The uniform Roe algebra of a uniformly discrete metric space 
of bounded geometry is
the operator-norm closure of the finite-propagation
bounded operators on 
. Finite propagation means
that 
when 
for some finite 
. The matrix entries lie in 
. For a finite
space with 
points, the algebra is 
. The Roe algebra instead
uses 
and requires local compactness of its operators, with
compact operators as matrix
entries. That is the version used in the coarse assembly
maps discussed here.
Uniform Roe Algebra
See also
Bounded Geometry, Compact Operator, Roe Algebra, Uniformly Discrete Metric SpaceExplore with Wolfram|Alpha
References
Willett, R. and Yu, G. "Higher Index Theory for Certain Expanders and Gromov Monster Groups, I." Adv. Math. 229, 1380-1416, 2012. https://doi.org/10.1016/j.aim.2011.10.024.Willett, R. and Yu, G. Higher Index Theory. Cambridge, England: Cambridge University Press, 2020. https://doi.org/10.1017/9781108867351.Cite this as:
Weisstein, Eric W. "Uniform Roe Algebra." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/UniformRoeAlgebra.html