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Abstract: This paper addresses the isomorphism problem for the universalnonself-adjoint operator algebras generated by a row contraction subject tohomogeneous polynomial relations. We find that two such algebras areisometrically isomorphic if and only if the defining polynomial relations arethe same up to a unitary change of variables, and that this happens if and onlyif the associated subproduct systems are isomorphic. The proof makes use of thecomplex analytic structure of the character space, together with some recentresults on subproduct systems. Restricting attention to commutative operatoralgebras defined by radical relations yields strong resemblances with classicalalgebraic geometry. These commutative operator algebras turn out to be algebrasof analytic functions on algebraic varieties. We prove a projectiveNullstellensatz connecting closed ideals and their zero sets. Under sometechnical assumptions, we find that two such algebras are isomorphic asalgebras if and only if they are similar, and we obtain a clear geometricalpicture of when this happens. This result is obtained with tools from algebraicgeometry, reproducing kernel Hilbert spaces, and some new complex-geometricrigidity results of independent interest. The C*-envelopes of these algebrasare also determined. The Banach-algebraic and the algebraic classificationresults are shown to hold for the weak-operator closures of these algebras aswell.

Autor: Kenneth R. Davidson, Christopher Ramsey, Orr Shalit


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