Roe bimodules as morphisms of discrete metric spaces

Abstract

For two discrete metric spaces, $X$ and $Y$ we consider metrics on $X\bigsqcup Y$ compatible with the metrics on $X$ and $Y$. As morphisms from $X$ to $Y$ we consider the Roe bimodules, i.e. the norm closures of bounded finite propagation operators from $l^2(X)$ to $l^2(Y)$. We study the corresponding category $\mathcal{M}$, which is also a 2-category. We show that almost isometries determine morphisms in $\mathcal{M}$. We also consider the case $Y=X$, when there is a richer algebraic structure on the set of morphisms of $\mathcal{M}$: it is a partially ordered semigroup with the neutral element, with involution, and with a lot of idempotents. We also give a condition when a morphism is a $C^{\ast }$-algebra.

10 pages

Cited by in corpus (2)

• Inverse semigroups of metrics on doubles related to certain subsets
• On the inverse semigroup of bimodules over a C*-algebra