# A Lefschetz fixed-point formula for noncompact fixed-point sets

Peter Hochs111Radboud University<EMAIL_ADDRESS>
###### Abstract

We obtain an equivariant index theorem, or Lefschetz fixed-point formula, for
isometries from complete Riemannian manifolds to themselves. The fixed-point
set of such an isometry may be noncompact. We build on techniques developed by
Roe. Key new ingredients are a localised functional on operators with bounded
smooth kernels, and an algebra (reminiscent of Yu’s localisation algebra) of
“asymptotically local” operators, on which this functional has an asymptotic
trace property. As consequences, we show that some earlier indices used are
special cases of the one we introduce here, and obtain an obstruction to
positive scalar curvature.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Dirac operators
    2. 2.2 Exhaustions
    3. 2.3 A functional
  3. 3 Asymptotically local operators
    1. 3.1 Uniform operators
    2. 3.2 Asymptotically local operators
    3. 3.3 A subalgebra
  4. 4 A $K$-theoretic index
    1. 4.1 Functional calculus
    2. 4.2 Heat operators
    3. 4.3 The index
  5. 5 An asymptotic trace property
    1. 5.1 Estimates of partial integrals
    2. 5.2 Proof of Theorem 5.1
    3. 5.3 The trace of the index of $D$
  6. 6 An index theorem
    1. 6.1 The main result
    2. 6.2 Consequences and applications

##  1 Introduction

### Background

A Lefschetz fixed-point theorem, or equivariant index theorem, expresses the
graded trace of a geometric operator on cohomology, or on the kernel of an
elliptic operator, in topological terms involving the fixed-point set of the
map inducing such an operator. Important early results are the ones by Atiyah
and Bott [1, 2] and by Atiyah, Singer and Segal [3, 4]. Since then, more than
100 papers on this subject have appeared, with various generalisations and
applications.

Our focus in this paper is a Lefschetz fixed point formula on noncompact
manifolds. Such results have been obtained before, but, to our knowledge, in
most of these results it is either assumed that the fixed-point set in
question is compact, as for example in [7, 10], or one considers a proper
group action with compact quotient, as for example in [9, 11, 14]. The main
result in this paper is a Lefschetz fixed-point theorem without such
compactness assumptions.

The approach is to build on Roe’s index theorem on noncompact maniofolds [12,
13]. He used functionals associated to certain exhaustions of manifolds by
compact sets. Roe proved that these functionals are traces on certain smooth
precursors of the Roe algebra, which allowed him to extract a number from a
$K$-theoretic index, and obtain an index formula.

A key point in our equivariant setting, is that we need to use _localised_
versions of Roe’s functionals, to obtain a nontrivial result. (More on this
below.) This means that the relevant analogue of Roe’s functional is no longer
a trace. Our solution is to define a suitable algebra of _asymptotically
local_ paths of operators, on which this functional has an asymptotic trace
property. Constructing the right algebra for this purpose was the main
technical challenge in constructing an index and obtaining our main result:
the algebra should be large enough so that its $K$-theory contains indices of
Dirac operators, and small enough for the asymptotic trace property to hold.

The algebras of “uniform operators” in [12] later led to the development of
the Roe algebra, which has found applications in many places. We found it
interesting that our search for an algebra on which our localised functional
has an asymptotic trace property naturally led to a smooth analogue of Yu’s
localisation algebras. See [15] for an overview of index theory based on Roe
algebras and localisation algebras.

### Results

Let $M$ be a complete Riemannian manifold of bounded geometry, and
$\varphi\colon M\to M$ an isometry. Let $D$ be a Dirac-type operator on a
vector bundle $S\to M$, and assume that $D$ commutes with a lift $\Phi$ of
$\varphi$ to $S$. Suppose that $D$ is odd with respect to a $\Phi$-invariant
grading on $S$. Let $(M_{j})_{j=1}^{\infty}$ be an increasing sequence of
compact subsets of $M$ whose union is all of $M$. We use a neighbourhood $U$
of the fixed-point set $M^{\varphi}$ of $\varphi$, and a functional $I$ on
bounded densities on $M$ associated to the sequence of averages

$\frac{1}{\operatorname{vol}(M_{j}\cap U)}\int_{M_{j}\cap U}\alpha,$

for bounded densities $\alpha$. For an operator $A$ with smooth, bounded
Schwartz kernel $\kappa$, we define the trace-like number
$\operatorname{Tr}^{U}_{\Phi}(A)$ as the value of $I$ on the density
$m\mapsto\operatorname{tr}(\Phi\kappa(\varphi^{-1}(m),m))dm$, where $dm$ is
the Riemannian density.

A central result in this paper is the construction of an algebra
$\mathcal{A}_{-\infty}^{L}(S)^{\Phi}$ (see Definition 3.7) of paths of
asymptotically local operators $t\mapsto A(t)$, on which
$\operatorname{Tr}^{U}_{\Phi}$ is a trace “in the limit $t\downarrow 0$” (see
Theorem 5.1). The Dirac operator $D$ has an index

$\operatorname{index}^{L}(D)\in K_{0}(\mathcal{A}_{-\infty}^{L}(S)^{\Phi}).$

The asymptotic trace property of $\operatorname{Tr}^{U}_{\Phi}$ allows us to
apply this functional to $\operatorname{index}^{L}(D)$. Our main result is an
equivariant index formula of the form

$\operatorname{Tr}^{U}_{\Phi}(\operatorname{index}^{L}(D))=\lim_{j\to\infty}\frac{1}{\operatorname{vol}(M_{j}\cap
U)}\int_{(M_{j}\cap U)^{\varphi}}\operatorname{AS}_{\Phi}(D),$ (1.1)

where $\operatorname{AS}_{\Phi}(D)$ is the classical Atiyah–Segal–Singer
integrand for $D$, and we assume that the right hand side converges. See
Theorem 6.3 for the precise formulation, assumptions are that $\varphi$
preserves an orientation and lies in a compact group of isometries.

Both sides of (1.1) depend on $U$. This is deliberate, and it is important to
allow $U$ to be different from $M$. Indeed, as $\operatorname{AS}_{\Phi}(D)$
is bounded in our setting, the right hand side of (1.1) for $U=M$ is bounded
by a constant times

$\frac{\operatorname{vol}(M_{j}^{\varphi})}{\operatorname{vol}(M_{j})}.$

In general, we expect this ratio to go to zero as $j\to\infty$ in many cases,
because $M^{\varphi}$ has lower dimension than $M$ in general. Taking $U$ to
be a tubular neighbourhood of $M^{\varphi}$, for example, is more likely to
lead to nontrivial results.

As consequences of (1.1), we show that the indices from [10, 12] are special
cases of the index we consider here. In the case where $M^{\varphi}$ and $U$
are compact, we find that

$\operatorname{Tr}^{U}_{\Phi}(\operatorname{index}^{L}(D))=\frac{1}{\operatorname{vol}(U)}\int_{M^{\varphi}}\operatorname{AS}_{\Phi}(D).$

And if $M$ is $\operatorname{Spin}$ and has uniformly positive scalar
curvature, then the limit

$\lim_{j\to\infty}\frac{1}{\operatorname{vol}(M_{j}\cap U)}\int_{(M_{j}\cap
U)^{\varphi}}\frac{\hat{A}(M^{\varphi})}{\det(1-\Phi e^{-R})^{1/2}}$

equals zero if it converges. (Here $R$ is the curvature of the connection on
the normal bundle to $M^{\varphi}$ induced by the Levi–Civita connection.) The
latter result is purely geometric, and its statement does not involve the
algebras and functionals developed in this paper.

### Outline

In Section 2, we introduce the geometric setting we consider, and the
functional $\operatorname{Tr}^{U}_{\Phi}$ that will be used to define a
numerical index of Dirac operators. This functional has an asymptotic trace
property on the algebra $\mathcal{A}_{-\infty}^{L}(S)^{\Phi}$ introduced in
Section 3. This trace property is Theorem 5.1, which is proved in Section 5.
We then use this trace property to obtain a numerical index from a
$K$-theoretic index of Dirac operators, and obtain an index theorem, in
Section 6.

### Acknowledgements

This research was partially supported by the Australian Research Council,
through Discovery Project DP200100729, and by NWO ENW-M grant OCENW.M.21.176.

##  2 Preliminaries

We start by briefly stating a standard definition of the Dirac operators that
we consider. Then we introduce a certain type of functional on operators with
bounded, smooth kernels (Definition 2.6). In Section 5, we show that this
functional has a trace-like property (Theorem 5.1) on an algebra introduced in
Section 3. That will allow us to extract a numerical index (Definition 5.11)
from a $K$-theoretic index (Definition 4.10) and obtain an index theorem
(Theorem 6.3).

###  2.1 Dirac operators

Let $M$ be a complete Riemannian manifold. Let $S\to M$ be a Hermitian vector
bundle.

Let $c\colon TM\to\operatorname{End}(S)$ be a vector bundle homomorphism such
that for all $v\in TM$,

$c(v)^{2}=-\|v\|^{2}.$

Let $\nabla^{TM}$ be the Levi–Civita connection on $TM$. Let $\nabla$ be a
Hermitian connection on $S$ such that for all vector fields $v$ and $w$ on
$M$, and all $s\in\Gamma^{\infty}(S)$,

$\nabla_{v}(c(w)s)=c(w)\nabla_{v}s+c(\nabla^{TM}_{v}w)s.$

Consider the Dirac operator $D\colon\Gamma^{\infty}(S)\to\Gamma^{\infty}(S)$
defined as the composition

$D\colon\Gamma^{\infty}(S)\xrightarrow{\nabla}\Gamma^{\infty}(T^{*}M\otimes
S)\xrightarrow{\cong}\Gamma^{\infty}(TM\otimes
S)\xrightarrow{c}\Gamma^{\infty}(S).$

The isomorphism in the middle is induced by the isomorphism $T^{*}M\cong TM$
defined by the Riemannian metric.

We assume that $S$ is $\mathbb{Z}/2\mathbb{Z}$-graded, that $\nabla$ preserves
this grading, and that $c(v)$ reverses it, for all $v\in TM$. Then $D$ is odd
with respect to the grading.

We assume that $M$ and $S$ have _bounded geometry_ , i.e. $M$ has positive
injectivity radius, and the curvature tensors of $\nabla^{TM}$ and $\nabla$,
and all their covariant derivatives, are bounded.

Let $\varphi\colon M\to M$ be an isometry. Let $\Phi\colon S\to S$ be a smooth
map such that for all $m\in M$, the restriction $\Phi_{m}$ of $\Phi$ to the
fibre $S_{m}$ at $m$ is a linear map from $S_{m}$ to $S_{\varphi(m)}$ that
preserves the metric and the grading. Then $\varphi$ and $\Phi$ define a map
from $\Gamma^{\infty}(S)$ to itself by

$(\Phi(s))(m)=\Phi(s(\varphi^{-1}(m)))$ (2.1)

for all $s\in\Gamma^{\infty}(S)$ and $m\in M$. We suppose that $D$ commutes
with this map.

###  2.2 Exhaustions

We denote the Riemannian distance on $M$ by $d$. Let $U$ be an open
neighbourhood of the fixed-point set $M^{\varphi}$ such that

  1. 1.

$U$ is invariant under $\varphi$; and

  2. 2.

there is a $\delta>0$ such that $d(\varphi(m),m)\geq\delta$ for all $m\in
M\setminus U$.

