All those properties are the foundation of the following more abstract definition:

\begin{definition}

\begin{definition}\label{def:roe_algebra}

Let $X$ be a proper metric space and $\Hilb$ a Hilbert space together with a *-homomorphism $\rho\colon C_0(X)\to\BH$.

\begin{itemize}

\item An operator $T \in\BH$ is called \Index{locally compact}, if $T f, fT \sim0$ for all $f \in C_0(X)$ (see \cite[Def.~6.3.2]{higson_roe}).

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@@ -85,6 +85,23 @@ This element is called \Index{coarse index}.

\]

\end{remark}

The next main player of coarse index theory is the so-called \emph{localisation algebra}, which we are going to motivate now.

In the same setting as above we now look at the family $\varphi\enbrace*{\frac{1}{t} D}$ for $t \in[1,\infty)$.\todo{in what sense is this \enquote{local}?}

If $\supp\widehat{\varphi}\subseteq(-r,r)$ we once again get