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partial translation algebras for certain discrete metric spaces

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
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.

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Published date: June 2010
Organisations: University of Southampton

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Local EPrints ID: 170227
URI: https://eprints.soton.ac.uk/id/eprint/170227
PURE UUID: 33e9d084-a8a3-442b-bef5-15bb6a68c1e0

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Date deposited: 17 Jan 2011 16:47
Last modified: 18 Jul 2017 12:17

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