# H-Unitality of Smooth Groupoid Algebras

Michael Francis

###### Abstract

We show that the convolution algebra of smooth, compactly supported functions
on a Lie groupoid is homologically unital in the sense of Wodzicki. We also
prove homological unitality of certain ideals associated to invariant closed
subsets of the unit space, leading to an excision property in Hochschild and
cyclic homology around invariant, closed subsets. This extends previous work
of the author establishing the Dixmier-Malliavin theorem in this setting.

##  1 Introduction

Let $A$ denote an associative algebra over $\mathbb{C}$. We do not assume $A$
is commutative or unital. Let us say that $A$ has the _weak factorization
property_ if every $a\in A$ can be expressed as a finite sum $a=\sum
b_{i}c_{i}$, where $b_{i},c_{i}\in A$. Notice that every unital algebra has
the weak factorization property, so this notion is only of interest in the
nonunital setting.

Recall that, given a Lie group $G$ equipped with Haar measure, the space
$C_{c}^{\infty}(G)$ of smooth, compactly-supported functions on $G$ becomes an
algebra with respect to convolution. This algebra is nonunital unless $G$ is
discrete (the unit wants to be the Dirac mass at $1$). In a 1978 paper,
Dixmier-Malliavin proved the following striking result.

######  Theorem 1.1 (3.1 Théorème, [7]).

For any Lie group $G$, the smooth convolution algebra $C_{c}^{\infty}(G)$ has
the weak factorization property.

The main technical ingredient of their proof is the following lemma whose own
proof is an intricate piece of hard analysis.

######  Lemma 1.2 (2.5 Lemme, [7]).

For any sequence $(c_{m})_{m=0}^{\infty}$ of positive scalars, there exist
$f_{0},f_{1}\in C_{c}^{\infty}(\mathbb{R})$ and scalars $(a_{m})$ with
$|a_{m}|\leq c_{m}$ such that
$\delta_{0}=f_{0}+\sum_{m=0}^{\infty}a_{m}f_{1}^{(m)}$. Here, $\delta_{0}$
denotes the Dirac mass at $0$ and the series converges in the sense of
compactly-supported distributions. The functions $f_{0},f_{1}$ may be chosen
to be supported in any fixed neighbourhood of the origin.

The (already nontrivial) $G=\mathbb{R}$ case of Theorem 1.1 follows quite
directly from Lemma 1.2. Given any $\varphi\in C_{c}^{\infty}(\mathbb{R})$, by
choosing the coefficients $a_{m}$ to converge sufficiently quickly to $0$, one
has that $\varphi_{1}\colonequals\sum_{m}a_{m}\varphi^{(k)}$ belongs to
$C_{c}^{\infty}(\mathbb{R})$ and so

$\varphi=\delta_{0}*\varphi=f_{0}*\varphi+\sum_{m=0}^{\infty}a_{m}f_{1}^{(m)}*\varphi=f_{0}*\varphi+f_{1}*\varphi_{1}.$

To prove Theorem 1.1 for a general Lie group, one writes $G$ (locally) as a
product of 1-parameter subgroups and factors one group at a time.

In [9], the author extended Dixmier-Malliavin’s result to the setting of Lie
groupoids:

######  Theorem 1.3 (Theorem 5.2, [9]).

For any Lie groupoid $G$, the smooth convolution algebra $C_{c}^{\infty}(G)$
has the weak factorization property.

We always assume our Lie groupoids $G$ are equipped with a smooth Haar system
so that a convolution product on $C_{c}^{\infty}(G)$ is defined (this
arbitrary choice can be avoided by working with appropriate densities in place
of functions, see [3], Section 2.5). The total space of $G$ (but not its unit
space) is permitted to be non-Hausdorff. As usual, in the non-Hausdorff case,
$C_{c}^{\infty}(G)$ is defined somewhat differently as the span of all
$C_{c}^{\infty}(U)$, where $U$ is a chart neighbourhood in $G$ (see, e.g. [3],
[2]. [11]).

In the case of groupoids, there is an additional phenomenon (absent in the
group case) of ideals associated to invariant subsets of the unit space. If
$Z$ is a closed, invariant subset of the unit space of $G$ and $G_{Z}\subseteq
G$ denotes the closed subgroupoid consisting of arrows whose source and target
lie in $Z$, we denote by $J_{Z}^{\infty}\subseteq C_{c}^{\infty}(G)$ the ideal
(with respect to convolution) of functions which vanish to infinite order on
$G_{Z}$. Weak factorization was also established by the author for these
ideals, in the case where $Z$ is a closed invariant submanifold:

######  Theorem 1.4 (Theorem 7.1, [9]).

For any Lie groupoid $G$, for any closed, invariant submanifold $Z$ of the
unit space of $G$, the corresponding ideal $J_{Z}^{\infty}\subseteq
C_{c}^{\infty}(G)$ has the weak factorization property.

The purpose of this article is to strengthen Theorem 1.3 and Theorem 1.4 by
showing that $C_{c}^{\infty}(G)$ and $J^{\infty}_{Z}$ are in fact
homologically unital in the sense defined by Wodzicki. This immediately
implies excision results for the Hochschild and cyclic homology of these
algebras.

The present article is also more general than [9] in that $Z$ is permitted to
be an arbitrary invariant, closed subset of the unit space, and not
necessarily a submanifold. Furthermore, an increased amount of care is taken
to ensure that the results and arguments apply in the case of non-Hausdorff
Lie groupoids; the results in [9] do apply in the non-Hausdorff setting, but
the verification of this is mostly left to the reader.

We give a brief review of the homological notions at play. For simplicity, we
only consider complex scalars, $\mathbb{C}$ being the ground field for all
algebras of interest here. In general, this restriction is not necessary,
though working with a field does streamline definitions somewhat.

