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.)