recently there has been considerable interest in studying algebras of composition operators, often modulo the ideal of compact operators (see, for example, [6][7][8][9][10]). In this direction, the question of when a difference C ϕ − C ψ is compact naturally arises.…”