Our main result, Theorem 6.3 depends on the choice of $U$. This flexibility
allows us to choose $U$ such that the main equality (6.7) yields interesting
information. This is crucial, as the most straightforward choice $U=M$ leads
to an index theorem that reduces to the equality $0=0$ in many cases, see
Remark 6.5. In fact, much of the work done in this paper is to allow the
possibility to take $U$ different from $M$, and this is a key difference with
[12]. (Compare for example Theorem 5.1 in this paper with Proposition 6.7 in
[12].)

For every $j\in\mathbb{N}$, let $M_{j}\subset M$ be a compact subset. Suppose
that

  1. 1.

$M_{j}\subset M_{j+1}$ for all $j\in\mathbb{N}$; and

  2. 2.

$\bigcup_{j=1}^{\infty}M_{j}=M$.

For every $j$, we write

$U_{j}:=U\cap M_{j}.$

For $j\in\mathbb{N}$ and $r>0$, define

$\operatorname{Pen}_{U}^{-}(U_{j},r):=\\{m\in U_{j};d(m,M\setminus M_{j})\geq
r\\}.$

######  Definition 2.1.

The sequence $(M_{j})_{j=1}^{\infty}$ is a _$U$ -regular exhaustion_ of $M$ if
for all $r>0$,

$\lim_{j\to\infty}\frac{\operatorname{vol}(U_{j})-\operatorname{vol}(\operatorname{Pen}_{U}^{-}(U_{j},r))}{\operatorname{vol}(U_{j})}=0.$
(2.2)

A $U$-regular exhaustion exists for example if volumes of balls in $M$
increase slower than exponentially in their radii, via an argument analogous
to the proof of Proposition 6.2 in [12].

######  Example 2.2.

Suppose that $M=\mathbb{R}^{n}$, with the Euclidean metric. Let
$k\in\\{1,\ldots,n\\}$, and define $\varphi\colon M\to M$ by

$\varphi(x)=(-x_{1},\ldots,-x_{k},x_{k+1},\ldots,x_{n})$

for $x=(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}$. Then
$M^{\varphi}=\\{0_{\mathbb{R}^{k}}\\}\times\mathbb{R}^{n-k}$. Take
$U=(-1,1)^{k}\times\mathbb{R}^{n-k}$. Then we make take $\delta=2$ in the
second assumption on $U$.

Take $M_{j}=[-j,j]^{n}$. Then $U_{j}=(-1,1)^{k}\times[-j,j]^{n-k}$. If $r>0$
and $j\geq r+1$, then

$\operatorname{Pen}_{U}^{-}(U_{j},r)=(-1,1)^{k}\times[-j+r,j-r]^{n-k}.$

So for such $j$,

$\frac{\operatorname{vol}(U_{j})-\operatorname{vol}(\operatorname{Pen}_{U}^{-}(U_{j},r))}{\operatorname{vol}(U_{j})}=1-\left(\frac{j-r}{j}\right)^{n-k},$

which goes to $0$ as $j\to\infty$. So $(M_{j})_{j=1}^{\infty}$ is a
$U$-regular exhaustion.

###  2.3 A functional

Let $|\Omega_{b}|(M)$ be the Banach space of bounded, continuous densities on
$M$.

######  Definition 2.3.

A continuous linear functional $I\colon|\Omega_{b}|(M)\to\mathbb{R}$ is
_associated_ with the exhaustion $(U_{j})_{j=1}^{\infty}$ of $U$ if for all
$\alpha\in|\Omega_{b}|(M)$,

$\liminf_{j\to\infty}\left|\langle
I,\alpha\rangle-\frac{1}{\operatorname{vol}(U_{j})}\int_{U_{j}}\alpha\right|=0.$

######  Lemma 2.4.

There exists a functional associated to $(U_{j})_{j=1}^{\infty}$.

###### Proof.

This is analogous to the proof of Proposition 6.4 in [12]. ∎

######  Example 2.5.

Let $M=\mathbb{R}$, and let $\varphi$ be the identity map. Then we must take
$U=M$. Let $U_{j}=M_{j}=[-2^{j+1},2^{j+1}]$. Let $\chi\in
C^{\infty}_{c}(\mathbb{R})$ be supported in $[0,1]$, and such that
$\int_{0}^{1}\chi(x)\,dx=1$. Define the bounded function $\zeta\in
C^{\infty}(\mathbb{R})$ by

$\zeta(x):=\sum_{j=0}^{\infty}(-1)^{j}\chi(2^{-j}x-1).$

Then the term corresponding to $j$ is supported in $[2^{j},2^{j+1}]$, and its
integral equals $(-1)^{j}2^{j}$. It follows from the truncated geometric
series that

$\frac{1}{\operatorname{vol}(U_{j})}\int_{U_{j}}\zeta(x)\,dx=\frac{1}{3\cdot
2^{j+2}}+\frac{(-1)^{j}}{6}.$

This does not converge as $j\to\infty$. But

$\liminf_{j\to\infty}\left|\frac{1}{6}-\frac{1}{\operatorname{vol}(U_{j})}\int_{U_{j}}\zeta(x)\,dx\right|=0,$

and also

$\liminf_{j\to\infty}\left|\frac{-1}{6}-\frac{1}{\operatorname{vol}(U_{j})}\int_{U_{j}}\zeta(x)\,dx\right|=0.$

This suggests that there exist different functionals associated to
$(U_{j})_{j=1}^{\infty}$, taking the values $\pm 1/6$ on $\zeta dx$.

We denote the Riemannian density on $M$ by $dm$.

######  Definition 2.6.

Let $I$ be a functional associated to $(U_{j})_{j=1}^{\infty}$. Let
$A\colon\Gamma^{\infty}_{c}(S)\to\Gamma^{\infty}(S)$ be a linear operator with
a bounded, smooth Schwartz kernel $\kappa$. Define $\alpha\in|\Omega_{b}|(M)$
by

$\alpha(m):=\operatorname{tr}\bigl{(}\Phi_{\varphi^{-1}(m)}\circ\kappa(\varphi^{-1}(m),m)\bigr{)}dm.$

Then we define

$\operatorname{Tr}^{U}_{\Phi}(A):=\langle I,\alpha\rangle.$ (2.3)

######  Remark 2.7.

The number (2.3) depends on the functional $I$, which is not unique in
general.

##  3 Asymptotically local operators

We will see in Theorem 5.1 that the number (2.3) has an “asymptotic trace
property” on a suitable algebra of operators. In this section, we construct
this algebra. We then define an index of $D$ in the $K$-theory of that
algebra.

We do not consider an exhaustion or a functional associated to it until
Section 5.

###  3.1 Uniform operators

We briefly review some definitions from Section 5 of [12].

For $k\in\mathbb{Z}_{\geq 0}$, consider the norm $\|\cdot\|_{W^{k}}$ on
$\Gamma^{\infty}_{c}(S)$ defined by

$\|s\|_{W^{k}}^{2}=\sum_{j=0}^{k}\|D^{j}s\|_{L^{2}(S)}^{2},$

for $s\in\Gamma_{c}^{\infty}(S)$. Let $W^{k}(S)$ be the completion of
$\Gamma^{\infty}_{c}(S)$ in this norm, and let $W^{-k}(S)$ be the continuous
dual of $W^{k}(S)$.

For a subset $X\subset M$, and $s\in W^{k}(S)$, we define

$\|s\|_{W^{k},X}:=\inf\bigl{\\{}\|s^{\prime}\|_{W^{k}};\text{$s^{\prime}\in
W^{k}(S)$ is equal to $s$ in a neighbourhood of $X$}\bigr{\\}}.$

For a subset $X\subset M$ and $r\geq 0$, we write

$\operatorname{Pen}^{+}(X,r):=\overline{\\{m\in M;d(m,X)\leq r\\}}.$

######  Definition 3.1.

Let $k\in\mathbb{Z}$. An operator
$A\colon\Gamma_{c}^{\infty}(S)\to\Gamma^{\infty}(S)$ is _uniform of order at
most $k$_ if

  1. 1.

for all $l\in\mathbb{Z}$, $A$ extends to a bounded operator from $W^{l}(S)$ to
$W^{l-k}(S)$, also denoted by $A$; and

  2. 2.

for all $l\in\mathbb{Z}$, there is a function
$\mu_{l}\colon(0,\infty)\to(0,\infty)$ such that

    1. (a)

$\lim_{r\to\infty}\mu_{l}(r)=\infty$; and

    2. (b)

for all compact subsets $K\subset M$, all $r>0$ and all $s\in W^{l}(S)$
supported in $K$,

$\|As\|_{W^{l-k},M\setminus\operatorname{Pen}^{+}(K,r)}\leq\mu_{l}(r)\|s\|_{W^{k}}.$

We denote the vector space of all such operators by $\mathcal{U}_{k}(S)$.
Furthermore, we write

$\begin{split}\mathcal{U}(S)&:=\bigcup_{k\in\mathbb{Z}}\mathcal{U}_{k}(S);\\\
\mathcal{U}_{-\infty}(S)&:=\bigcap_{k\in\mathbb{Z}}\mathcal{U}_{k}(S).\end{split}$

The following fact implies that $\mathcal{U}(S)$ is an algebra and
$\mathcal{U}_{-\infty}(S)$ is an ideal in $\mathcal{U}(S)$.

######  Lemma 3.2.

If $A\in\mathcal{U}_{k}(S)$ and $B\in\mathcal{U}_{l}(S)$, then
$AB\in\mathcal{U}_{k+l}(S)$.

This is Proposition 5.2 in [12].

######  Proposition 3.3.

Every operator in $\mathcal{U}_{-\infty}(S)$ has a bounded, smooth Schwartz
kernel.

This is the first part of Proposition 5.4 in [12].

The algebras $\mathcal{U}(S)$ and $\mathcal{U}_{-\infty}(S)$ contain the
subalgebras $\mathcal{U}(S)^{\Phi}\subset\mathcal{U}(S)$ and
$\mathcal{U}_{-\infty}(S)^{\Phi}\subset\mathcal{U}_{-\infty}(S)$ of operators
that commute with the map (2.1) on sections of $S$ defined by $\Phi$ and
$\varphi$. Note that $\mathcal{U}_{-\infty}(S)^{\Phi}$ is an ideal in
$\mathcal{U}(S)^{\Phi}$.

###  3.2 Asymptotically local operators

######  Definition 3.4.

A path of operators $A\colon(0,\infty)\to\mathcal{U}_{k}(S)$ is
_asymptotically local_ of order at most $k$ if for all $l\in\mathbb{Z}$,

  * •

there are $C_{l},a_{l}>0$ such that for all $t\in(0,1]$, the norm of $A(t)$ as
an operator from $W^{l}(S)$ to $W^{l-k}(S)$ satisfies

$\|A(t)\|\leq C_{l}t^{-a_{l}},$ (3.1)

and

  * •

there is a function $\mu_{l}\colon(0,\infty)\times(0,\infty)\to(0,\infty)$
such that

    1. 1.

for all $t>0$, $\lim_{r\to\infty}\mu_{l}(r,t)=0$;

    2. 2.

for all $r>0$ and $a\geq 0$, the function $t\mapsto t^{-a}\mu_{l}(r,t)$ is
bounded and non-decreasing on $(0,1]$;

    3. 3.

for all compact subsets $K\subset M$, all $r,t>0$ and all $s\in W^{l}(S)$
supported in $K$,

$\|A(t)s\|_{W^{l-k},M\setminus\operatorname{Pen}^{+}(K,r)}\leq\mu_{l}(r,t)\|s\|_{W^{l}}.$

We denote the vector space of all such operators by $\mathcal{U}_{k}^{L}(S)$.
We write

$\mathcal{U}_{-\infty}^{L}(S):=\bigcap_{k\in\mathbb{Z}}\mathcal{U}_{k}^{L}(S).$

If $A\in\mathcal{U}_{k}^{L}(S)$ and $B\in\mathcal{U}_{l}^{L}(S)$, then we
define the path $AB\colon(0,\infty)\to\mathcal{U}_{k+l}(S)$ by
$(AB)(t)=A(t)B(t)$ for $t>0$. Here we use Lemma 3.2.

