In this chapter we cover some of the basics of coarse index theory.

The material in here is for most parts a recollection of my discussions with Johannes.

They might be enriched by

\begin{itemize}

\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}

\end{itemize}

\section{Motivation}% (fold)

\label{sec:motivation}

Clearly we motivate this geometrically via manifolds and their Dirac operators.

So let $M^d$ be a complete Riemannian manifold with a $\Spin$-structure and $D$ the Dirac operator, which we view as a self-adjoint operator acting on $L^2(M;S)$, where $S$ denotes the Spinor bundle.\footnote{a more general setting using some operator of Dirac type is also possible}

Now let $u$ be a smooth section with compact support and consider the \Index{wave equation}\footnote{the classical one from physics is recovered from this by taking the second order derivatives and plugging in the definition of $D$}

\[

\diff{u}{t} u = i D u

\]

We write $u_t = e^{i D t} u$ for the solution of the wave equation.

The main feature of a solution -- which is well known to physicists -- is the property of having \Index{finite propagation}, i.e. the support of $u_t$ is contained in the $t$-neighbourhood of the support of $u$,

\[

\supp(u_t)\subseteq N_t \enbrace*{\supp u}

\]

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 $f \in C_c(\mathbb{R})$.

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 $r>0$.

For the general case we use, that $\varphi\in C_0(\mathbb{R})$ can be approximated arbitrarily well in the $C_0$-norm by a Schwartz function $\psi$, such that $\widehat{\psi}$ has compact support.

We summarise.

\begin{lemma}

$\forall\varepsilon > 0$ and $\varphi\in C_0(\mathbb{R})$ there is an operator $T$ of $L^2(M,S)$ such that $\norm*{\varphi(D)- T} < \varepsilon$ and there exists $r >0$ with