M.Sc Student M.Sc Thesis Gurevich Maxim Ideals in Non-Commutative Hardy Algebras and Subproduct Systems Department of Mathematics PROFESSOR EMERITUS Baruch Solel

Abstract

The classical Hardy algebra H has an operator-theoretical interpretation as the weak-* closed unital operator algebra generated by the right shift operator on l2. Given a W*-correspondence (bi-module) E over a von-Neumann algebra, we can define a non-commutative Hardy algebra H(E) by taking the weak-* closure of the shift operators on the tensor powers of E. This construction gives a large class of non-self-adjoint operator algebras that encompasses many examples of independent interest, such as the analytic non-commutative Toeplitz algebras or graph operator algebras. We study the ideal structure of these algebras, and the corresponding quotient algebras.

We show that when the non-commutative Hardy algebra is represented on a Hilbert space in a canonical way, the lattice of its weak-* closed ideals becomes isomorphic to the lattice of subspaces of this Hilbert space that are invariant for both the algebra and its commutant. Furthermore, we show that the compression of the Hardy algebra onto the complement of such subspace gives a completely isometric and weak-* homeomorphic realization of the corresponding quotient algebra. Some implications for graph algebras are exhibited.

We point to a tight relation of some of these quotient algebras to algebras arising from subproduct systems. A subproduct system over N (the monoid of natural numbers) is a sequence of W*-correspondences { X(n) }n , such that X(m n) is a subspace of the tensor product of X(m) and X(n). This structure gives rise to shift operators between the X(n)'s, which generate operator algebras. We show that the weak-* closure of these algebras serve as a realization of homogeneous quotients of Hardy algebras.

We also examine to what extent similar results remain valid for subproduct systems of Hilbert spaces over N x N. The basics of this theory are developed, with emphasis on the tensor algebra of a given subproduct system, i.e. the norm-closed algebra generated by the shifts. By comparing it to a quotient of a better known algebra, we find that the character spaces of these tensor algebras are homeomorphic to Euclidean algebraic varieties intersected with a unit ball. This identification is applied to achieve some results regarding subproduct systems whose tensor algebras are isomorphic.