partial translation algebras for certain discrete metric spaces

partial translation algebras for certain discrete metric spaces

The notion of a partial translation algebra was introduced by Brodzki, Niblo

and Wright in [11] to provide an analogue of the reduced group C*-algebra

for metric spaces. Such an algebra is constructed from a partial translation

structure, a structure which any bounded geometry uniformly discrete metric

space admits; we prove that these structures restrict to subspaces and are

preserved by uniform bijections, leading to a new proof of an existing theorem.

We examine a number of examples of partial translation structures and

the algebras they give rise to in detail, in particular studying cases where

two different algebras may be associated with the same metric space. We

introduce the notion of a map between partial translation structures and use

this to describe when a map of metric spaces gives rise to a homomorphism

of related partial translation algebras. Using this homomorphism, we construct

a C*-algebra extension for subspaces of groups, which we employ to

compute K-theory for the algebra arising from a particular subspace of the

integers. We also examine a way to form a groupoid from a partial translation

structure, and prove that in the case of a discrete group the associated

C*-algebra is the same as the reduced group C*-algebra. In addition to

this we present several subsidiary results relating to partial translations and

cotranslations and the operators these give rise to.

Putwain, Rosemary Johanna

f3a91697-f258-4fdf-a024-e5de09c2522c

June 2010

Putwain, Rosemary Johanna

f3a91697-f258-4fdf-a024-e5de09c2522c

Brodzki, Jacek

b1fe25fd-5451-4fd0-b24b-c59b75710543

Putwain, Rosemary Johanna
(2010)
partial translation algebras for certain discrete metric spaces.
*University of Southampton, School of Mathematics, Doctoral Thesis*, 146pp.

Record type:
Thesis
(Doctoral)

## Abstract

The notion of a partial translation algebra was introduced by Brodzki, Niblo

and Wright in [11] to provide an analogue of the reduced group C*-algebra

for metric spaces. Such an algebra is constructed from a partial translation

structure, a structure which any bounded geometry uniformly discrete metric

space admits; we prove that these structures restrict to subspaces and are

preserved by uniform bijections, leading to a new proof of an existing theorem.

We examine a number of examples of partial translation structures and

the algebras they give rise to in detail, in particular studying cases where

two different algebras may be associated with the same metric space. We

introduce the notion of a map between partial translation structures and use

this to describe when a map of metric spaces gives rise to a homomorphism

of related partial translation algebras. Using this homomorphism, we construct

a C*-algebra extension for subspaces of groups, which we employ to

compute K-theory for the algebra arising from a particular subspace of the

integers. We also examine a way to form a groupoid from a partial translation

structure, and prove that in the case of a discrete group the associated

C*-algebra is the same as the reduced group C*-algebra. In addition to

this we present several subsidiary results relating to partial translations and

cotranslations and the operators these give rise to.

## More information

Published date: June 2010

Organisations:
University of Southampton

## Identifiers

Local EPrints ID: 170227

URI: https://eprints.soton.ac.uk/id/eprint/170227

PURE UUID: 33e9d084-a8a3-442b-bef5-15bb6a68c1e0

## Catalogue record

Date deposited: 17 Jan 2011 16:47

Last modified: 18 Jul 2017 12:17

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## Contributors

Author:
Rosemary Johanna Putwain

## University divisions

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