Given an associative algebra $A$ over $\mathbb{C}$, _bar complex_ of $A$ is
the chain complex

${\ldots}$${A^{\otimes 3}}$${A^{\otimes
2}}$${A}$${0,}$$\scriptstyle{d^{\prime}}$$\scriptstyle{d^{\prime}}$$\scriptstyle{d^{\prime}}$

where the differential $d^{\prime}$ is determined by

$\displaystyle d^{\prime}(a_{0}\otimes\ldots\otimes a_{n})$
$\displaystyle=\sum_{i=0}^{n-1}(-1)^{i}a_{0}\otimes\ldots\otimes
a_{i}a_{i+1}\otimes\ldots\otimes a_{n}$ $\displaystyle n\geq 1.$

######  Remark 1.5.

If one modifies the above definition of $d^{\prime}$ to include the “wrap
around term” $a_{n}a_{0}\otimes\ldots\otimes a_{n-1}$, then one gets the
differential $d$ involved in the definition of Hochschild and cyclic homology.

The following terminology was introduced by Wodzicki.

######  Definition 1.6 ([20]).

An associative algebra $A$ over $\mathbb{C}$ is said to be _homologically
unital_ or _H-unital_ if its bar complex is acyclic. That is, if

${\ldots}$${A^{\otimes 3}}$${A^{\otimes
2}}$${A}$${0}$$\scriptstyle{d^{\prime}}$$\scriptstyle{d^{\prime}}$$\scriptstyle{d^{\prime}}$

is an exact sequence.

######  Example 1.7.

Every unital algebra $A$ is H-unital. Indeed, $x\mapsto 1\otimes x:A^{\otimes
n}\to A^{\otimes(n+1)}$ defines a contracting chain homotopy for the bar
complex. More generally, if $A$ is _locally unital_ in the sense that, for any
finite set of elements, there is an element acting as a unit for those
elements, then $A$ is H-unital. Any Banach algebra with a bounded approximate
unit is H-unital. In particular, all C*-algebras are H-unital. More in the
spirit of the present article, if $M$ is a smooth manifold and $Z\subseteq M$
is any closed subset, then then ideal $I_{Z}^{\infty}\subseteq C^{\infty}(M)$
of smooth functions which vanish with all their derivatives on $Z$ is
H-unital.

From Wodzicki’s groundbreaking paper on excision in cyclic homology and
algebraic K-theory, H-unitality has the following relationship to excision.

######  Theorem 1.8 (3.1 Theorem, [20]).

Let $A$ be an associative algebra over $\mathbb{C}$. Then, the following are
equivalent:

  1. 1.

$A$ has the excision property for Hochschild homology.

  2. 2.

$A$ has the excision property for cyclic homology.

  3. 3.

$A$ is H-unital.

For a precise explanation of the terminology above, one may see [20], or
Section 1.4 of [12]. The key point is the following: if $I$ is an H-unital
algebra, and $B$ contains $I$ as an ideal, then the short exact sequence of
$\mathbb{C}$-algebras

${0}$${I}$${B}$${B/I}$${0}$

induces a corresponding long exact sequence in Hochschild/cyclic homology.

Note that, if $A$ is H-unital, then, in particular, the mapping $A\otimes A\to
A$ is surjective. That is, H-unital algebras have the weak factorization
property.

In this article, we prove the following result which may be viewed as a
generalization of the Dixmier-Malliavin theorem for Lie groupoids of [9].

######  Theorem 1.9.

For any Lie groupoid $G$, the smooth convolution algebra $C_{c}^{\infty}(G)$
is H-unital.

######  Remark 1.10.

The author would like to thank Xiang Tang for pointing out the similarity of
Theorem 1.9 above to Proposition 2 of [6]. We note, however, that, because the
tensor products in this article are algebraic tensor products and the tensor
products in [6] are completed tensor products $\overline{\otimes}$ satisfying
$C_{c}^{\infty}(M)\overline{\otimes}C_{c}^{\infty}(N)=C_{c}^{\infty}(M\times
N)$ for all smooth manifolds $M$ and $N$, the results are distinct.

The second purpose of this article is to prove the following generalization of
Theorem 1.4.

######  Theorem 1.11.

For any Lie groupoid $G$, for any closed, invariant subset $Z$ of the unit
space of $G$, the corresponding ideal $J_{Z}^{\infty}\subseteq
C_{c}^{\infty}(G)$ is H-unital.

Theorems 1.8, 1.9 and 1.11 have the following important corollary.

######  Corollary 1.12.

The algebras $C_{c}^{\infty}(G)$ and $J_{Z}^{\infty}$ have the excision
property in Hochschild homology and cyclic homology.

The main practical consequence of our results is that any extension of the
form

${0}$${J_{Z}^{\infty}}$${C_{c}^{\infty}(G)}$${C_{c}^{\infty}(G)/J_{Z}^{\infty}}$${0}$

induces a corresponding long exact sequence in Hochschild/cyclic homology
(since $\mathbb{C}$ is a field, the purity hypothesis in [20] is automatically
satisfied). It is hoped that this will lead to an improved understanding of
localization around invariant subsets in calculations of the cyclic and
Hochschild homology of convolution algebras of Lie groupoids. Examples of
calculations utilizing this excision principle will be considered elsewhere.
Such calculations fall squarely within Connes’ noncommutative geometry
program. One may see [14] for recent progress in this area.

We now discuss the criterion that will be used to establish H-unitality. As
mentioned above, if an algebra $A$ is unital, then $x\mapsto 1\otimes
x:A^{\otimes n}\to A^{\otimes(n+1)}$ defines a contracting homotopy for the
bar complex. More generally, if there exists a $\mathbb{C}$-linear map
$\phi:A\to A\otimes A$ which is right $A$-linear (where the right $A$-module
structure on $A\otimes A$ is such that $(a\otimes b)\cdot c=a\otimes(bc)$ for
$a,b,c\in A$) and makes the diagram

${A}$${A\otimes A}$${A}$$\scriptstyle{\phi}$multiplication

commutative, then a simple calculation shows that
$\phi\otimes\mathrm{id}:A^{\otimes n}\to A^{\otimes(n+1)}$ gives a contracting
homotopy for the bar complex. In fact, because any cycle only involves
finitely any elements of $A$, it is not actually necessary to have a globally-
defined map $\phi$; one can get by with a family of locally-defined maps. This
is formalized in the following result from [20] (specialized to the case
$k=\mathbb{C}$) and we refer the reader to that source for the proof.

