C*-algebras of the planar crystal groups and their irreducible *-representations
type: CreativeWork
description: The main result of this thesis is the explicit construction of the group C*-Algebra C*(G) for each of the seventeen planar crystal groups G. A secondary result is the explicit description of all the irreducible *-representations of these C*-Algebras. Two further applications are described in the concluding paragraph below. The study of generalized representation theory arose from the study of unitary representations of groups. Quantum theory, for example, is where representation theory of groups finds its most important physical application ([6], p.29). Moving up to the general representation theory on *algebras can often solve a problem which yields a solution in the group case by restriction. Thus, one finds, for example, titles such as "Operator Algebras and Quantum Statistical Mechanics", and there is a strong connection between representations of groups and *-representations of *-algebras. Chapter two is basic in nature. The definitions and results for Hilbert spaces, C*-Algebras, and C*-Bundles are laid down. The major result is the Gelfand-Neumark theorem which says, basically, that C*-Algebras are not more general than, and, in fact, are subsumed by, operator *-algebras on Hilbert spaces. A large portion of the chapter is devoted to finite-dimensional C*-Algebras since this case is required in later chapters for finding the structure set of C*(G). Chapter three introduces representations of *-algebras. One of the long range objectives of representation theory is to classify all the *-representations of a given *-algebra ([5], p.416). Any *-algebra gives rise to a C*-algebra, and the *-representations of each have a natural one to one correspondence. So, "from a representation theoretic viewpoint", *-algebras are not more general than C*-algebras. In the finite dimensional case, a C*-Algebra has a corresponding finite condition on its irreducible representations. Chapter three also gives a development for the result that the group C*-Algebra for a locally com…
abstract: The main result of this thesis is the explicit construction of the group C*-Algebra C*(G) for each of the seventeen planar crystal groups G. A secondary result is the explicit description of all the irreducible *-representations of these C*-Algebras. Two further applications are described in the concluding paragraph below. The study of generalized representation theory arose from the study of unitary representations of groups. Quantum theory, for example, is where representation theory of groups finds its most important physical application ([6], p.29). Moving up to the general representation theory on *algebras can often solve a problem which yields a solution in the group case by restriction. Thus, one finds, for example, titles such as "Operator Algebras and Quantum Statistical Mechanics", and there is a strong connection between representations of groups and *-representations of *-algebras. Chapter two is basic in nature. The definitions and results for Hilbert spaces, C*-Algebras, and C*-Bundles are laid down. The major result is the Gelfand-Neumark theorem which says, basically, that C*-Algebras are not more general than, and, in fact, are subsumed by, operator *-algebras on Hilbert spaces. A large portion of the chapter is devoted to finite-dimensional C*-Algebras since this case is required in later chapters for finding the structure set of C*(G). Chapter three introduces representations of *-algebras. One of the long range objectives of representation theory is to classify all the *-representations of a given *-algebra ([5], p.416). Any *-algebra gives rise to a C*-algebra, and the *-representations of each have a natural one to one correspondence. So, "from a representation theoretic viewpoint", *-algebras are not more general than C*-algebras. In the finite dimensional case, a C*-Algebra has a corresponding finite condition on its irreducible representations. Chapter three also gives a development for the result that the group C*-Algebra for a locally com…
author: Pohorecky, Eric M.
language: en_US
date published: 1990-08-01
date modified: 2026-07-24T04:26:52Z
provider: University of Saskatchewan
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: University of Saskatchewan

PropertyValue
property ID: Handle
value: https://hdl.handle.net/10388/etd-06122012-135102