Commit 88ee9807 by Jannes Bantje

### introduce the coarse index

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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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