######  Proposition 1.13 (4.1 Proposition, [20]).

Let $A$ be an associative $\mathbb{C}$-algebra. Suppose that, for every finite
set $\mathscr{P}\subseteq A$, there exists a right ideal $A_{0}\subseteq A$
such that $\mathscr{P}\subseteq A_{0}$ and a $\mathbb{C}$-linear map
$\phi:A_{0}\to A\otimes A$ such that:

  1. 1.

$\phi$ is a map of right $A$-modules (the right $A$-module structure on
$A\otimes A$ is such that $(a\otimes b)\cdot c=a\otimes(bc)$ for $a,b,c\in
A$).

  2. 2.

The diagram

${A_{0}}$${A\otimes A}$${A}$inclusion$\scriptstyle{\phi}$multiplication

is commutative.

One straightforward consequence of Proposition 1.13 is H-unitality of “locally
unital” algebras; if every finite subset of $A$ admits a common left unit,
then $A$ is H-unital. While the latter statement suffices for many algebras of
interest in noncommutative geometry, the algebras considered here are not
locally unital. Already, the convolution algebra $C_{c}^{\infty}(\mathbb{R})$
is not locally unital (under Fourier transform, it is an algebra of analytic
functions under pointwise multiplication). Nonetheless, our proofs of
H-unitality will pass through the sufficient condition of Proposition 1.13. As
an instructive example, we establish H-unitality for
$C_{c}^{\infty}(\mathbb{R})$ below.

######  Example 1.14.

Let $X=\frac{d}{dx}$, considered as an operator on
$C_{c}^{\infty}(\mathbb{R})$. Given a formal series $P(z)=\sum_{m\geq
0}a_{m}z^{m}\in\mathbb{C}[[z]]$, one may define
$P(X):\operatorname{dom}(P(X))\to C_{c}^{\infty}(\mathbb{R})$ by

$\displaystyle\operatorname{dom}(P(X))$ $\displaystyle=\\{\varphi\in
C_{c}^{\infty}(\mathbb{R}):{\textstyle\sum_{m\geq
0}}|a_{m}|\left\|X^{m+r}\varphi\right\|<\infty\text{ for all }r\geq 0\\}$
$\displaystyle P(X)\varphi$ $\displaystyle=\sum_{m\geq 0}a_{m}X^{m}\varphi,$

where $\|\cdot\|$ denotes the uniform norm. One may check that
$\operatorname{dom}(P(X))$ is an ideal in $C_{c}^{\infty}(\mathbb{R})$. Given
any finite subset $\mathscr{P}\subseteq C_{c}^{\infty}(\mathbb{R})$, one may
use Lemma 1.2 to find a representation
$\delta_{0}=f_{0}+\sum_{m=0}^{\infty}a_{m}f_{1}^{(m)}$ of the Dirac measure in
terms of $f_{0},f_{1}\in C_{c}^{\infty}(\mathbb{R})$ such that
$P(z)=\sum_{m\geq 0}a_{m}z^{m}$ has
$\mathscr{P}\subseteq\operatorname{dom}(P(X))$. Then, the map

$\varphi\mapsto f_{0}\otimes\varphi+f_{1}\otimes
P(X)\varphi:\operatorname{dom}(P(X))\to C_{c}^{\infty}(\mathbb{R})\otimes
C_{c}^{\infty}(\mathbb{R})$

satisfies the conditions of Proposition 1.13 by the same calculation given
below the statement of Lemma 1.2, so $C_{c}^{\infty}(\mathbb{R})$ is H-unital
by Proposition 1.13.

We give a brief summary of the contents of this article. Section 2 reviews
certain aspects of the theory of non-Hausdorff smooth manifolds. A number of
examples are included with the intention being to clarify where complications
can arise, especially as pertains to supports of functions and flows of vector
fields. Section 3 is a technical section devoted to domains of operators
obtained by taking power series in a fixed set of vector fields. The goal here
is to confirm that, by taking series coefficients to be sufficiently small,
domains of such operators can be made to contain any finite set of bump
functions. In Section 4, we establish conventions for Lie groupoids and smooth
actions of Lie groupoids. Section 5 is concerned with our first main result,
the H-unitality of the smooth convolution algebras of groupoids. Section 6 is
concerned with factorization with respect to pointwise multiplication of
smooth functions vanishing to infinite order on a closed set, relative to a
submersion. Section 7 contains our second main result, the H-unitality of
infinite-order vanishing ideals in smooth convolution algebras of groupoids.

##  2 Non-Hausdorff smooth manifolds

It is frequently necessary in noncommutative geometry to work with non-
Hausdorff manifolds. In this section, we highlight some of the complications
that can arise and review the standard workarounds. Readers who are only
interested in Hausdorff Lie groupoids may safely ignore this section. Readers
who are already well-versed in the techniques needed to deal with non-
Hausdorff Lie groupoids may also prefer to skip this material.

In this article, a smooth manifold refers to a second-countable topological
space $M$ equipped with a smooth atlas (more precisely, an equivalence class
of smooth atlases or, alternatively, a maximal smooth atlas) of some fixed
dimension. The topology of $M$ is not required to be Hausdorff. Of course, $M$
is locally Hausdorff, being locally Euclidean.

######  Remark 2.1.

Actually, it would do little harm to drop the assumption of second-countablity
as well, provided we at least assume that every _component_ of $M$ is second-
countable. This amounts to working with possibly uncountable disjoint unions
of second countable manifolds.

Much of the basic theory of smooth manifolds goes through unchanged without
the Hausdorff hypothesis. For example, one can talk about smooth vector fields
on and smooth maps between non-Hausdorff smooth manifolds by requiring
smoothness in every chart. Many of the issues that do arise are due to the
nonexistence of partitions of unity. Ultimately, this stems from an
undersupply of scalar-valued functions that look smooth in _every_ chart. To
some extent, it is possible to define one’s way around this issue by modifying
the definition of bump functions. The simple definition below is due to
Connes. More sophisticated approaches with better functorial properties are
also possible. See the unpublished manuscript [5].