######  Lemma 3.5.

For all $A\in\mathcal{U}_{k}^{L}(S)$ and $B\in\mathcal{U}_{l}^{L}(S)$, the
path $AB$ lies in $\mathcal{U}_{k+l}^{L}(S)$.

###### Proof.

The condition involving estimates of the form (3.1) is clearly preserved under
composition.

Let $n\in\mathbb{Z}$. Let $\mu_{n}^{B}$ be as in Definition 3.4, applied to
$B$ as an operator from $W^{n}(S)$ to $W^{n-l}(S)$. Let
$C^{B}_{n},a^{B}_{n}>0$ be such that for all $t\in(0,1]$, the norm of $B(t)$
as an operator from $W^{n}(S)$ to $W^{n-l}(S)$ is at most
$C^{B}_{n}t^{-a^{B}_{n}}$. Similarly, let $\mu_{n-l}^{A}$ be as in Definition
3.4, applied to $A$ as an operator from $W^{n-l}(S)$ to $W^{n-l-k}(S)$. Let
$C^{A}_{n-l},a^{A}_{n-l}>0$ be such that for all $t\in(0,1]$, the norm of
$A(t)$ as an operator from $W^{n-l}(S)$ to $W^{n-l-k}(S)$ is at most
$C^{A}_{n-l}t^{-a^{A}_{n-l}}$. For $t>0$, define

$\begin{split}F_{A}(t)&:=\max\\{\|A(t)\|,C^{A}_{n-l}t^{-a^{A}_{n-l}}\\};\\\
F_{B}(t)&:=\max\\{\|B(t)\|,C^{B}_{n}t^{-a^{B}_{n}}\\}.\end{split}$

Here $\|A(t)\|$ is the norm of $A(t)$ as an operator from $W^{n-l}(S)$ to
$W^{n-l-k}(S)$, and $\|B(t)\|$ is the norm of $B(t)$ as an operator from
$W^{n}(S)$ to $W^{n-l}(S)$. (Note that if $t\in(0,1]$, then
$F_{A}(t)=C^{A}_{n-l}t^{-a^{A}_{n-l}}$ and
$F_{B}(t)=C^{B}_{n}t^{-a^{B}_{n}}$.)

Define $\mu_{n}^{AB}\colon(0,\infty)\times(0,\infty)\to(0,\infty)$ by

$\mu_{n}^{AB}(r,t):=2F_{B}(t)\mu_{n-l}^{A}(r/2,t)\\\
+\mu_{n}^{B}(r/2,t)(F_{A}(t)+2\mu_{n-l}^{A}(r/2,t)),$

for $r,t>0$. It follows from the definitions that this function satisfies the
first two conditions in Definition 3.4. In the proof of Proposition 5.2 in
[12], it is shown that a version of the function $\mu_{n}^{AB}$ with
$F_{A}(t)$ replaced by $\|A(t)\|$ and $F_{B}(t)$ replaced by $\|B(t)\|$
satisfies the third condition on $\mu_{n}$ in Definition 3.4 for the operators
$A(t)B(t)\colon W^{n}(S)\to W^{n-l-k}(S)$. This implies that the function
$\mu_{n}^{AB}$ also has this property. ∎

If $r>0$ and $m\in M$, then we write $B(m,r)\subset M$ for the open ball with
centre $m$ and radius $r$.

######  Theorem 3.6.

For all $A\in\mathcal{U}_{-\infty}^{L}(S)$, there is a function
$v\colon(0,\infty)\times(0,\infty)\to(0,\infty)$ such that

  1. 1.

for all $t>0$, $\lim_{r\to\infty}v(r,t)=0$;

  2. 2.

for all $r>0$ and $a\geq 0$, the function $t\mapsto t^{-a}v(r,t)$ is bounded
and non-decreasing on $(0,1]$;

  3. 3.

for all $m\in M$, and $r,t>0$, if $\kappa_{t}$ is the Schwartz kernel of
$A(t)$, then

$\begin{split}\int_{M\setminus
B(m,r)}\|{\kappa_{t}(m,m^{\prime})}\|^{2}\,dm^{\prime}&\leq
v(r,t);\quad\text{and}\\\ \int_{M\setminus
B(m,r)}\|{\kappa_{t}(m^{\prime},m)}\|^{2}\,dm^{\prime}&\leq
v(r,t).\end{split}$

###### Proof.

In the proof of Proposition 5.4 in [12], it is shown that there are
$l,l^{\prime}\in\mathbb{Z}$ and $C,C^{\prime}>0$ such that for all $t,r>0$,

$\begin{split}\int_{M\setminus
B(m,r)}\|{\kappa_{t}(m,m^{\prime})}\|^{2}\,dm^{\prime}&\leq
C\mu_{l}(r,t)^{2};\quad\text{and}\\\ \int_{M\setminus
B(m,r)}\|{\kappa_{t}(m^{\prime},m)}\|^{2}\,dm^{\prime}&\leq
C^{\prime}\mu_{l^{\prime}}(r,t)^{2},\end{split}$

with $\mu_{l}$ and $\mu_{l^{\prime}}$ as in Definition 3.4. So we can take
$v=C\mu_{l}^{2}+C^{\prime}\mu_{l^{\prime}}^{2}$. ∎

We write $\mathcal{U}_{-\infty}^{L}(S)^{\Phi}$ for the subalgebra of
$\mathcal{U}_{-\infty}^{L}(S)$ of paths of operators with values in
$\mathcal{U}_{-\infty}(S)^{\Phi}$.

###  3.3 A subalgebra

Lemma 3.5 implies that $\mathcal{U}_{-\infty}^{L}(S)$ is an algebra. We will
use a subalgebra of this algebra. By Proposition 3.3, every operator in
$\mathcal{U}_{-\infty}(S)$ has a smooth, bounded Schwartz kernel.

######  Definition 3.7.

The vector space $\mathcal{A}_{-\infty}^{L}(S)$ consists of those
$A\in\mathcal{U}_{-\infty}^{L}(S)$ with the following property. Let
$\kappa_{t}\in\Gamma(S\boxtimes S^{*})$ be the smooth kernel of $A(t)$. Then
$A\in\mathcal{A}_{-\infty}^{L}(S)$ if there are $C,a>0$ such that for all
$m\in M$ and $t\in(0,1]$,

$\begin{split}\int_{M}\|\kappa_{t}(m,m^{\prime})\|^{2}\,dm^{\prime}&\leq
Ct^{-a};\quad\text{and}\\\
\int_{M}\|\kappa_{t}(m^{\prime},m)\|^{2}\,dm^{\prime}&\leq Ct^{-a}.\\\
\end{split}$ (3.2)

######  Remark 3.8.

Because of Theorem 3.6, the condition in Definition 3.7 may be replaced by:
there exist $C,a,r>0$ such that for all $m\in M$ and $t\in(0,1]$,

$\begin{split}\int_{B(m,r)}\|\kappa_{t}(m,m^{\prime})\|^{2}\,dm^{\prime}&\leq
Ct^{-a};\quad\text{and}\\\
\int_{B(m,r)}\|\kappa_{t}(m^{\prime},m^{\prime})\|^{2}\,dm^{\prime}&\leq
Ct^{-a}.\end{split}$ (3.3)

######  Proposition 3.9.

The space $\mathcal{A}_{-\infty}^{L}(S)$ is closed under composition, and
hence an algebra.

###### Proof.

Let $A,B\in\mathcal{A}_{-\infty}^{L}(S)$. Then
$AB\in\mathcal{U}_{-\infty}^{L}(S)$ by Lemma 3.5. Let $C_{A},a_{A}$ be as in
Definition 3.7 for $A$, and let $C_{B},a_{B}$ be as in Definition 3.7 for $B$.
Let $m\in M$ and $t\in(0,1]$. Let $\kappa_{t}$ be the Schwartz kernel of
$A(t)$, $\lambda_{t}$ the Schwartz kernel of $B(t)$, and $\mu_{t}$ the
Schwartz kernel of $A(t)B(t)$. Then by the Cauchy–Schwartz inequality,

$\int_{B(m,1)}\|\mu_{t}(m,m^{\prime})\|^{2}\,dm^{\prime}\\\
\leq\int_{B(m,1)}\left(\int_{M}\|\kappa_{t}(m,m^{\prime\prime})\|^{2}\,dm^{\prime\prime}\int_{M}\|\lambda_{t}(m^{\prime\prime},m^{\prime})\|^{2}\,dm^{\prime\prime}\right)\,dm^{\prime}\\\
\operatorname{vol}(B(m,1))C_{A}C_{B}t^{-(a_{A}+a_{B})}.$

Because $M$ has bounded geometry, $\operatorname{vol}(B(m,1))$ is bounded in
$m$. By Remark 3.8, it follows that $AB\in\mathcal{A}_{-\infty}^{L}(S)$. ∎

######  Lemma 3.10.

Suppose that $A\in\mathcal{U}_{-\infty}^{L}(S)$, and that there is a single
function $\mu\colon(0,\infty)\times(0,\infty)\to(0,\infty)$ such that for all
$l\in\mathbb{Z}$, $\mu_{l}=\mu$ has the properties in Definition 3.4, and so
does the function $(r,t)\mapsto\mu(1,t)^{-1/2}\mu(r,t)$. Let
$B\in\mathcal{U}_{k}(S)$. Then $AB$ and $BA$ lie in
$\mathcal{U}_{-\infty}^{L}(S)$.

If, furthermore, $A$ lies in $\mathcal{A}_{-\infty}^{L}(S)$ and commutes with
$B$, and $k\leq 0$, then $AB$ and $BA$ lie in $\mathcal{A}_{-\infty}^{L}(S)$.

###### Proof.

Define the path of operators $\tilde{B}\colon(0,\infty)\to\mathcal{U}_{k}(S)$
by $\tilde{B}(t)=\mu(1,t)^{1/2}B$. Then $\tilde{B}\in\mathcal{U}_{k}^{L}(S)$.
And the path $t\mapsto\mu(1,t)^{-1/2}A(t)$ lies in
$\mathcal{U}_{-\infty}^{L}(S)$ by the assumption on $\mu$. Hence the path
$t\mapsto A(t)B=\mu(1,t)^{-1/2}A(t)\tilde{B}(t)$ lies in
$\mathcal{U}_{-\infty}^{L}(S)$ by Lemma 3.5. The same argument applies to
$BA$.

