Friday, June 25, 2010

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.)

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