######  Definition 2.2.

Let $M$ be a possibly non-Hausdorff smooth manifold. Then, $C_{c}^{\infty}(M)$
denotes the linear span of all functions $\varphi:M\to\mathbb{C}$ that are
given as the extension by zero of a function of the form
$f\circ\chi:U\to\mathbb{C}$ where $U\subseteq M$ is open,
$\chi:U\to\mathbb{R}^{d}$ is a diffeomorphism, and $f\in
C_{c}^{\infty}(\mathbb{R}^{d})$.

If $M$ is Hausdorff, the usual meaning of $C_{c}^{\infty}(M)$ is recovered. On
the other hand, if $M$ is non-Hausdorff, the notation is $C_{c}^{\infty}(M)$
is misleading in several ways:

  1. (i)

A function $\varphi\in C_{c}^{\infty}(M)$ need not be smooth in every chart
(or even continuous).

  2. (ii)

The support of $\varphi\in C_{c}^{\infty}(M)$, defined in the usual way as the
complement of the largest open set where $\varphi$ is zero (equivalently, the
closure of the nonvanishing locus of $\varphi$), need not be compact.

  3. (iii)

$C_{c}^{\infty}(M)$ need not be closed under the operation of pointwise
product.

######  Example 2.3.

Let
$M=\Big{(}(-\infty,0)\times\\{0\\}\Big{)}\cup\bigcup_{n\in\mathbb{Z}}\Big{(}[0,\infty)\times\\{n\\}\Big{)}$,
with smooth manifold structure determined by the atlas
$\\{\chi_{n}:U_{n}\to\mathbb{R}\\}_{n\in\mathbb{Z}}$ defined by

$\displaystyle
U_{n}=\left((-\infty,0)\times\\{0\\}\right)\cup\left([0,\infty)\times\\{n\\}\right)$
$\displaystyle\chi_{n}=\mathrm{proj}_{1}|_{U_{n}}$ $\displaystyle
n\in\mathbb{Z}.$

Choose some $f\in C_{c}^{\infty}(\mathbb{R})$ with $\mathrm{supp}(f)=[-1,1]$
and define $\varphi_{n}$ to be the extension by zero to all of $M$ of
$f\circ\chi_{n}$, so that $\varphi_{n}\in C_{c}^{\infty}(M)$ by definition.
Then,

  1. (i)

The restriction of $\varphi_{m}$ to $U_{n}$ is not smooth if $m\neq n$.

  2. (ii)

The support of $\varphi_{0}$ is
$\left([-1,1]\times\\{0\\}\right)\cup\\{(0,n):n\in\mathbb{Z}\\}$, which is not
compact.

  3. (iii)

The pointwise product of $\varphi_{m}$ and $\varphi_{n}$ does not belong to
$C_{c}^{\infty}(M)$ if $m\neq n$.

$\vdots$$\vdots$

Note as well that the space in the example above does arise naturally in
geometry. For example, foliate the lower half of the cylinder
$S^{1}\times\mathbb{R}$ into circles and the upper half into spirals so that
there is one-sided holonomy at the equator $S^{1}\times\\{0\\}$. The
restriction of the holonomy groupoid of this foliation to a vertical line
transversal is diffeomorphic to $M$.

Although it can occur that
$\operatorname{supp}(\varphi)\colonequals\overline{\varphi^{-1}(\mathbb{C}^{\times})}$
is noncompact for $\varphi\in C_{c}^{\infty}(M)$ when $M$ is noncompact, we do
at least have the following.

######  Proposition 2.4.

Let $\tau:M\to B$ be a smooth map of smooth manifolds, where $B$ is Hausdorff
and $M$ is possibly non-Hausdorff. Then, for all $\varphi\in
C_{c}^{\infty}(M)$, we have that $\tau(\operatorname{supp}(\varphi))$ is
compact.

###### Proof.

Write $\varphi=\sum_{i=1}^{n}(f_{i}\circ\chi_{i})_{0}$ where
$\chi_{i}:U_{i}\to\mathbb{R}^{d}$ are charts, $f_{i}\in
C_{c}^{\infty}(\mathbb{R}^{d})$, and the $0$ subscripts denote the operation
of extension by zero. Then,
$K=\bigcup_{i=1}^{n}\chi_{i}^{-1}(\operatorname{supp}(f_{i}))$ is a compact
(but possibly not closed) subset of $M$ containing
$\varphi^{-1}(\mathbb{C}^{\times})$. The conclusion follows from the
elementary point-set topological lemma below, with
$S=\varphi^{-1}(\mathbb{C}^{\times})$. ∎

######  Lemma 2.5.

Let $f:X\to Y$ be a continuous map of topological spaces, where $Y$ is
Hausdorff and $X$ is possibly non-Hausdorff. Suppose that $S\subseteq
K\subseteq X$ and $K$ is compact (but possibly not closed). Then,
$f(\overline{S})=\overline{f(S)}$ and this is a compact subset of $Y$.

###### Proof.

We have
$f\left(\overline{S}\right)\subseteq\overline{f(S)}\subseteq\overline{f\left(\overline{S}\cap
K\right)}=f\left(\overline{S}\cap K\right)\subseteq
f\left(\overline{S}\right)$, so all these sets are equal to the compact set
$f(\overline{S}\cap K)$. ∎

Another issue with Definition 2.2 is the poor control that the support of an
element $\varphi\in C_{c}^{\infty}(M)$ exerts over the supports of the
summands in the possible decompositions $\varphi$ into functions coming from
charts. One might expect that, if $U$ is open and
$\mathrm{supp}(\varphi)\subseteq U$, then $\varphi\in
C_{c}^{\infty}(U)\subseteq C_{c}^{\infty}(M)$, i.e. $\varphi$ can be expressed
in terms of bump functions defined on chart neighbourhoods contained in $U$.
In general, this is not assured, as the example below illustrates.

