partial translation algebras for certain discrete metric spaces

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

Identifiers

ORCID for Jacek Brodzki: orcid.org/0000-0002-4524-1081

Catalogue record

Export record