Now suppose that $A\in\mathcal{A}_{-\infty}^{L}(S)$. If $k\leq 0$, then
$\mathcal{U}_{k}(S)\subset\mathcal{B}(L^{2}(S))$. Let $\|B\|$ be the operator
norm of $B\colon L^{2}(S)\to L^{2}(S)$. Let $t>0$. Let $\kappa_{A(t)}$ be the
Schwartz kernel of $A(t)$ and let $\kappa_{BA(t)}$ be the Schwartz kernel of
$BA(t)$. Let $m\in M$. For $v\in S_{m}$, let $\delta_{m}v$ be the
distributional section of $S$ mapping $s\in\Gamma^{\infty}_{c}(S^{*})$ to
$\langle s(m),v\rangle$. Then $\delta_{m}v\in W^{l}(S)$ for some
$l\in\mathbb{Z}$, so $A(t)\delta_{m}v\in L^{2}(S)$. Let
$\\{e_{1},\ldots,e_{r}\\}$ be an orthonormal basis of $S_{m}$. Then, up to a
multiplicative constant we absorb in the definition of the norms on
$\operatorname{Hom}(S_{m},S_{m^{\prime}})$, for $m^{\prime}\in M$, we have

$\begin{split}\int_{M}\|\kappa_{BA(t)}(m^{\prime},m)\|^{2}\,dm^{\prime}&=\sum_{j=1}^{r}\int_{M}\|\kappa_{BA(t)}(m^{\prime},m)e_{j}\|^{2}\,dm\\\
&=\sum_{j=1}^{r}\|BA(t)\delta_{m}e_{j}\|_{L^{2}(S)}^{2}\\\
&\leq\|B\|^{2}\sum_{j=1}^{r}\|A(t)\delta_{m}e_{j}\|_{L^{2}(S)}^{2}\\\
&=\|B\|^{2}\int_{M}\|\kappa_{A(t)}(m^{\prime},m)\|^{2}\,dm^{\prime}.\end{split}$
(3.4)

Furthermore, if we use stars to denote $L^{2}$-adjoints, then, because $B$ and
$A(t)$ commute,

$\begin{split}\int_{M}\|\kappa_{BA(t)}(m,m^{\prime})\|^{2}\,dm^{\prime}&=\int_{M}\|\kappa_{A(t)B}(m,m^{\prime})\|^{2}\,dm^{\prime}\\\
&=\int_{M}\|\kappa_{(A(t)B)^{*}}(m^{\prime},m)\|^{2}\,dm^{\prime}\\\
&=\int_{M}\|\kappa_{B^{*}A(t)^{*}}(m^{\prime},m)\|^{2}\,dm^{\prime}.\end{split}$

As in (3.4), the latter integral is bounded above by

$\|B^{*}\|^{2}\int_{M}\|\kappa_{A(t)^{*}}(m^{\prime},m)\|^{2}\,dm^{\prime}=\|B\|^{2}\int_{M}\|\kappa_{A(t)}(m,m^{\prime})\|^{2}\,dm^{\prime}.$

∎

We write $\mathcal{A}_{-\infty}^{L}(S)^{\Phi}$ for the subalgebra of
$\mathcal{A}_{-\infty}^{L}(S)$ of paths of operators with values in
$\mathcal{U}_{-\infty}(S)^{\Phi}$.

##  4 A $K$-theoretic index

Our next goal is to show that the Dirac operator $D$ has an index in the
$K$-theory of the algebra $\mathcal{A}_{-\infty}^{L}(S)^{\Phi}$, see
Definition 4.10.

###  4.1 Functional calculus

For $a\in\mathbb{R}$, let $\mathcal{S}^{a}(\mathbb{R})$ be the space of
functions $f\in C^{\infty}(\mathbb{R})$ such that for all $l=0,1,2,\ldots$,
there is a $C_{l}>0$ such that for all $x\in\mathbb{R}$,

$|f^{(l)}(x)|\leq C_{l}(1+|x|)^{a-l}.$ (4.1)

For a function $f\in\mathcal{S}^{a}(\mathbb{R})$, the operator $f(D)$ defined
by functional calculus lies in $\mathcal{U}_{k}(S)$ by Theorem 5.5 in [13]. We
extend this argument to paths of operators in Proposition 4.2.

######  Lemma 4.1.

Let $a\in\mathbb{R}$ and $f\in\mathcal{S}^{a}(\mathbb{R})$. Let $g\in
C^{\infty}_{c}(\mathbb{R})$. For $\varepsilon>0$, define the function
$f_{\varepsilon}\in C^{\infty}(\mathbb{R})$ by
$f_{\varepsilon}(x)=f(x)g(\varepsilon x)$. Then for all $l\geq 0$ and all
$j\geq a+l+2$, there is a $C_{l,j}>0$ such that for all $\varepsilon\in(0,1]$
and $\xi\in\mathbb{R}$,

$|\widehat{f_{\varepsilon}}^{(l)}(\xi)|\leq C_{l,j}(1+|\xi|)^{-j}.$

###### Proof.

Let $l\geq 0$. Consider the function
$f_{\varepsilon,l}(x):=x^{l}f_{\varepsilon}(x)$. For all $j\geq 0$, the
Leibniz rule implies that there is a $C>0$ such that for all
$\varepsilon\in(0,1]$ and all $x\in\mathbb{R}$,

$|f_{\varepsilon,l}^{(j)}(x)|\leq C(1+|x|)^{a+l-j}.$

If $j\geq a+l+2$, then this becomes

$|f_{\varepsilon,l}^{(j)}(x)|\leq C(1+|x|)^{-2}.$

So for all $\xi\in\mathbb{R}$,

$|\xi|^{j}|\widehat{f_{\varepsilon}}^{(l)}(\xi)|=|\widehat{f_{\varepsilon,l}^{(j)}}(\xi)|\leq
C\int_{\mathbb{R}}(1+|x|)^{-2}\,dx.$

∎

######  Proposition 4.2.

Let $k\in\mathbb{Z}$ and $f\in\mathcal{S}^{k}(\mathbb{R})$. We write
$\tilde{k}:=\max(-k,0)$. Then the path of operators $t\mapsto f(tD)$ lies in
$\mathcal{U}_{k}^{L}(S)$. Specifically, there is a single function
$\mu\colon(0,\infty)\times(0,\infty)\to(0,\infty)$ such that for all
$l\in\mathbb{Z}$, $\mu_{l}=\mu$ has the properties in Definition 3.4, and for
all $b\geq 2\tilde{k}+2$ there is a constant $C_{b}>0$, independent of $l$,
such that for all $r,t>0$,

$\mu(r,t)\leq C_{b}r^{\tilde{k}-b+1}t^{b-\tilde{k}-1}.$ (4.2)

###### Proof.

For $t>0$, define $f_{t}\in C^{\infty}(\mathbb{R})$ by $f_{t}(x)=f(tx)$. Let
$g\in C^{\infty}_{c}(\mathbb{R})$ be such that $g(0)=1$. For $\varepsilon>0$,
define $f_{t,\varepsilon}\in C^{\infty}(\mathbb{R})$ by
$f_{t,\varepsilon}(x)=f_{t}(x)g(\varepsilon x)$. Let $h\in
C^{\infty}(\mathbb{R})$ be such that $h(\mathbb{R})\subset[0,1]$, and

$h(\xi)=\left\\{\begin{array}[]{ll}1&\text{if $|\xi|\leq 1/2$};\\\ 0&\text{if
$|\xi|\geq 1$}.\end{array}\right.$

For $r>0$, define $h_{r}\in C^{\infty}(\mathbb{R})$ by $h_{r}(\xi)=h(\xi/r)$.
By Theorem 5.5 in [12], the operator $f(tD)$ lies in $\mathcal{U}_{k}(S)$,
where the function $\mu_{l}$ in Definition 3.1 may be taken to be

$\mu_{l}(r,t)=\mu(r,t):=\frac{1}{2\pi}\sup_{\varepsilon\in(0,t]}\sum_{n=0}^{\tilde{k}}\int_{\mathbb{R}}\left|\frac{d^{n}}{d\xi^{n}}\left(\widehat{f_{t,\varepsilon}}(\xi)(1-h_{r}(\xi))\right)\right|\,d\xi,$
(4.3)

for all $l\in\mathbb{Z}$. (The supremum over $\varepsilon$ may be taken over
$(0,a]$ for an arbitrarily small $a>0$, because the limit
$\varepsilon\downarrow 0$ is taken in the end. The choice $a=t$ is convenient
in the current proof.)

By a direct computation,

$\widehat{f_{t,\varepsilon}}(\xi)=\frac{1}{t}\widehat{f_{1,\varepsilon/t}}(\xi/t)$

for all $\xi\in\mathbb{R}$. So for all $n$ and $\xi$,

$\left|\frac{d^{n}}{d\xi^{n}}\left(\widehat{f_{t,\varepsilon}}(\xi)(1-h_{r}(\xi))\right)\right|\leq\frac{1}{t}\sum_{p=0}^{n}{n\choose
p}\left|\frac{d^{p}}{d\xi^{p}}\widehat{f_{1,\varepsilon/t}}(\xi/t)\right|\cdot\left|\frac{d^{n-p}}{d\xi^{n-p}}(1-h_{r}(\xi))\right|.$

Now the factor
$\left|\frac{d^{p}}{d\xi^{p}}\widehat{f_{1,\varepsilon/t}}(\xi/t)\right|$ is
bounded by a constant times

$\frac{1}{t^{p}}\left|\widehat{f_{1,\varepsilon/t}}^{(p)}(\xi/t)\right|.$

And the factor $\left|\frac{d^{n-p}}{d\xi^{n-p}}(1-h_{r}(\xi))\right|$ equals
$|1-h(\xi/r)|$ if $p=n$, and is bounded by a constant times
$r^{-(n-p)}h^{(n-p)}(\xi/r)$ if $p<n$. In all cases, it is bounded by a
constant times $r^{-(n-p)}$ times a function that vanishes on $[-r/2,r/2]$.

We find that if $t\in(0,1]$ and $r\geq 1$, (4.3) is smaller than or equal to a
constant times

$t^{-\tilde{k}-1}r^{\tilde{k}}\sup_{\varepsilon\in(0,t]}\sum_{n=0}^{\tilde{k}}\sum_{p=0}^{n}\int_{\mathbb{R}\setminus[-r/2,r/2]}|\widehat{f_{1,\varepsilon/t}}^{(p)}(\xi/t)|\,d\xi\\\
=t^{-\tilde{k}-1}r^{\tilde{k}}\sup_{\varepsilon\in(0,1]}\sum_{n=0}^{\tilde{k}}\sum_{p=0}^{n}\int_{\mathbb{R}\setminus[-r/2,r/2]}|\widehat{f_{1,\varepsilon}}^{(p)}(\xi/t)|\,d\xi$
(4.4)

Lemma 4.1 implies that for all $p\geq 0$ and all $b\geq k+p+2$, there is a
$C_{p,b}>0$ such that for all $\xi\in\mathbb{R}$ and all
$\varepsilon\in(0,1]$,

$|\widehat{f_{1,\varepsilon}}^{(p)}(\xi)|\leq C_{p,b}(1+|\xi|)^{-b}.$

So if $t\in(0,1]$ and $r\geq 1$, and $b\geq 2\tilde{k}+2$, then (4.4) is
smaller than or equal to a constant times

$t^{-\tilde{k}-1}r^{\tilde{k}}\sum_{n=0}^{\tilde{k}}\sum_{p=0}^{n}\int_{\mathbb{R}\setminus[-r/2,r/2]}(1+|\xi/t|)^{-b}\,d\xi.$

The latter expression is smaller than or equal to a constant times

$t^{-\tilde{k}}r^{\tilde{k}}(1+(r/2t))^{-b+1}<2^{1-b}t^{-\tilde{k}+b-1}r^{\tilde{k}-b+1}.$

∎

###  4.2 Heat operators

For $t>0$, consider the heat operator $e^{-tD^{2}}$ associated to $D$.