######  Example 2.6.

Fix a smooth function $\theta:\mathbb{R}\to\mathbb{R}$ with $\theta(x)=0$ for
$x\leq 0$ that restricts to a diffeomorphism
$\theta_{+}:(0,\infty)\to(0,\infty)$. Consider the non-Hausdorff smooth
manifold $M=\mathbb{R}\cup_{\theta_{+}}\mathbb{R}$ obtained by using
$\theta_{+}$ to glue two copies of $\mathbb{R}$ along $(0,\infty)$. More
precisely,

$\displaystyle M$
$\displaystyle=\left((-\infty,0]\times\\{-1,1\\}\right)\cup\left((0,\infty)\times\\{0\\}\right)$
$\displaystyle U_{+}$
$\displaystyle=\left((-\infty,0]\times\\{1\\}\right)\cup\left((0,\infty)\times\\{0\\}\right)$
$\displaystyle U_{-}$
$\displaystyle=\left((-\infty,0]\times\\{-1\\}\right)\cup\left((0,\infty)\times\\{0\\}\right)$
$M$$U_{+}$$U_{-}$

with non-Hausdorff smooth manifold structure determined by the two charts:

$\displaystyle\chi_{+}:U_{+}\to\mathbb{R}$ $\displaystyle\chi_{+}(x,y)$
$\displaystyle=x$ $\displaystyle\chi_{-}:U_{-}\to\mathbb{R}$
$\displaystyle\chi_{-}(x,y)$ $\displaystyle=\begin{cases}x&x\leq 0\\\
\theta_{+}(x)&x>0.\end{cases}$

Fix $g\in C_{c}^{\infty}(\mathbb{R})$ with $\mathrm{supp}(g)=[-1,1]$ such that
$g(x)=x$ for all $x$ in a neighbourhood of $0$. Define $f\in
C_{c}^{\infty}(\mathbb{R})$ by $f(x)=g(\theta(x))$ for $x>0$ and $f(x)=0$ for
$x\leq 0$. Let $\varphi\in C_{c}^{\infty}(M)$ be given by
$\varphi=(g\circ\chi_{-})_{0}-(f\circ\chi_{+})_{0}$, where the subscript $0$
denotes extension by zero.

$f$$g$$\varphi$

Then, $\mathrm{supp}(\varphi)=[-1,0]\times\\{-1\\}$ (note the point $(0,1)$
does not belong to the support because $\varphi$ vanishes identically on
$U_{+}$). In particular, $\mathrm{supp}(\varphi)\subseteq U_{-}$. However,
$\varphi$ does not belong to $C_{c}^{\infty}(U_{-})\subseteq
C_{c}^{\infty}(M)$.

Another issue with Definition 2.2 of relevance to us relates to smooth vector
fields. On the one hand, there is no difficulty defining the tangent bundle
$TM$ of a non-Hausdorff smooth manifold $M$ and defining a vector field $X$ to
be a section of $TM$ which is smooth in every chart. On the other hand,
integral curves of $X$ may not be unique, preventing one from talking about
the flow of $X$. In fact things already go wrong at infinitesimal level, i.e.
one does not even have a well-defined linear map $X:C_{c}^{\infty}(M)\to
C_{c}^{\infty}(M)$. The operators defined by $X$ in charts need not patch
together. This issue arises already in the simplest examples, as shown in
Example 2.7 below.

######  Example 2.7.

Let $M=\mathbb{R}^{\times}\cup\\{0_{1},0_{2}\\}$, the “line with two origins”
with noncommutative smooth manifold structure coming from the two obvious
charts $\chi_{1},\chi_{2}:M\to\mathbb{R}$. Let $X$ be the (global) smooth
vector field on $M$ which coincides with $\frac{d}{dx}$ in both charts. Fix
$f\in C_{c}^{\infty}(\mathbb{R})$ with $f(0)=0$ and $f^{\prime}(0)=1$. For
$i=1,2$, let $\varphi_{i}\in C_{c}^{\infty}(M)$ be the extension by zero of
$f\circ\chi_{i}$ and let $\psi_{i}\in C_{c}^{\infty}(M)$ be the extension by
zero of $f^{\prime}\circ\chi_{i}$. Then, $\varphi_{1}-\varphi_{2}=0$, but
$\psi_{1}-\psi_{2}$ is the function which is zero on $\mathbb{R}^{\times}$,
$1$ at $0_{1}$ and $-1$ at $0_{2}$. Thus, $\varphi_{1}-\varphi_{2}=0$ and
$\psi_{1}-\psi_{2}\neq 0$. This shows that the result of applying $X$ to the
zero function is not unambiguously defined.

A flow for a smooth vector field $X$ on a smooth manifold $M$ is a smooth map
$(t,m)\mapsto\phi_{t}:W\to M$ where $W\subseteq\mathbb{R}\times M$ is an open
set containing $\\{0\\}\times M$ whose intersection with
$\mathbb{R}\times\\{m\\}$ is connected for all $m\in M$ such that
$\phi_{0}(m)=m$ for all $m\in M$ and $\frac{d}{dt}\phi_{t}(m)=X(m)$ for all
$(t,m)\in W$. It is easy to see that, if $X$ has a flow, then any two flows
agree on the intersection of their domains, and there is a unique maximal flow
$(t,m)\mapsto\phi_{t}(m):\mathbb{R}\times M$ which furthermore satisfies
$\phi_{s+t}(m)=\phi_{s}(\phi_{t}(m))$ whenever $(t,m),(s,\phi_{t}(m))\in W$.

######  Definition 2.8.

Let $M$ be a smooth manifold, not necessarily Hausdorff, and let $X$ be a
smooth vector field on $M$. We say that $X$ is non-branching if it has a flow.

A non-branching vector field $X$ with flow $\phi$ does determine a well-
defined linear map $X:C_{c}^{\infty}(M)\to C_{c}^{\infty}(M)$. One way to see
this is to note that the locally defined operators on charts agree with the
global one defined by $(Xf)(p)=\lim_{t\to 0}(f(\phi_{t}(p)-f(p))/t$, so they
patch together. Indeed, one may check that existence of the flow of $X$ is
equivalent to its well-definedness as an operator on $C_{c}^{\infty}(M)$. Note
the equation $(Xf)(p)=\lim_{t\to 0}(f(\phi_{t}(p)-f(p))/t$ can be also be used
to justify the following expected fact that $X$, viewed as an operator on
$C_{c}^{\infty}(M)$, does not increase supports.

