M.Sc Thesis | |

M.Sc Student | Gurevich Maxim |
---|---|

Subject | Ideals in Non-Commutative Hardy Algebras and Subproduct Systems |

Department | Department of Mathematics |

Supervisor | PROFESSOR EMERITUS Baruch Solel |

Full Thesis text |

The classical Hardy algebra *H*** ^{∞}** has an operator-theoretical
interpretation as the weak-* closed unital operator algebra generated by the
right shift operator on

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.