######  Proposition 4.3.

For all $j\geq 0$, the path of operators $t\mapsto D^{j}e^{-t^{2}D^{2}}$ lies
in $\mathcal{U}_{-\infty}^{L}(S)^{\Phi}$.

###### Proof.

For all $t>0$ and $j,l\geq 0$, the function $f_{j+l,t}$ on $(0,\infty)$
defined by $f_{j+l,t}(x)=x^{j+l}e^{-t^{2}x^{2}}$ has maximum value

$\|f_{j+l,t}\|_{\infty}=\left(\frac{j+l}{2t^{2}}\right)^{(j+l)/2}e^{-(j+l)/2}.$

So for all $k\in\mathbb{Z}$, the right hand side is an upper bound for the
norm of $D^{j}e^{-t^{2}D^{2}}$ as an operator from $W^{k}(S)$ to $W^{k+l}(S)$.
It follows that $t\mapsto D^{j}e^{-t^{2}D^{2}}$ has the first property in
Definition 3.4, for all $k\in\mathbb{Z}$.

Let $\tilde{\mu}$ be as in Proposition 4.2, with $f(x)=x^{j}e^{-x^{2}}$. This
function has the properties of the functions $\mu_{l}$ in Definition 3.4 for
the path of operators $t\mapsto t^{j}D^{j}e^{-t^{2}D^{2}}$. It follows that
the function $\mu(r,t)=t^{-j}\tilde{\mu}(r,t)$ has these properties for the
path of operators $t\mapsto D^{j}e^{-t^{2}D^{2}}$. ∎

######  Proposition 4.4.

For all $j\geq 0$, the path of operators $t\mapsto D^{j}e^{-t^{2}D^{2}}$ lies
in $\mathcal{A}_{-\infty}^{L}(S)^{\Phi}$.

###### Proof.

For $t>0$, let $\kappa_{t}$ be the Schwartz kernel of $D^{j}e^{-t^{2}D^{2}}$.
By Proposition 4.2 in [6], the fact that $M$ has bounded geometry implies that
there are $C,a>0$ such that for all $m,m^{\prime}\in M$ and all $t\in(0,1]$,

$\|\kappa_{t}(m,m^{\prime})\|\leq
Ct^{-(\dim(M)+j)}e^{-ad(m,m^{\prime})^{2}/t^{2}}\leq Ct^{-(\dim(M)+j)}.$

So for all $m\in M$ and $\varepsilon>0$, the left hand sides of (3.3) are less
than or equal to

$C^{2}\operatorname{vol}(B(m,r))t^{-2(\dim(M)+j)}.$

Because $M$ has bounded geometry, the volume of $B(m,r)$ is bounded in $m$. By
Remark 3.8 and Proposition 4.3, this implies the claim. ∎

###  4.3 The index

######  Definition 4.5.

Let $\mathcal{A}_{-\infty}^{L}(S;D)\subset\mathcal{A}_{-\infty}^{L}(S)^{\Phi}$
be the linear subspace of $A\in\mathcal{A}_{-\infty}^{L}(S)$ for which

  1. 1.

there are functions $f\colon(0,\infty)\to(0,\infty)$ and
$g\in\mathcal{S}(\mathbb{R})$ such that for all $t>0$,

$A(t)=f(t)g(tD).$

  2. 2.

for all $j=0,1,2,\ldots$, there are $C_{j},a_{j}>0$ such that for all $m\in M$
and $t\in(0,1]$,

$\begin{split}\int_{M}\|\kappa_{j,t}(m,m^{\prime})\|^{2}\,dm^{\prime}&\leq
C_{j}t^{-a_{j}};\quad\text{and}\\\
\int_{M}\|\kappa_{j,t}(m^{\prime},m)\|^{2}\,dm^{\prime}&\leq
C_{j}t^{-a_{j}},\end{split}$

where $\kappa_{j,t}$ is the Schwartz kernel of $D^{j}A(t)$.

Let $\mathcal{U}^{L}(S;D)$ be the vector space of all paths
$A\colon(0,\infty)\to\mathcal{U}(S)$ such that

$\begin{split}A\mathcal{A}_{-\infty}^{L}(S;D)&\subset\mathcal{A}_{-\infty}^{L}(S;D);\quad\text{and}\\\
\mathcal{A}_{-\infty}^{L}(S;D)A&\subset\mathcal{A}_{-\infty}^{L}(S;D).\end{split}$

######  Lemma 4.6.

The space $\mathcal{U}^{L}(S;D)$ is an algebra, and
$\mathcal{A}_{-\infty}^{L}(S;D)$ is a two-sided ideal in
$\mathcal{U}^{L}(S;D)$.

###### Proof.

The only nontrivial point to check is that $\mathcal{A}_{-\infty}^{L}(S;D)$ is
closed under composition. The first point in Definition 4.5 is clearly
preserved under composition. One can show that the second point in Definition
4.5 is preserved under composition in a similar way to the proof of
Proposition 3.9. ∎

######  Lemma 4.7.

The algebra $\mathcal{U}^{L}(S;D)$ contains the constant path $D$ and the path
$Q(t)=\frac{1-e^{-t^{2}D^{2}}}{D}$.

###### Proof.

It follows from Definition 4.5 that $D\in\mathcal{U}^{L}(S;D)$. To see that
$Q\in\mathcal{U}^{L}(S;D)$, consider the function
$h(x)=\frac{1-e^{-x^{2}}}{x}$. Let $A\in\mathcal{A}_{-\infty}^{L}(S;D)$, and
let $f\colon(0,\infty)\to(0,\infty)$ and $g\in\mathcal{S}(\mathbb{R})$ be such
that for all $t>0$, we have $A(t)=f(t)g(tD)$. Then

$A(t)Q(t)=f(t)t(hg)(tD).$

The Fourier transform of $h$ is

$\hat{h}(\xi)=-i\pi\left(\operatorname{erf}(-\xi/2)+\operatorname{sgn}(\xi)\right).$

This function decays faster than any rational function in $\xi$. So $h$ is
smooth, and all its derivatives are (square integrable, and hence) bounded. So
$hg\in\mathcal{S}(\mathbb{R})$, and therefore $AQ$ is of the desired form.

It follows from Proposition 4.2 that $AQ\in\mathcal{U}_{-\infty}^{L}(S)$. The
second point in Definition 4.5 follows because $Q(t)$ is bounded and commutes
with $A(t)$, analogously to the proof of Lemma 3.10.∎

######  Remark 4.8.

Lemma 4.7 is the reason why we use the subalgebra
$\mathcal{A}_{-\infty}^{L}(S;D)\subset\mathcal{A}_{-\infty}^{L}(S)^{\Phi}$. It
is less obvious to us if $Q$ is a multiplier of
$\mathcal{A}_{-\infty}^{L}(S)^{\Phi}$.

######  Proposition 4.9.

The image of $D$ in $\mathcal{U}^{L}(S;D)/\mathcal{A}_{-\infty}^{L}(S;D)$ is
invertible.

###### Proof.

Let $Q\in\mathcal{U}^{L}(S;D)$ be as in Lemma 4.7. Then for all $t>0$,

$1-DQ(t)=e^{-t^{2}D^{2}}.$

So $1-DQ\in\mathcal{A}_{-\infty}^{L}(S;D)$ by Proposition 4.4. ∎

By Proposition 4.9, the restriction of $D$ to even-graded sections defines a
class

$[D^{+}]\in K_{1}(\mathcal{U}^{L}(S;D)/\mathcal{A}_{-\infty}^{L}(S;D)).$ (4.5)

######  Definition 4.10.

The class

$\operatorname{index}_{D}^{L}(D)\in K_{0}(\mathcal{A}_{-\infty}^{L}(S;D))$
(4.6)

is the image of (4.5) under the boundary map

$\partial\colon K_{1}(\mathcal{U}^{L}(S;D)/\mathcal{A}_{-\infty}^{L}(S;D))\to
K_{0}(\mathcal{A}_{-\infty}^{L}(S;D)).$ (4.7)

The class

$\operatorname{index}^{L}(D)\in K_{0}(\mathcal{A}_{-\infty}^{L}(S)^{\Phi})$
(4.8)

is the image of (4.6) under the map

$\iota_{*}\colon K_{0}(\mathcal{A}_{-\infty}^{L}(S;D))\to
K_{0}(\mathcal{A}_{-\infty}^{L}(S)^{\Phi})$

induced by the inclusion map
$\iota\colon\mathcal{A}_{-\infty}^{L}(S;D)\to\mathcal{A}_{-\infty}^{L}(S)^{\Phi}$.

######  Remark 4.11.

As in [12], we use _algebraic_ $K$-theory to define the indices (4.6) and
(4.8), because we do not consider topologies on the algebras used.

######  Remark 4.12.

The index (4.6) lies in a $K$-theory group depending on $D$, and is therefore
less suitable for constructing canonical invariants of $M$. The index (4.8)
lies in a $K$-theory group independent of $D$; we expect that this can even be
mapped to a $K$-theory group independent of $S$ analogously to Section 7 in
[12]. However, it is a priori possible that the index (4.6) is nonzero in
cases when (4.8) is zero, so (4.6) could potentially be more refined, for
example as an obstruction to positive scalar curvature.

######  Proposition 4.13.

If $D$ is invertible, then $\mathcal{U}^{L}(S;D)$ contains the constant path
$D^{-1}$.

###### Proof.

Fix $c>0$ such that $[-c,c]$ is disjoint from the spectrum of $D$. Let $f\in
C^{\infty}(\mathbb{R})$ be such that

$f(x)=\left\\{\begin{array}[]{ll}0&\text{if $|x|\leq c/2$};\\\ 1&\text{if
$|x|\geq c$}.\end{array}\right.$

Then $x\mapsto f(x)/x$ lies in $\mathcal{S}^{-1}(\mathbb{R})$, so
$D^{-1}=f(D)/D\in\mathcal{U}_{-1}(S)$ by Theorem 5.5 in [12]. By Proposition
4.2, an element $A\in\mathcal{A}_{-\infty}^{L}(S;D)$ satisfies the assumptions
in Lemma 3.10. So $D^{-1}A=AD^{-1}$ lies in $\mathcal{A}_{-\infty}^{L}(S)$.
And this path of operators also satisfies the conditions in Definition 4.5,
where for the second condition we use $L^{2}$-boundedness of $D^{-1}$,
analogously to the proof of Lemma 3.10. So $D^{-1}A=AD^{-1}$ lies in
$\mathcal{A}_{-\infty}^{L}(S;D)$. ∎

######  Proposition 4.14.

If $D$ is invertible, then the indices (4.6) and (4.8) equal zero.

