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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 $G(0)$ of $\omega\times\omega$ matrices (over a field) of constant bandwidth. An extension of these ideas shows that for all real numbers $r$ in the unit interval $[0,1]$, the growth algebras $G(r)$(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 $r$in $[0,1]$.


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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

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