######  Proposition 2.9.

If $X$ is a smooth, non-branching vector field on a possibly non-Hausdorff
smooth manifold $M$, then
$\operatorname{supp}(X\psi)\subseteq\operatorname{supp}(\psi)$ for all
$\psi\in C_{c}^{\infty}(M)$. ∎

If $X$ is a _complete_ non-branching vector field, we consider its flow as a
smooth, 1-parameter group of diffeomorphisms of $M$ and denote it by $t\mapsto
e^{tX}$. In other words:

$\displaystyle\tfrac{d}{dt}e^{tX}m=X(m)$ $\displaystyle m\in M$

In spite of the shortcomings highlighted in Examples 2.3, 2.6 and 2.7,
Definition 2.2 does allow one to bypass many issues relating to nonexistence
of partitions of unity. The following simple lemma will suffice for many
purposes.

######  Lemma 2.10 (Cf. [10], Lemma 1.3.).

Let $M$ be a possibly non-Hausdorff smooth manifold. Let $(U_{i})_{i\in I}$ be
a cover of $M$ by Hausdorff open sets. Then, every $\varphi\in
C_{c}^{\infty}(M)$ can be expressed as a finite sum $\sum\varphi_{i}$ where
$\varphi_{i}$ is the extension by zero to all of $M$ of some $f_{i}\in
C_{c}^{\infty}(U_{i})$.

###### Proof.

Without loss of generality, $\varphi$ is the extension by zero to all of $M$
of some $f\in C_{c}^{\infty}(U)$, where $U\subseteq M$ is open and Hausdorff.
Then, by usual theory of Hausdorff smooth manifolds, we can write $f$ as a
finite sum $\sum f_{i}$ where $f_{i}\in C_{c}^{\infty}(U_{i}\cap U)\subseteq
C_{c}^{\infty}(U_{i})$. Then, letting $\varphi_{i}$ be the extension by zero
to all of $M$ of $f_{i}$, we have $\varphi=\sum\varphi_{i}$. ∎

The following proposition shows some other important respects in which
$C_{c}^{\infty}(M)$ is well-behaved.

######  Proposition 2.11.

  1. 1.

If $M$ and $N$ are possibly non-Hausdorff smooth manifolds, $f\in
C_{c}^{\infty}(M)$, $g\in C_{c}^{\infty}(N)$, then $f\otimes g\in
C_{c}^{\infty}(M\times N)$ (here, $(f\otimes g)(m,n)=f(m)g(n)$).

  2. 2.

If $M$ is a possibly non-Hausdorff smooth manifold, and $N\subseteq M$ is a
closed submanifold, then restriction gives a surjective map
$C_{c}^{\infty}(M)\to C_{c}^{\infty}(N)$.

  3. 3.

If $M$ is a possibly non-Hausdorff smooth manifold and $\theta:M\to M$ is a
self-diffeomorphism, then pullback along $\theta$ determines a linear
bijection $C_{c}^{\infty}(M)\to C_{c}^{\infty}(M)$.

  4. 4.

If $\tau:M\to N$ is a smooth map of manifolds, with $N$ Hausdorff, then
$C_{c}^{\infty}(M)$ is a $C^{\infty}(N)$-module with respect to
$f\cdot\varphi\colonequals(f\circ\tau)g$, $f\in C^{\infty}(N)$, $\varphi\in
C_{c}^{\infty}(M)$.

∎

The above statements are either immediate from Definition 2.2, or follow from
an application of Lemma 2.10.

##  3 Infinite series of vector fields

The following section is somewhat technical, but more or less elementary. The
results obtained will be somewhat stronger than strictly necessary, but will
enable us to streamline arguments later on. Essentially we need elaborated
versions of the following elementary fact: if $(f_{n})_{n\geq 0}$ is a
sequence in $C_{c}^{\infty}(\mathbb{R})$ with uniformly bounded supports, then
there exist positive scalars $(c_{n})_{n\geq 0}$ such that, for any sequence
of scalars $(a_{n})_{n\geq 0}$ with $|a_{n}|\leq c_{n}$, the series
$\sum_{n\geq 0}a_{n}f_{n}$ converges absolutely and uniformly to a function in
$C_{c}^{\infty}(\mathbb{R})$. To see this, one may take, for instance,
$c_{n}=\min_{k\leq n}\left(2^{n}\|f^{(k)}_{n}\|\right)^{-1}$.

The following definition is well-suited to our purposes.

######  Definition 3.1.

Let $P(z)=\sum_{m\geq 0}a_{m}z^{m}\in\mathbb{C}[[z]]$ be a formal series and
$X$ a smooth vector field on $\mathbb{R}^{d}$, thought of as a differential
operator $X:C_{c}^{\infty}(\mathbb{R}^{d})\to C_{c}^{\infty}(\mathbb{R}^{d})$.
Then, we define $\operatorname{dom}(P(X))$ to be the set of all $\varphi\in
C_{c}^{\infty}(\mathbb{R}^{d})$ such that $\sum_{m\geq
0}|a_{m}|\|\tfrac{\partial^{\alpha}}{\partial
x^{\alpha}}X^{m}\varphi\|<\infty$ for every multi-index
$\alpha\in\mathbb{N}^{d}$ and put

$P(X)\varphi=\sum_{m\geq 0}a_{m}X^{m}\varphi$

for all $\varphi\in\operatorname{dom}(P(X))$.

Some basic consequences of this definition are collected in the following
proposition.

######  Proposition 3.2.

Suppose $P(z)=\sum_{m\geq 0}a_{m}z^{m}\in\mathbb{C}[[z]]$ and $X$ is a smooth
vector field on $\mathbb{R}^{d}$. Define $P(X):\operatorname{dom}(P(X))\to
C_{c}^{\infty}(\mathbb{R}^{d})$ as above.

  1. 1.