###### Proof.

If $D$ is invertible, then $D^{+}$ defines a class in
$K_{1}(\mathcal{U}^{L}(S;D))$ by Proposition 4.13. The image of this class in
$K_{1}(\mathcal{U}^{L}(S;D)/\mathcal{A}_{-\infty}^{L}(S;D))$ is (4.5). Then
the image of the latter class under (4.7) is now zero, because the composition
of two consecutive maps in the $K$-theory exact sequence is zero. ∎

##  5 An asymptotic trace property

We return to the situation of Subsections 2.2 and 2.3. Let $U$ be as in
Subsection 2.2, fix a $U$-regular exhaustion $(M_{j})_{j=1}^{\infty}$ of $M$,
and let $I$ be a functional on $|\Omega_{b}|(M)$ associated to
$(U_{j})_{j=1}^{\infty}$. By Proposition 3.3, every path of operators
$A\in\mathcal{A}_{-\infty}^{L}(S)$ defines a function

$t\mapsto\operatorname{Tr}^{U}_{\Phi}(A(t))$

as in (2.3).

The main result of this section is the following “asymptotic trace property”
of this construction. This will allow us to extract a numerical index from the
$K$-theoretic index of Definition 4.10.

######  Theorem 5.1.

For all $A\in\mathcal{A}_{-\infty}^{L}(S)^{\Phi}$ and
$B\in\mathcal{A}_{-\infty}^{L}(S)$,

$\lim_{t\downarrow 0}\operatorname{Tr}_{\Phi}^{U}(A(t)B(t)-B(t)A(t))=0.$

###  5.1 Estimates of partial integrals

Let $A,B\in\mathcal{A}_{-\infty}^{L}(S)$. For $t>0$, let $\kappa_{t}$ be the
Schwartz kernel of $A(t)$, and let $\lambda_{t}$ be the Schwartz kernel of
$B(t)$.

Let $\varepsilon>0$. Let $v_{A}$ and $v_{B}$ be the functions in Theorem 3.6,
applied to $A$ and $B$, respectively. Let $r>0$ be such that
$\max(v_{A}(r,1),v_{B}(r,1))<\varepsilon$. Because of the third and fourth
points in Theorem 3.6, we then have for all $m\in M$ and $t\in(0,1]$,

$\begin{split}\int_{M\setminus
B(m,r)}\|{\kappa_{t}(m,m^{\prime})}\|^{2}\,dm^{\prime}&<\varepsilon;\quad\text{and}\\\
\int_{M\setminus
B(m,r)}\|{\lambda_{t}(m^{\prime},m)}\|^{2}\,dm^{\prime}&<\varepsilon.\end{split}$
(5.1)

For $j\in\mathbb{N}$, consider the following subsets of $U_{j}\times M$:

$\begin{split}V_{j}&:=U_{j}\times U_{j};\\\ W_{j}&:=\\{(m,m^{\prime})\in
U_{j}\times(M\setminus M_{j});\text{$d(m,m^{\prime})<r$ or
$d(\varphi^{-1}(m),m^{\prime})<r$}\\};\\\ X_{j}&:=\\{(m,m^{\prime})\in
U_{j}\times(M\setminus M_{j});\text{$d(m,m^{\prime})\geq r$ and
$d(\varphi^{-1}(m),m^{\prime})\geq r$}\\};\\\
Y_{j}&:=U_{j}\times(M_{j}\setminus U_{j}).\end{split}$ (5.2)

These sets are disjoint, and their union is $U_{j}\times M$.

######  Lemma 5.2.

If $A\in\mathcal{A}_{-\infty}^{L}(S)^{\Phi}$, then for all $t>0$ and
$j\in\mathbb{N}$,

$\int_{V_{j}}\operatorname{tr}\left(\Phi\kappa_{t}(\varphi^{-1}(m),m^{\prime})\lambda_{t}(m^{\prime},m)-\Phi\lambda_{t}(\varphi^{-1}(m),m^{\prime})\kappa_{t}(m^{\prime},m)\right)\,dm^{\prime}\,dm=0.$

###### Proof.

Let $t>0$ and $j\in\mathbb{N}$. Because $A(t)$ commutes with $\Phi$, we have
for all $m,m^{\prime}\in M$,

$\Phi\kappa_{t}(\varphi^{-1}(m),m^{\prime})=\kappa_{t}(m,\varphi(m^{\prime}))\Phi.$

Hence by the trace property of the fibre-wise trace,

$\int_{V_{j}}\operatorname{tr}\left(\Phi\kappa_{t}(\varphi^{-1}(m),m^{\prime})\lambda_{t}(m^{\prime},m)\right)\,dm^{\prime}\,dm\\\
=\int_{U_{j}}\int_{U_{j}}\operatorname{tr}\left(\Phi\lambda_{t}(m^{\prime},m)\kappa_{t}(m,\varphi(m^{\prime}))\right)\,dm\,dm^{\prime}.$
(5.3)

Here we also used the Fubini–Tonelli theorem to interchange the integrals; we
have absolute convergence by compactness of $\overline{U_{j}}$ and boundedness
of $\kappa_{t}$ and $\lambda_{t}$ (see Proposition 3.3). Substituting
$m^{\prime\prime}=\varphi(m^{\prime})$ and using $\varphi$-invariance of $dm$,
we see that the right hand side of (5.3) equals

$\int_{V_{j}}\operatorname{tr}\left(\Phi\lambda_{t}(\varphi^{-1}(m^{\prime\prime}),m)\kappa_{t}(m,m^{\prime\prime})\right)\,dm\,dm^{\prime\prime}.$

∎

######  Remark 5.3.

Lemma 5.2 is the one place in the proof of Theorem 5.1 where we use
$\Phi$-equivariance of $A$.

######  Lemma 5.4.

For all $t>0$, there is an $N>0$ such that for all $j\geq N$,

$\left|\frac{1}{\operatorname{vol}(U_{j})}\int_{W_{j}}\operatorname{tr}\left(\Phi\kappa_{t}(\varphi^{-1}(m),m^{\prime})\lambda_{t}(m^{\prime},m)\right)\,dm^{\prime}\,dm\right|<\varepsilon.$
(5.4)

###### Proof.

We claim that for all $j$,

$W_{j}\subset\bigl{(}(U_{j}\setminus\operatorname{Pen}_{U}^{-}(U_{j},r)\bigr{)}\times
M.$ (5.5)

Indeed, suppose that $(m,m^{\prime})\in W_{j}$. If $d(m,m^{\prime})<r$, then
$d(m,M\setminus M_{j})<r$, so $m\not\in\operatorname{Pen}_{U}^{-}(U_{j},r)$.
And if $d(\varphi^{-1}(m),m^{\prime})=d(m,\varphi(m^{\prime}))<r$, then
$m\not\in\operatorname{Pen}_{U}^{-}(U_{j},r)$ because $M_{j}$ is
$\varphi$-invariant. So (5.5) follows.

Because $M$ has bounded geometry, there is a $V>0$ such that for all $m\in M$,
the volume of $B(m,r)$ is at most $V$. It follows from (5.5) and the fact that
$\Phi$ preserves the metric that the left hand side of (5.4) is smaller than
or equal to

$\frac{1}{\operatorname{vol}(U_{j})}\int_{U_{j}\setminus\operatorname{Pen}_{U}^{-}(U_{j},r)}\int_{B(m,r)\cup
B(\varphi^{-1}(m),r)}\|\kappa_{t}(\varphi^{-1}(m),m^{\prime})\|\|\lambda_{t}(m^{\prime},m)\|\,dm^{\prime}\,dm\\\
\leq
2V\frac{\operatorname{vol}(U_{j})-\operatorname{vol}(\operatorname{Pen}_{U}^{-}(U_{j},r))}{\operatorname{vol}(U_{j})}\|\kappa_{t}\|_{\infty}\|\lambda_{t}\|_{\infty}.$

Here we used boundedness of $\kappa_{t}$ and $\lambda_{t}$, Proposition 3.3.
So the claim follows from (2.2). ∎

######  Lemma 5.5.

For all $t\in(0,1]$ and all $j\in\mathbb{N}$,

$\left|\frac{1}{\operatorname{vol}(U_{j})}\int_{X_{j}}\operatorname{tr}\left(\Phi\kappa_{t}(\varphi^{-1}(m),m^{\prime})\lambda_{t}(m^{\prime},m)\right)\,dm^{\prime}\,dm\right|<\varepsilon.$
(5.6)

###### Proof.

Let $t\in(0,1]$ and all $j\in\mathbb{N}$. Then the left hand side of (5.6) is
smaller than or equal to

$\frac{1}{\operatorname{vol}(U_{j})}\int_{U_{j}}\int_{M\setminus(B(m,r)\cup
B(\varphi^{-1}(m),r))}\|\kappa_{t}(\varphi^{-1}(m),m^{\prime})\lambda_{t}(m^{\prime},m)\|\,dm^{\prime}\,dm\\\
\leq\frac{1}{\operatorname{vol}(U_{j})}\int_{U_{j}}\left(\int_{M\setminus
B(\varphi^{-1}(m),r)}\|\kappa_{t}(\varphi^{-1}(m),m^{\prime})\|^{2}\,dm^{\prime}\right)^{1/2}\\\
\left(\int_{M\setminus
B(m,r)}\|\lambda_{t}(m^{\prime},m)\|\,dm^{\prime}\right)^{1/2}\,dm.$ (5.7)

By (5.1), the right hand side is smaller than $\varepsilon$. ∎

######  Lemma 5.6.

There is a $T>0$ such that for all $t\in(0,T)$ and $j\in\mathbb{N}$,

$\left|\frac{1}{\operatorname{vol}(U_{j})}\int_{Y_{j}}\operatorname{tr}\left(\Phi\kappa_{t}(\varphi^{-1}(m),m^{\prime})\lambda_{t}(m^{\prime},m)\right)\,dm^{\prime}\,dm\right|<\varepsilon.$
(5.8)

###### Proof.

Let $\delta>0$ be as in the second assumption on $U$ in Subsection 2.2. If
$(m,m^{\prime})\in Y_{j}$, then $m^{\prime}\not\in U$, so by the triangle
inequality, we either have $d(m,m^{\prime})\geq\delta/2$ or
$d(\varphi^{-1}(m),m^{\prime})\geq\delta/2$. So the left hand side of (5.8) is
at most equal to

$\frac{1}{\operatorname{vol}(U_{j})}\int_{U_{j}}\int_{M\setminus
B(m,\delta/2)}\|\kappa_{t}(\varphi^{-1}(m),m^{\prime})\|\|\lambda_{t}(m^{\prime},m)\|\,dm^{\prime}\,dm\\\
+\frac{1}{\operatorname{vol}(U_{j})}\int_{U_{j}}\int_{M\setminus
B(\varphi^{-1}(m),\delta/2)}\|\kappa_{t}(\varphi^{-1}(m),m^{\prime})\|\|\lambda_{t}(m^{\prime},m)\|\,dm^{\prime}\,dm.$
(5.9)

And for all $m\in M$,

$\int_{M\setminus
B(m,\delta/2)}\|\kappa_{t}(\varphi^{-1}(m),m^{\prime})\|\|\lambda_{t}(m^{\prime},m)\|\,dm^{\prime}\\\
\leq\left(\int_{M\setminus
B(m,\delta/2)}\|\kappa_{t}(\varphi^{-1}(m),m^{\prime})\|^{2}\,dm^{\prime}\right)^{1/2}\left(\int_{M}\|\lambda_{t}(m^{\prime},m)\|^{2}\,dm^{\prime}\right)^{1/2}\\\
\leq v_{A}(\delta/2,t)C_{B}t^{-a_{B}},$

where $v_{A}$ is the function $v$ in Theorem 3.6 for $A$, and $C_{B}$ and
$a_{B}$ are the constants $C$ and $a$ in Definition 3.7, respectively, for
$B$. Via a similar estimate for the second term in (5.9), we find that (5.9)
is bounded above by

$v_{A}(\delta/2,t)C_{B}t^{-a_{B}}+v_{B}(\delta/2,t)C_{A}t^{-a_{A}}.$

This is independent of $j$, and goes to zero as $t\downarrow 0$. ∎

######  Remark 5.7.

The end of the proof of Lemma 5.6 is the main place where we use that the
functions $\mu_{l}$ in Definition 3.4, and hence the function $v$ in Theorem
3.6, vanish to all orders in $t$ as $t\downarrow 0$. (We also use this in the
proof of Lemma 3.5.)

