Gromov translation algebras over discrete trees are exchange rings

Authors:
P. Ara, K. C. O'Meara and F. Perera

Journal:
Trans. Amer. Math. Soc. **356** (2004), 2067-2079

MSC (2000):
Primary 16E50, 16D70, 16S50

DOI:
https://doi.org/10.1090/S0002-9947-03-03372-5

Published electronically:
November 12, 2003

MathSciNet review:
2031053

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Abstract: It is shown that the Gromov translation ring of a discrete tree over a von Neumann regular ring is an exchange ring. This provides a new source of exchange rings, including, for example, the algebras of matrices (over a field) of constant bandwidth. An extension of these ideas shows that for all real numbers in the unit interval , the growth algebras (introduced by Hannah and O'Meara in 1993) are exchange rings. Consequently, over a countable field, countable-dimensional exchange algebras can take any prescribed bandwidth dimension in .

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Additional Information

**P. Ara**

Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Bellaterra (Barcelona), Spain

Email:
para@mat.uab.es

**K. C. O'Meara**

Affiliation:
Department of Mathematics, University of Canterbury, Christchurch, New Zealand

Address at time of publication:
Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269

Email:
K.OMeara@math.canterbury.ac.nz, staf198@ext.canterbury.ac.nz

**F. Perera**

Affiliation:
Department of Pure Mathematics, Queen’s University Belfast, Belfast, BT7 1NN, Northern Ireland

Address at time of publication:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain

Email:
perera@qub.ac.uk, perera@mat.uab.es

DOI:
https://doi.org/10.1090/S0002-9947-03-03372-5

Keywords:
Translation algebra,
exchange ring,
von Neumann regular ring,
infinite matrices,
bandwidth dimension

Received by editor(s):
September 27, 2002

Received by editor(s) in revised form:
April 15, 2003

Published electronically:
November 12, 2003

Additional Notes:
The first and third authors were partially supported by DGESIC, and by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya. The initial ideas for this paper were discussed while the second author was visiting the Centre de Recerca Matemàtica, Institut d’Estudis Catalans in Barcelona, and he thanks this institution for its support and hospitality.

Article copyright:
© Copyright 2003
American Mathematical Society