Commit 88ee9807 authored by Jannes Bantje's avatar Jannes Bantje

introduce the coarse index

parent 67dc224b
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......@@ -10,9 +10,11 @@ They might be enriched by
\item the book \citetitle{WilletYu} by \textcite{WilletYu}
\item the notes from Rudi's lecture on that topic
\item references to \citetitle{higson_roe} by \textcite{higson_roe}
\item references to \citetitle{Higson_guentner} by \textcite{Higson_guentner}
\item references to \citetitle{Higson_guentner} by \textcite{Higson_guentner} for the spectral picture of \K-theory, which is central to coarse index theory.
\end{itemize}
\todo[inline]{coarse index theory can deal with open manifolds rather well --- this is not the case for the classical theory!}
\section{Motivation} % (fold)
\label{sec:motivation}
......@@ -30,7 +32,7 @@ The main feature of a solution -- which is well known to physicists -- is the pr
See \cite[Prop.~10.3.1]{higson_roe}.
In this construction we used the functional calculus with $e^{ixt}$ and we want to generalise the finite propagation property to other functions, namely to functions $\varphi \in \mathcal{S} = C_0(\mathbb{R})$.
These are the functions that come up in the spectral picture of \K-theory.
Since $\varphi$ is a $C_0$ function we know, that $\varphi(D) \cdot f$ and $ f \cdot \varphi(D)$ are compact for $f \in C_c(\mathbb{R})$.
Since $\varphi$ is a $C_0$ function we know, that $\varphi(D) \cdot f$ and $ f \cdot \varphi(D)$ are compact for $f \in C_c(\mathbb{R})$ (see \cite[Prop.~10.5.1]{higson_roe}).
For the time being we assume, that $\varphi$ is of Schwartz class such that its Fourier transform has compact support, $\supp(\widehat{\varphi}) \subseteq \benbrace*{-r,r}$ for some $r>0$.
Hence we may write
......@@ -54,7 +56,35 @@ We summarise.
\supp(T u) \subseteq N_R \enbrace*{\supp u}
\]
\end{lemma}
All those properties are the foundation of the following more abstract definition:
\begin{definition}
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}).
\item $T \in \BH$ has \Index{finite propagation} $\prop(T) \le R$, if for all $u \in \Hilb$
\[
\supp (Tu) \subseteq N_R \enbrace*{\supp u}
\]
\end{itemize}
The \Index{Roe-Algebra} is the \Cstar-algebra
\[
\Roe(X;\Hilb) \coloneqq \overline{\set*{T \in \BH \given T \text{ locally compact and } \prop(T) < \infty}}
\]
\end{definition}
In the setting of $M^d$ being a complete $\Spin$-manifold, the homomorphism $\varphi \mapsto \varphi(D)$ gives an element in the spectral picture of the \K-theory of the Roe-algebra associated to $M$.
This element is called \Index{coarse index}.
\begin{remark}
If $X$ is compact any locally compact operator is compact already and every operator $T$ fulfils $\prop(T) < \diam(X) < \infty$ and we see that the Roe-algebra ist just the compact operators.
\[
\Roe(X,\Hilb) = \mathcal{K}(\Hilb)
\]
\end{remark}
% section motivation (end)
\newpage
\section{The General Theory} % (fold)
\label{sec:the_general_theory}
......
......@@ -118,6 +118,8 @@
\newcommand{\HA}{\mathcal{H}_{A}}
\newcommand{\HB}{\mathcal{H}_{B}}
\newcommand{\plus}{{\hspace{-1.6pt}+}}
\newcommand{\Roe}{\Cstar_{\mathrm{Roe}}}
\DeclareMathOperator{\prop}{prop}
\DeclarePairedDelimiterX\bra[1]{\langle}{\rvert}{#1\,}
\DeclarePairedDelimiterX\ket[1]{\lvert}{\rangle}{\,#1}
......@@ -221,6 +223,8 @@
\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\im}{im}
\newcommand{\Hilb}{\mathcal{H}}
\newcommand{\Bop}{\mathcal{B}}
\newcommand{\BH}{\Bop(\Hilb)}
\newcommand{\Inf}{I\mkern-.1mu n\mkern-.5mu f}
\newcommand{\Infhat}{I\mkern-.1mu \widehat{n}\mkern-.5mu f}
......
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