Commit ad32adb2 authored by Jannes Bantje's avatar Jannes Bantje

gave a definition of the localisation algebra

parent 9a63215f
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......@@ -58,7 +58,7 @@ We summarise.
\end{lemma}
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 $T f, fT \sim 0$ for all $f \in C_0(X)$ (see \cite[Def.~6.3.2]{higson_roe}).
......@@ -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
\[
\varphi \enbrace*{\frac{1}{t} D} = \frac{1}{\sqrt{2 \pi_n}} \int\limits_{-r}^r \widehat{\varphi}(\xi) e^{i \xi \frac{1}{t} D} \, \mathd \xi = \frac{1}{\sqrt{2 \pi_n}} \int\limits_{-r/t}^{r/t} \widehat{\varphi}(t \cdot \eta) e^{i \xi \eta D} \, \mathd \xi
\]
\begin{definition}
In the same setting as in \cref{def:roe_algebra} we define the \Index{localisation algebra} of $(X,\Hilb)$ to be
\[
\Cstar_L(X; \Hilb) \coloneqq \overline{\set*{L \colon [1,\infty) \to \BH \text{ uniformly cont., $L(t)$ loc. compact and } \prop(L(t)) \grenzw{t \to \infty} 0 }}
\]
\end{definition}
There is an obvious *-homomorphism $\Cstar_L(X;\Hilb) \to \Roe(X;\Hilb)$ given by evaluation at $t=1$.
In our motivational setting this should be seen as a map from the local index to the coarse index.
% section motivation (end)
\newpage
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