Integral Equations and Operator Theory

, Volume 58, Issue 3, pp 341–362

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


    • Department of MathematicsUniversity of Florida

DOI: 10.1007/s00020-007-1494-0

Cite this article as:
Jury, M.T. Integr. equ. oper. theory (2007) 58: 341. doi:10.1007/s00020-007-1494-0


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 47B33Secondary 47A53, 47L80


Composition operatorToeplitz operatorFredholm operatorFredholm index
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© Birkhäuser Verlag Basel/Switzerland 2007