Representations of the Polydisc Algebra

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Abstract Ferguson The proposed research is to study representations of the polydisc algebra on Hilbert space using cohomological tools together with techniques from operator theory and operator algebras. Problems in dilation theory, as well as, joint similarity problems for commuting N-tuples of operators, can be formulated and studied in the general framework of bounded cohomology. The cohomological groups can be realized concretely as quotients of bounded operators and thus these groups, as well as, the techniques used to compute them, should be of interest to both operator theorists and operator algebraists. A significant portion of the cohomology theory developed for Banach modules is not, in general, applicable in the study of Hilbert modules over operator algebras. Consequently, there are no standard techniques one can use to compute cohomology groups. The methods employed so far involve homological algebra together with operator theoretic techniques, two seemingly disparate areas of mathematics. Consequently, computations of these groups leads to new insight and new techniques in operator theory. Also important is that certain problems in operator theory when formulated in the general context of bounded cohomology become more transparent and simple algebraic computation often leads to a significant reduction in the problem. For this reason alone, bounded cohomolgy for Hilbert modules will likely become a powerful tool for those working in operator theory.