###  5.2 Proof of Theorem 5.1

######  Lemma 5.8.

Consider maps $a\colon\mathbb{N}\times(0,\infty)\to\mathbb{C}$ and
$b\colon(0,\infty)\to\mathbb{C}$. For every $\varepsilon>0$, let
$a_{1,\varepsilon},a_{2,\varepsilon}\colon\mathbb{N}\times(0,\infty)\to\mathbb{C}$
be such that $a=a_{1,\varepsilon}+a_{2,\varepsilon}$. Suppose that for all
$\varepsilon>0$,

  1. 1.

there is a $T>0$ such that for all $t\in(0,T)$ and $j\in\mathbb{N}$, we have
$|a_{1,\varepsilon}(t,j)|<\varepsilon$;

  2. 2.

for all $t\in(0,1]$, there is an $N>0$ such that for all $j\geq N$, we have
$|a_{2,\varepsilon}(t,j)|<\varepsilon$;

  3. 3.

for all $t>0$,

$\liminf_{j\to\infty}|b(t)-a(t,j)|=0.$

Then $\lim_{t\downarrow 0}b(t)=0$.

###### Proof.

Let $\varepsilon>0$. Let $T>0$ be as in the first point. Let
$t\in(0,T)\cap(0,1]$. Let $N$ be as in the second point, for this value of
$t$. By the third point, there is an $N^{\prime}\geq 0$ such that for all
$n\geq N^{\prime}$, there is a $j\geq n$ such that
$|b(t)-a(t,j)|<\varepsilon$. Let $j\in\mathbb{N}$ have this property for
$n=\max(N,N^{\prime})$. Then

$|b(t)|\leq|b(t)-a(t,j)|+|a_{1,\varepsilon}(t,j)|+|a_{2,\varepsilon}(t,j)|<3\varepsilon.$

∎

###### Proof of Theorem 5.1.

Let $A\in\mathcal{A}_{-\infty}^{L}(S)^{\Phi}$ and
$B\in\mathcal{A}_{-\infty}^{L}(S)$. As before, we write $\kappa_{t}$ for the
Schwartz kernel of $A(t)$, and $\lambda_{t}$ for the Schwartz kernel of
$B(t)$. For $t>0$ and $j\in\mathbb{N}$, write

$a(t,j):=\\\
\frac{1}{\operatorname{vol}(U_{j})}\int_{U_{j}}\operatorname{tr}\left(\Phi\kappa_{t}(\varphi^{-1}(m),m^{\prime})\lambda_{t}(m^{\prime},m)-\Phi\lambda_{t}(\varphi^{-1}(m),m^{\prime})\kappa_{t}(m^{\prime},m)\right)\,dm^{\prime}\,dm,$

and

$b(t):=\operatorname{Tr}_{\Phi}^{U}(A(t)B(t)-B(t)A(t)).$

Furhermore, if $\varepsilon>0$, then we choose $r>0$ as at the start of
Subsection 5.1, and use the sets (5.2) to write

$a_{1,\varepsilon}(t,j):=\\\
\frac{1}{\operatorname{vol}(U_{j})}\int_{Y_{j}}\operatorname{tr}\left(\Phi\kappa_{t}(\varphi^{-1}(m),m^{\prime})\lambda_{t}(m^{\prime},m)-\Phi\lambda_{t}(\varphi^{-1}(m),m^{\prime})\kappa_{t}(m^{\prime},m)\right)\,dm^{\prime}\,dm,$

and

$a_{2,\varepsilon}(t,j):=\\\
\frac{1}{\operatorname{vol}(U_{j})}\int_{V_{j}\cup W_{j}\cup
X_{j}}\operatorname{tr}\left(\Phi\kappa_{t}(\varphi^{-1}(m),m^{\prime})\lambda_{t}(m^{\prime},m)-\Phi\lambda_{t}(\varphi^{-1}(m),m^{\prime})\kappa_{t}(m^{\prime},m)\right)\,dm^{\prime}\,dm.$

Then the first condition in Lemma 5.8 holds by Lemma 5.6 and the analogous
statement with $\kappa_{t}$ and $\lambda_{t}$ interchanged. The second
condition in Lemma 5.8 holds by Lemmas 5.2, 5.4 and 5.5, and the versions of
Lemmas 5.4 and 5.5 with $\kappa_{t}$ and $\lambda_{t}$ interchanged. The third
condition in Lemma 5.8 holds by Definition 2.3. So the claim follows by Lemma
5.8. ∎

###  5.3 The trace of the index of $D$

We extend $\operatorname{Tr}^{U}_{\Phi}$ to matrices with entries in
$\mathcal{A}_{-\infty}^{L}(S)^{\Phi}$, by combining it with the matrix trace
in the usual way.

######  Corollary 5.9.

Let $e,f\in M_{\infty}(\mathcal{A}_{-\infty}^{L}(S)^{\Phi})$ be idempotents
such that

$\lim_{t\downarrow
0}\bigl{(}\operatorname{Tr}_{\Phi}^{U}(e(t))-\operatorname{Tr}_{\Phi}^{U}(f(t))\bigr{)}$
(5.10)

converges. Let $e^{\prime},f^{\prime}\in
M_{\infty}(\mathcal{A}_{-\infty}^{L}(S)^{\Phi})$ be idempotents such that we
have Murray–von Neumann equivalences $e\sim e^{\prime}$ and $f\sim
f^{\prime}$. Then

$\lim_{t\downarrow
0}\bigl{(}\operatorname{Tr}_{\Phi}^{U}(e^{\prime}(t))-\operatorname{Tr}_{\Phi}^{U}(f^{\prime}(t))\bigr{)}$

converges, and equals (5.10).

###### Proof.

This follows directly from Theorem 5.1. ∎

Let $\widetilde{K}_{0}(\mathcal{A}_{-\infty}^{L}(S)^{\Phi})\subset
K_{0}(\mathcal{A}_{-\infty}^{L}(S)^{\Phi})$ be the subgroup of classes that
can be represented by idempotents as in Corollary 5.9. Then by this corollary,
we have a well-defined map

$\operatorname{Tr}_{\Phi}^{U}\colon\widetilde{K}_{0}(\mathcal{A}_{-\infty}^{L}(S)^{\Phi})\to\mathbb{C}$

given by

$\operatorname{Tr}_{\Phi}^{U}(x):=\lim_{t\downarrow
0}\bigl{(}\operatorname{Tr}_{\Phi}^{U}(e(t))-\operatorname{Tr}_{\Phi}^{U}(f(t))\bigr{)}.$

if $x=[e]-[f]$, with $e$ and $f$ as in Corollary 5.9.

Let $\gamma$ be the grading operator on $S$.

######  Proposition 5.10.

Suppose that $\lim_{t\downarrow 0}\operatorname{Tr}_{\Phi}^{U}(\gamma
e^{-tD^{2}})$ converges. Then the class $\operatorname{index}^{L}(D)\in
K_{0}(\mathcal{A}_{-\infty}^{L}(S)^{\Phi})$ in Definition 4.10 lies in
$\widetilde{K}_{0}(\mathcal{A}_{-\infty}^{L}(S)^{\Phi})$. Furthermore,

$\operatorname{Tr}_{\Phi}^{U}(\operatorname{index}^{L}(D))=\lim_{t\downarrow
0}\operatorname{Tr}_{\Phi}^{U}(\gamma e^{-tD^{2}}).$ (5.11)

###### Proof.

Let

$\begin{split}e(t)&:=\begin{pmatrix}e^{-t^{2}D^{-}D^{+}}&e^{-\frac{t^{2}}{2}D^{-}D^{+}}\frac{1-e^{-t^{2}D^{-}D^{+}}}{D^{-}D^{+}}D^{-}\\\
e^{-\frac{t^{2}}{2}D^{+}D^{-}}D^{+}&1-e^{-t^{2}D^{+}D^{-}}\end{pmatrix};\quad\text{and}\\\
f(t)&=\begin{pmatrix}0&0\\\ 0&1\end{pmatrix}.\end{split}$

Here $D^{+}$ is the restriction of $D$ to even-graded sections, and $D^{-}$ is
its restriction to odd-graded sections. The entries of $e$ lie in the
unitisation of $\mathcal{A}_{-\infty}^{L}(S;D)$ by Proposition 4.4 and Lemmas
4.6 and 4.7.

By a standard form of the boundary map (4.7) (see e.g. page 356 of [8]),

$\operatorname{index}^{L}(D)=[e]-[f].$

Here we use the parametrix $Q$ of $D$ from Proposition 4.9. Extending
$\operatorname{Tr}_{\Phi}^{U}$ to the unitisation of
$\mathcal{A}_{-\infty}^{L}(S;D)$ in the usual way, by defining it to map the
unit element to $0$, we see that (5.11) holds. ∎

######  Definition 5.11.

If
$\operatorname{index}^{L}(D)\in\widetilde{K}_{0}(\mathcal{A}_{-\infty}^{L}(S)^{\Phi})$,
then the _localised $\Phi$-index_ of $D$ is the number

$\operatorname{index}_{\Phi}^{U}(D):=\operatorname{Tr}_{\Phi}^{U}(\operatorname{index}^{L}(D)).$

Proposition 4.14 has an important consequence.

######  Corollary 5.12.

If $D$ is invertible, then $\operatorname{index}_{\Phi}^{U}(D)=0$.

##  6 An index theorem

###  6.1 The main result

We will use some standard heat kernel asymptotics. We denote the
Atiyah–Segal–Singer integrand associated to $D$ and $\Phi$ by
$\operatorname{AS}_{\Phi}(D)$. See e.g. Theorem 6.16 in [5] for a general
expression.

######  Proposition 6.1.

Let $\kappa_{t}$ be the Schwartz kernel of $e^{-tD^{2}}$. There is a function
$R\colon(0,\infty)\to(0,\infty)$ such that $\lim_{t\downarrow 0}R(t)=0$, and
for all relatively compact open subsets $V\subset M$, the following hold.

  1. (a)

If there is a $\delta>0$ such that $d(\varphi(m),m)\geq\delta$ for all $m\in
M$, then

$\left|\int_{V}\operatorname{tr}(\gamma\Phi\kappa_{t}(\varphi^{-1}(m),m))\,dm\right|\leq\operatorname{vol}(V)R(t).$

  2. (b)

If $\varphi$ preserves an orientation on $M$, and is contained in a compact
group of isometries of $M$, and $\Phi$ is contained in a compact group of
isometries of $S$, then

$\left|\int_{V}\operatorname{tr}(\gamma\Phi\kappa_{t}(\varphi^{-1}(m),m))\,dm-\int_{V^{\varphi}}\operatorname{AS}_{\Phi}(D)\right|\leq\operatorname{vol}(V)R(t).$
(6.1)

###### Proof.

Part (a) follows from the global Gaussian upper bound for heat kernels in
manifolds of bounded geometry that was also used in the proof of Proposition
4.4.

