# Exchange rings and translation algebras

Recall that a ring $$R$$ is von Neumann regular if given any $$x \in R$$ there is $$y\in R$$ such that $$xyx=x$$. (Examples: fields; matrix algebras; various rings of unbounded operators, where $$y$$ is “the inverse of $$x$$ away from the kernel”.) A ring $$R$$ is called an exchange ring if, for every $$x\in R$$, there is an idempotent $$e\in R$$ such that $$e\in xR$$ and $$(1-e)\in (1-x)R$$. Von Neumann regular rings are examples of exchange rings. There is a developed algebraic theory of these rings, with links to real rank 0 C*-algebras among other things.

I just became aware of the paper Ara, P., K. C O’Meara, and F. Perera. Gromov translation algebras over discrete trees are exchange rings. Transactions of the American Mathematical Society 356, no. 5 (2004): 2067–2079 (electronic). In this paper it is shown that the algebraic translation algebra associated to a tree, with coefficients in a von Neumann regular ring, must be an exchange ring. The tree hypothesis is used in various places, but the authors don’t know, apparently, whether there are examples of metric spaces $$X$$ for which the translation algebra is not an exchange ring. (The plane might be a good example to start with.)