Composition Operators and Endomorphisms

If b is an inner function, then composition with b induces an endomorphism, β, of L^￥(\mathbbT){L^\infty({\mathbb{T}})} that leaves H^￥(\mathbbT){H^\infty({\mathbb{T}})} invariant. We investigate the structure of the endomorphisms of B(L²(\mathbbT)){B(L^2({\mathbb{T}}))} and B(H²(\mathbbT)){B(H^2({\mathbb{T}}))} that implement β through the representations of L^￥(\mathbbT){L^\infty({\mathbb{T}})} and H^￥(\mathbbT){H^\infty({\mathbb{T}})} in terms of multiplication operators on L²(\mathbbT){L^2({\mathbb{T}})} and H²(\mathbbT){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.