Noncommutative Hardy algebras, multipliers, and quotients

The principal objects of study in this thesis are the noncommutative Hardy algebras introduced by Muhly and Solel in 2004, also called simply “Hardy algebras,” and their quotients by ultraweakly closed ideals. The Hardy algebras form a class of nonselfadjoint dual operator algebras that generalize the classical Hardy algebra, the noncommutative analytic Toeplitz algebras introduced by Popescu in 1991, and other classes of operator algebras studied in the literature. It is known that a quotient of a noncommutative analytic Toeplitz algebra by a weakly closed ideal can be represented completely isometrically as the compression of the algebra to the complement of the range of the ideal, but there is no known general extension of this result to Hardy algebras. An analogous problem on representing quotients of Hardy algebras as compressions of images of induced representations is considered in Chapter 2. Using Muhly and Solel’s generalization of Beurling’s theorem together with factorizations of weakly continuous linear functionals on infinite multiplicity operator spaces, it is shown that compressing onto the complement of the range of an ultraweakly closed ideal in the space of an infinite multiplicity induced representation yields a completely isometric isomorphism of the quotient. A generalization of Pick’s interpolation theorem for elements of Hardy algebras evaluated on their spaces of representations was proved by Muhly and Solel. In Chapter 3, a general theory of reproducing kernel W ∗-correspondences and their multipliers is developed, generalizing much of the classical theory of reproducing kernel Hilbert space. As an application, it is shown using the generalization of Pick’s theorem that the function space representation of a Hardy algebra is isometrically isomorphic (with its quotient norm) to the multiplier algebra of a reproducing kernel