The authors consider the following \(\ell_p\)-analogue of Roe algebras:
“Let \((X,d)\) be a metric space with bounded geometry and \(1\leq p<\infty\). For an operator \(T=(T_{xy})_{x,y\in X}\in B(\ell^p(X))\), where \(T_{xy}=(T\delta_y)(x)\), we define the propagation of \(T\) to be \[ \text{prop}(T)=\sup\{ d(x,y):x,y\in X,\,T_{xy}\neq 0 \}\in[0,\infty]. \] We denote by \(\mathbb{C}_u^p[X]\) the unital algebra of all bounded operators on \(\ell^p(X)\) with finite propagation. The \(\ell^p\) uniform Roe algebra, denoted by \(B^p_u(X)\), is defined to be the operator norm closure of \(\mathbb{C}_u^p[X]\) in \(B(\ell^p(X))\).”
Metric spaces considered in the paper are not assumed to have Yu’s property \(A\) or finite decomposition complexity.
The main result of the paper on the uniform Roe algebras is (Theorem 1.7):
“Let \(X\) and \(Y\) be metric spaces with bounded geometry, and let \(p\in [1,\infty)\setminus\{2\}\).

(1)

\(X\) and \(Y\) are bijectively coarsely equivalent if and only if their \(\ell^p\) uniform Roe algebras are isometrically isomorphic.

(2)

\(X\) and \(Y\) are coarsely equivalent if and only if their \(\ell^p\) uniform Roe algebras are stably isometrically isomorphic.”

The authors also consider other Roe-type algebras and prove similar results for them.

Reviewer: Mikhail Ostrovskii (Flushing)

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