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