Now suppose that $\varphi$ and $\Phi$ lie in compact groups, and that
$\varphi$ preserves an orientation. Then it follows from standard heat kernel
asymptotics that for every relatively compact open subset $V\subset M$, there
is a function $R$ with the property in part (b), possibly depending on $V$.
See Theorems 6.11 and 6.16 in [5], where one uses that the asymptotic
expansion of the heat kernel holds with respect to supremum norms on compact
sets. The fact that $R$ may be chosen independent of $V$ follows from bounded
geometry. This implies that the heat kernel $\kappa_{t}$ and its derivatives
are uniformly bounded, which implies the desired independence of $V$ as in
Section 2 of [12]. Compare also Proposition 4.2 in [6]. ∎

In the non-equivariant setting of [12], the relevant trace of $\gamma
e^{-tD^{2}}$ is independent of $t$ by Proposition 8.1 in [12]. In the current
situation, we need to take the limit $t\downarrow 0$ into account explicitly.
This leads to an additional assumption.

######  Proposition 6.2.

  1. (a)

If there is a $\delta>0$ such that $d(\varphi(m),m)\geq\delta$ for all $m\in
M$, then

$\lim_{t\downarrow 0}\operatorname{Tr}_{\Phi}^{U}(\gamma e^{-tD^{2}})=0.$

  2. (b)

Suppose that $\varphi$ preserves an orientation on $M$, and that $\varphi$ and
$\Phi$ lie in compact groups of isometries. Suppose that

$\lim_{j\to\infty}\frac{1}{\operatorname{vol}(U_{j})}\int_{U_{j}^{\varphi}}\operatorname{AS}_{\Phi}(D)$
(6.2)

converges. Then

$\lim_{t\downarrow 0}\operatorname{Tr}_{\Phi}^{U}(\gamma e^{-tD^{2}})$

converges to the same value as (6.2).

###### Proof.

Let $R$ be as in Proposition 6.1. Let $\varepsilon>0$, and let $T>0$ be such
that $R(t)<\varepsilon$ for all $t\in(0,T)$. Fix $t\in(0,T)$.

In case (a), choose $j\in\mathbb{N}$ such that

$\left|\operatorname{Tr}_{\Phi}^{U}(\gamma
e^{-tD^{2}})-\frac{1}{\operatorname{vol}(U_{j})}\int_{U_{j}}\operatorname{tr}(\gamma\Phi\kappa_{t}(\varphi^{-1}(m),m))\,dm\right|<\varepsilon.$

Then by Proposition 6.1(a), with $V=U_{j}$,

$|\operatorname{Tr}_{\Phi}^{U}(\gamma e^{-tD^{2}})|<2\varepsilon.$

In case (b), Proposition 6.1(b) implies that for all $j\in\mathbb{N}$,

$\left|\frac{1}{\operatorname{vol}(U_{j})}\int_{U_{j}}\operatorname{tr}(\gamma\Phi\kappa_{t}(\varphi^{-1}(m),m))\,dm-\frac{1}{\operatorname{vol}(U_{j})}\int_{U_{j}^{\varphi}}\operatorname{AS}_{\Phi}(D)\right|<\varepsilon.$
(6.3)

Let $a$ be the value of the limit (6.2). Let $N\in\mathbb{N}$ be such that for
all $j\geq N$,

$\left|\frac{1}{\operatorname{vol}(U_{j})}\int_{U_{j}^{\varphi}}\operatorname{AS}_{\Phi}(D)-a\right|<\varepsilon.$
(6.4)

By definition of $\operatorname{Tr}_{\Phi}^{U}$, there is a $j\geq N$ such
that

$\left|\operatorname{Tr}_{\Phi}^{U}(\gamma
e^{-tD^{2}})-\frac{1}{\operatorname{vol}(U_{j})}\int_{U_{j}}\operatorname{tr}(\gamma\Phi\kappa_{t}(\varphi^{-1}(m),m))\,dm\right|<\varepsilon.$
(6.5)

The inequalities (6.3)–(6.5) for such a $j$ imply that

$\left|\operatorname{Tr}_{\Phi}^{U}(\gamma
e^{-tD^{2}})-a\right|<3\varepsilon.$

∎

Our main result is the following index theorem, which follows from
Propositions 5.10 and 6.2.

######  Theorem 6.3.

  1. (a)

If there is a $\delta>0$ such that $d(\varphi(m),m)\geq\delta$ for all $m\in
M$, then
$\operatorname{index}^{L}(D)\in\widetilde{K}_{0}(\mathcal{A}_{-\infty}^{L}(S)^{\Phi})$,
and

$\operatorname{index}_{\Phi}^{U}(D)=0.$

  2. (b)

Suppose that $\varphi$ preserves an orientation on $M$, and that $\varphi$ and
$\Phi$ lie in compact groups of isometries. Suppose that

$\lim_{j\to\infty}\frac{1}{\operatorname{vol}(U_{j})}\int_{U_{j}^{\varphi}}\operatorname{AS}_{\Phi}(D)$
(6.6)

converges. Then
$\operatorname{index}^{L}(D)\in\widetilde{K}_{0}(\mathcal{A}_{-\infty}^{L}(S)^{\Phi})$,
and

$\operatorname{index}_{\Phi}^{U}(D)=\lim_{j\to\infty}\frac{1}{\operatorname{vol}(U_{j})}\int_{U_{j}^{\varphi}}\operatorname{AS}_{\Phi}(D).$
(6.7)

######  Remark 6.4.

Convergence of the limit (6.6) depends on the choice of $U$, so it is possible
that $U$ can be adapted so that this condition holds.

######  Remark 6.5.

In the case where $U=M$,

$\frac{1}{\operatorname{vol}(U_{j})}\int_{U_{j}^{\varphi}}\operatorname{AS}_{\Phi}(D)=\frac{1}{\operatorname{vol}(M_{j})}\int_{M_{j}^{\varphi}}\operatorname{AS}_{\Phi}(D).$

Because of bounded geometry, the integrand is bounded. Hence the absolute
value of the right hand side is less than or equal to a constant times

$\frac{\operatorname{vol}(M_{j}^{\varphi})}{\operatorname{vol}(M_{j})}.$

If $\varphi$ is not the identity map, then goes to zero as $j\to\infty$ in
many cases, because $M^{\varphi}$ has lower dimension than $M$. In such cases,
Theorem 6.3 implies that $\operatorname{index}_{\Phi}^{M}(D)=0$. For this
reason, it is important to allow $U$ to be different from $M$. Taking $U$ to
be a tubular neighbourhood of $M^{\varphi}$ seems to be a natural choice.

###  6.2 Consequences and applications

Assume from now on that $\varphi$ preserves an orientation on $M$, and that
$\varphi$ and $\Phi$ lie in compact groups of isometries.

Our main interest is in the case where the fixed-point set $M^{\varphi}$ is
noncompact, but in the case where it is compact, we obtain a natural
consequence of Theorem 6.3.

######  Corollary 6.6.

Suppose that $M^{\varphi}$ and $\bar{U}$ are compact. Then
$\operatorname{index}^{L}(D)\in\widetilde{K}_{0}(\mathcal{A}_{-\infty}^{L}(S))$,
and

$\operatorname{index}_{\Phi}^{U}(D)=\frac{1}{\operatorname{vol}(U)}\int_{M^{\varphi}}\operatorname{AS}_{\Phi}(D).$

###### Proof.

If $M^{\varphi}$ is compact, then $U_{j}=U$ for large enough $j$. So the claim
follows from Theorem 6.3. ∎

Corollary 6.6 implies that the index defined in this paper generalises the one
constructed in [10].

######  Corollary 6.7.

Suppose that $M^{\varphi}$ and $\bar{U}$ are compact. Then
$\operatorname{index}_{\Phi}^{U}(D)$ equals the index of Definition 2.7 in
[10] divided by $\operatorname{vol}(U)$.

###### Proof.

In this setting, Corollary 6.6 in this paper and Theorem 2.16 in [10] imply
that the two indices equal the same topological expression, up to a factor
$1/\operatorname{vol}(U)$. ∎

The number $\operatorname{index}_{\Phi}^{U}(D)$ also generalises the index
from [12].

######  Corollary 6.8.

Suppose that $\varphi=\operatorname{Id}_{M}$ and $\Phi=\operatorname{Id}_{S}$
are the identity maps on $M$ and $S$, respectively. Then
$\operatorname{index}^{L}(D)\in\widetilde{K}_{0}(\mathcal{A}_{-\infty}^{L}(D)^{\Phi})$,
and

$\operatorname{index}_{\operatorname{Id}_{S}}^{M}(D)=\dim_{\tau}(\operatorname{Ind}(D)),$

where the right hand side is as in Section 7 and 8 of [12].

###### Proof.

Now $\operatorname{Tr}_{\operatorname{Id}_{S}}^{M}$ is the trace $\tau$ in
Theorem 6.7 in [12]. By Proposition 8.1 in [12],

$\operatorname{Tr}_{\operatorname{Id}_{S}}^{M}(\operatorname{Ind}(D))=\operatorname{Tr}_{\operatorname{Id}_{S}}^{M}(\gamma
e^{-tD^{2}})$

for all $t>0$. In particular, $\lim_{t\downarrow
0}\operatorname{Tr}_{\operatorname{Id}_{S}}^{M}(\gamma e^{-tD^{2}})$
converges. So the claim follows from Proposition 5.10. ∎

######  Corollary 6.9.

If $U$ has infinite volume and

$\int_{M^{\varphi}}\operatorname{AS}_{\Phi}(D)$

converges, then $\operatorname{index}_{\Phi}^{U}(D)=0$.

###### Proof.

In this case,

$\lim_{j\to\infty}\frac{1}{\operatorname{vol}(U_{j})}\int_{U_{j}^{\varphi}}\operatorname{AS}_{\Phi}(D)=0.$

∎

Finally, we obtain an obstruction to positive scalar curvature metrics. Note
that the statement does not involve the algebras, asymptotic trace and indices
defined in this paper; they are only used in its proof.

######  Corollary 6.10.

If $M$ is $\operatorname{Spin}$ and has a complete Riemannian metric with
uniformly positive scalar curvature, then for every isometry $\varphi\colon
M\to M$ that has a lift $\Phi$ to the spinor bundle and lies in a compact
group of isometries, the limit

$\lim_{j\to\infty}\frac{1}{\operatorname{vol}(U_{j})}\int_{U_{j}^{\varphi}}\frac{\hat{A}(M^{\varphi})}{\det(1-\Phi
e^{-R})^{1/2}}$

equals zero if it converges. Here $R$ is the curvature of the connection on
the normal bundle of $M^{\varphi}$ induced by the Levi–Civita connection.

###### Proof.

If $D$ is the $\operatorname{Spin}$-Dirac operator, then

$\operatorname{AS}_{\Phi}(D)=\frac{\hat{A}(M^{\varphi})}{\det(1-\Phi
e^{-R})^{1/2}}.$

So the claim follows from the Lichnerowicz formula, Proposition 4.14 and
Theorem 6.3. ∎

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