Abstract
If b is an inner function, then composition with b induces an endomorphism, β, of \({L^\infty({\mathbb{T}})}\) that leaves \({H^\infty({\mathbb{T}})}\) invariant. We investigate the structure of the endomorphisms of \({B(L^2({\mathbb{T}}))}\) and \({B(H^2({\mathbb{T}}))}\) that implement β through the representations of \({L^\infty({\mathbb{T}})}\) and \({H^\infty({\mathbb{T}})}\) in terms of multiplication operators on \({L^2({\mathbb{T}})}\) and \({H^2({\mathbb{T}})}\) . Our analysis, which is based on work of Rochberg and McDonald, will wind its way through the theory of composition operators on spaces of analytic functions to recent work on Cuntz families of isometries and Hilbert C*-modules.
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Communicated by Palle Jorgensen.
DC and SS were partially supported by the University of Iowa Department of Mathematics NSF VIGRE grant DMS-0602242.
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Courtney, D., Muhly, P.S. & Schmidt, S.W. Composition Operators and Endomorphisms. Complex Anal. Oper. Theory 6, 163–188 (2012). https://doi.org/10.1007/s11785-010-0075-4
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DOI: https://doi.org/10.1007/s11785-010-0075-4