# C*-algebras of the planar crystal groups and their irreducible *-representations

## Date

1990-08-01

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Masters

## 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 compact abelian group A is the space Cₒ(Â) for the pontryagin dual Â. This result is extended in
chapter five (see [14]) to a general locally compact group G that has an abelian normal subgroup of
finite index. In fact, C* (G) is tied to a wonderfully explicit formulation via [14].
Chapter five also ties together C*(G) with the section C*-Algebra of a C*-bundle. Evans' ([3]) analysis of compact symmetry groups is modified for this result. Then, with this bundle identification, the irreducible representations of C* (G) are readily accessible in the case of a crystal group's C*-Algebra.
Chapter four essentially lists the planar crystal groups for use in the sequel. Symmetry, in general, is one of the most fascinating areas in the known universe and finds its expression in various forms. Hilbert's eighteenth problem, which deals with bounded symmetry and was solved by Bieberbach around 1910, shows the restrictive nature of symmetry. Yet symmetry, balance and uniformity seem to be universal characteristics of all systems, in some sense.
In chapter six, this thesis concretely carries out part of the program of representation theory. The
C*-Algebra C*(G) is constructed for each of the seventeen planar crystal groups G. For each C*(G),
the irreducible *-representation set is described. These results form an important set of examples of
non-trivial, non-commutative C*-Algebras of infinite dimension and also provide interesting examples
of the structure sets IRR(C*(G)). The methods employed are not restricted to planar groups, but could be applied to any crystal group (although the computation time required would grow quite quickly).
A few concluding remarks will now be made. The structure set and the description of C* (G) lend themselves quite readily to calculation of the topology on the structure space, which is a further objective of representation theory. Lastly, there is a natural embedding of the group G itself into
L¹(G)- by point-mass functions x → δₓ -and then the unitary representations of G can be deduced from the structure set of C* (G) . So there are at least these two further applications which could
be carried out from the information provided in this thesis.

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Master of Science (M.Sc.)

## Department

Mathematics

## Program

Mathematics