Hilbert modules over semicrossed products of the disk algebra

Thumbnail Image
Date
1997
Authors
Buske, Dale
Major Professor
Advisor
Justin R. Peters
Committee Member
Journal Title
Journal ISSN
Volume Title
Publisher
Altmetrics
Authors
Research Projects
Organizational Units
Organizational Unit
Journal Issue
Is Version Of
Versions
Series
Department
Mathematics
Abstract

Given the disk algebra A( I !D) and an automorphism [alpha], there is associated a non-self-adjoint norm closed subalgebra doubz+x[alpha]A( I !D) of the crossed product doubzx[alpha]C( T) called the semicrossed product of A( I !D) with [alpha]. It is well known that the automorphisms of A( I !D) arise via composition with conformal bijections [phi] of I !D. These automorphisms are labeled according to the corresponding conformal maps as parabolic, hyperbolic, or elliptic and each case is studied. The contractive and completely contractive representations of doubz+x[alpha]A( I !D) on a Hilbert space H (i.e. contractive Hilbert modules) are found to be in a one-to-one correspondence with pairs of contractions S and T on H satisfying TS=S[phi](T). To this end, a noncommutative dilation result is obtained. It states that given a pair of contractions S and T on H satisfying TS=S[phi](T) there exist a pair of unitaries U and V on K supseteqH satisfying VU=U[phi](V) and dilating S and T respectively. Some concrete representations of doubz+x[alpha]A( I !D) are then found in order to compute the characters, the maximal ideal space, and the strong radical. The Shilov and orthoprojective Hilbert modules over doubz+x[alpha]A( I !D) are shown to correspond to pairs of isometries S and T satisfying TS=S[phi](T).

Comments
Description
Keywords
Citation
Source
Subject Categories
Keywords
Copyright
Wed Jan 01 00:00:00 UTC 1997