partial translation algebras for certain discrete metric spaces

The notion of a partial translation algebra was introduced by Brodzki, Niblo<br/>and Wright in [11] to provide an analogue of the reduced group C*-algebra<br/>for metric spaces. Such an algebra is constructed from a partial translation<br/>structure, a structure which any bounded geometry uniformly discrete metric<br/>space admits; we prove that these structures restrict to subspaces and are<br/>preserved by uniform bijections, leading to a new proof of an existing theorem.<br/>We examine a number of examples of partial translation structures and<br/>the algebras they give rise to in detail, in particular studying cases where<br/>two different algebras may be associated with the same metric space. We<br/>introduce the notion of a map between partial translation structures and use<br/>this to describe when a map of metric spaces gives rise to a homomorphism<br/>of related partial translation algebras. Using this homomorphism, we construct<br/>a C*-algebra extension for subspaces of groups, which we employ to<br/>compute K-theory for the algebra arising from a particular subspace of the<br/>integers. We also examine a way to form a groupoid from a partial translation<br/>structure, and prove that in the case of a discrete group the associated<br/>C*-algebra is the same as the reduced group C*-algebra. In addition to<br/>this we present several subsidiary results relating to partial translations and<br/>cotranslations and the operators these give rise to.

Name dc:type.qualificationname

Ph.D.

Level dc:type.qualificationlevel

doctoral

Grantor dc:publisher.institution

University of Southampton

Year dc:date.issued

2010

Author dc:creator

• Putwain, Rosemary Johanna

Advisor dc:contributor.advisor

• Brodzki, Jacek