$\frac{\partial^{\alpha}}{\partial x^{\alpha}}P(X)\varphi=\sum_{m\geq
0}\frac{\partial^{\alpha}}{\partial x^{\alpha}}X^{m}\varphi$ for all
$\alpha\in\mathbb{N}^{d}$, $\varphi\in C_{c}^{\infty}(\mathbb{R}^{d})$, where
the series converges absolutely and uniformly.

  2. 2.

$\operatorname{supp}(P(X)\varphi)\subseteq\operatorname{supp}(\varphi)$ for
all $\varphi\in\operatorname{dom}(P(X))$.

  3. 3.

The definition of $P(X)$ is coordinate-independent; if
$\theta:\mathbb{R}^{d}\to\mathbb{R}^{d}$ is a diffeomorphism, then
$\operatorname{dom}(P(\theta^{*}(X)))=\theta^{*}(\operatorname{dom}(P(X)))$
and $P(\theta^{*}(X))=\theta^{*}(P(X))$.

  4. 4.

If $P(z)=\sum_{m\geq 0}a_{m}z^{m},Q(z)=\sum_{m\geq
0}b_{m}z^{m}\in\mathbb{C}[[z]]$ and $|a_{m}|\leq|b_{m}|$ for $m\geq 0$, then
$\operatorname{dom}(Q(X))\subseteq\operatorname{dom}(P(X))$.

∎

The proof of the above proposition is straightforward and we omit it. For
example, property (3) comes down to the fact that
$\frac{\partial^{\alpha}}{\partial x^{\alpha}}(f\circ\theta)$ can be expressed
as a finite sum
$\sum_{|\beta|\leq|\alpha|}\left[(\frac{\partial^{\beta}}{\partial
x^{\beta}}f)\circ\theta\right]\cdot\theta_{\beta}$, where $\theta_{\beta}$ is
a smooth function made up of partial derivatives of components of $\theta$.

Property (3) permits us to make the following definition:

######  Definition 3.3.

Suppose that $M$ is a not-necessarily-Hausdorff smooth manifold, $X$ is a
smooth, non-branching (Definition 2.8) vector field on $M$ and
$P(z)=\sum_{m\geq 0}a_{m}z^{m}\in\mathbb{C}[[z]]$. Then, we define
$\operatorname{dom}(P(X))$ to be the linear span of functions $\varphi$ on $M$
given as the extension by zero of functions $f\circ\chi$, where
$\chi:U\to\mathbb{R}^{d}$ is a Euclidean chart and $f\in
C_{c}^{\infty}(\mathbb{R}^{d})$ belongs to $\operatorname{dom}(P(Y))$, where
$Y=\chi_{*}(X|_{U})$. We define $P(X):\operatorname{dom}(P(X))\to
C_{c}^{\infty}(M)$ by

$P(X)\varphi=\sum_{m\geq 0}a_{m}X^{m}\varphi,$

this series being absolutely and uniformly convergent for
$\varphi\in\operatorname{dom}(P(X)$.

The lemma below will be used to control the domains of compositions of
operators $P(X)$.

######  Lemma 3.4.

Suppose that $(a_{m})_{m\geq 0}$ is a sequence of nonnegative scalars. Then,
there exists a non-increasing sequence of positive scalars $(c_{m})_{m\geq 0}$
such that

$\sum_{m_{1},\ldots,m_{n}\geq 0}c_{m_{1}}\cdots
c_{m_{n}}a_{m_{1}+\ldots+m_{n}+r}<\infty$

for all integers $n,r$ with $n\geq 1$ and $r\geq 0$.

###### Proof.

Without loss of generality, the terms of $(a_{m})$ are positive. We make
repeated use of the fact that, given positive scalars $(a_{m}^{(i)})_{m,i\geq
0}$, there is a positive sequence $(b_{m})_{m\geq 0}$ such that $\sum_{m\geq
0}b_{m}a_{m}^{(i)}<\infty$ for every $i\geq 0$. One may use, for instance,
$b_{m}=\left(2^{m}\max_{i\leq m}a_{m}^{(i)}\right)^{-1}$.

Applying the aforementioned fact, there is a sequence of positive scalars
$(c^{(1)}_{m})_{m\geq 0}$ such that $a_{r}^{(2)}\colonequals\sum_{m\geq
0}c^{(1)}_{m}a_{m+r}<\infty$ for all $r\geq 0$. Applying this fact again,
there is a sequence of positive scalars $(c^{(2)}_{m})_{m\geq 0}$ such that
$a_{r}^{(3)}\colonequals\sum_{m\geq
0}c^{(2)}_{m}a_{m+r}^{(2)}=\sum_{m_{1},m_{2}\geq
0}c^{(1)}_{m_{1}}c^{(2)}_{m_{2}}a_{m_{1}+m_{2}+r}<\infty$ for all $r\geq 0$.
Continuing in this way, we obtain positive sequences $(c_{m}^{(i)})_{m\geq 0}$
for $i=1,\ldots,m$ such that

$\sum_{m_{1},\ldots,m_{n}\geq 0}c_{m_{1}}^{(1)}\cdots
c_{m_{n}}^{(n)}a_{m_{1}+\ldots+m_{n}+r}<\infty$

for all $r\geq 0$.

