A Roe algebra of a uniformly discrete metric space 
of bounded geometry is a completion of its algebra
of locally compact finite-propagation operators.
Use the Hilbert space 
.
Local compactness requires 
and 
to be compact
for every 
. Finite propagation
requires a bound 
with 
whenever 
.
The matrix entries are compact
operators, without a uniform bound on their ranks.
The reduced Roe algebra 
is the operator-norm closure of this algebra. The
maximal Roe algebra 
instead uses the universal 
-norm. There is a canonical quotient from the maximal completion
to the reduced one. These are locally compact Roe algebras, distinct from the uniform Roe algebra built using entries in 
on 
.
The different completions give different assembly maps
in coarse geometry.