Next, put $c_{m}\colonequals\min(c_{m}^{(1)},\ldots,c_{m}^{(m)})$. We show
that $(c_{m})$ satisfies the conclusion of the lemma by induction on $n\geq
1$. Since $c_{m}\leq c_{m}^{(1)}$ for all $m$, the statement holds for $n=1$.
Let $n\geq 2$. Then,

$\displaystyle\sum_{m_{1},\ldots,m_{n}\geq 0}c_{m_{1}}\cdots
c_{m_{n}}a_{m_{1}+\ldots+m_{n}+r}$ $\displaystyle\leq$
$\displaystyle\sum_{m_{1},\ldots,m_{n}\geq n}c_{m_{1}}\cdots
c_{m_{n}}a_{m_{1}+\ldots+m_{n}+r}+n\sum_{m=0}^{n-1}c_{m}\sum_{m_{1},\ldots,m_{n-1}\geq
0}c_{m_{1}}\cdots c_{m_{n-1}}a_{m_{1}+\ldots+m_{n-1}+m+r}.$

The first term is finite because $c_{m}\leq c^{(n)}_{m}$ for $m\geq n$. The
second term is finite by the induction hypothesis. ∎

######  Lemma 3.5.

Suppose $X_{1},X_{2},X_{3},\ldots$ is a sequence of smooth vector fields on
$\mathbb{R}^{d}$ and $\varphi\in C_{c}^{\infty}(\mathbb{R}^{d})$. Then, there
exists a sequence $(c_{m})_{m\geq 0}$ of positive real numbers such that, for
any formal series $P(z)=\sum_{m\geq 0}a_{m}z^{m}\in\mathbb{C}[[z]]$ satisfying
$|a_{m}|\leq c_{m}$ for all $m\geq 0$, one has
$\varphi\in\mathrm{dom}(P(X_{i_{1}})\cdots P(X_{i_{n}}))$ for all
$n,i_{1},\ldots,i_{n}\geq 1$.

###### Proof.

Set

$M_{m}=\max\left\\{\left\|\tfrac{\partial^{\alpha}}{\partial
x^{\alpha}}X_{i_{1}}^{m_{1}}\cdots
X_{i_{n}}^{m_{n}}\varphi\right\|:m_{1}+\ldots+m_{n}+|\alpha|\leq m\text{ and
}n,i_{1},\ldots,i_{n}\leq m\right\\},$

where $\|\cdot\|$ denotes the uniform norm. By the preceding lemma, there
exists a sequence $(c_{m})_{m\geq 0}$ of positive real numbers such that

$\sum_{m_{1},\ldots,m_{n}\geq 0}c_{m_{1}}\cdots
c_{m_{n}}M_{m_{1}+\ldots+m_{n}+r}<\infty.$

for all integers $n,r$ with $n\geq 1$ and $r\geq 0$. Then, provided
$|a_{m}|\leq c_{m}$, one has that

$\sum_{m_{1},\ldots,m_{n}\geq
0}|a_{m_{1}}|\ldots|a_{m_{n}}|\left\|\frac{\partial^{\alpha}}{\partial
x^{\alpha}}X_{i_{1}}^{m_{1}}\cdots X_{i_{n}}^{m_{n}}\varphi\right\|<\infty$

for all $n,i_{1},\ldots,i_{n}\geq 1$ and $\alpha\in\mathbb{N}^{d}$. Indeed,
for all but finitely many terms of the above sum, we have
$n,i_{1},\ldots,i_{n}\leq m_{1}+\ldots+m_{n}+|\alpha|$, whence
$\left\|\frac{\partial^{\alpha}}{\partial x^{\alpha}}X_{i_{1}}^{m_{1}}\cdots
X_{i_{n}}^{m_{n}}\varphi\right\|\leq M_{m_{1}+\ldots+m_{n}+|\alpha|}$, by
definition. It follows that $\varphi\in\operatorname{dom}(P(X_{1})\cdots
P(X_{n}))$. Indeed,

$P(X_{i_{1}})\cdots P(X_{i_{n}})\varphi=\sum_{m_{1},\ldots,m_{n}\geq
0}a_{m_{1}}\cdots a_{m_{n}}X^{m_{1}}_{i_{1}}\cdots X^{m_{n}}_{i_{n}}\varphi,$

where the above series is uniformly and absolutely convergent to a function
$C_{c}^{\infty}(\mathbb{R}^{d})$. ∎

As a direct corollary, we have:

######  Lemma 3.6.

Suppose $M$ is a possibly non-Hausdorff smooth manifold and
$X_{1},X_{2},X_{3},\ldots$ are smooth, non-branching vector fields on $M$. Fix
a finite subset $\mathscr{P}\subseteq C_{c}^{\infty}(M)$. Then, there exists a
sequence $(c_{m})_{m\geq 0}$ of positive real numbers such that, for any
formal series $P(z)=\sum_{m\geq 0}a_{m}z^{m}\in\mathbb{C}[[z]]$ satisfying
$|a_{m}|\leq c_{m}$ for all $m\geq 0$, one has
$\varphi\in\mathrm{dom}(P(X_{i_{1}})\cdots P(X_{i_{n}}))$ for all
$\varphi\in\mathscr{P}$ and all $n,i_{1},\ldots,i_{n}\geq 1$. ∎

##  4 Lie groupoids and Lie groupoid actions

In this somewhat lengthy section, we lay out the notations and conventions
that will be used for Lie groupoids and their actions. For the most part, our
conventions coincide with those of [13] and [15]. We identify sections of the
Lie algebroid with right-invariant vector fields, as is done in [13]. On the
other hand, it will be convenient to have our vector fields and measures
defined along the same fibers (namely the source fibers), so we will use right
Haar systems instead of the left Haar systems used in [15]. Although the
material to follow is rather standard, we do sometimes include proofs, mainly
in order to ensure that all arguments used make sense in the non-Hausdorff
setting as well.

###  4.1 Basic notions

We will denote a typical Lie groupoid by $G$ and its unit space by $B$. The
source and target maps are denoted $s,t:G\to B$. We always assume they are
submersions. Multiplication is a smooth map from $G^{(2)}\colonequals
G\times_{s,t}G$ to $G$ that is performed from right to left and denoted by
juxtaposition; given $\gamma_{1},\gamma_{2}\in G$, the product
$\gamma_{1}\gamma_{2}$ is defined if and only if
$s(\gamma_{1})=t(\gamma_{2})$. The inverse of $\gamma\in G$ is denoted by
$\gamma^{-1}$. For convenience, it is always assumed that $B$ is embedded in
$G$ as a closed submanifold. We write $k\colonequals\dim(G)-\dim(B)$. In other
words, $k$ is the dimension of the source and target fibers, for which we
shall use the standard notations $G_{x}\colonequals s^{-1}(x)$ and
$G^{x}\colonequals t^{-1}(x)$, $x\in B$.

We allow the arrow space $G$, but not the unit space $B$, of a Lie groupoid to
be non-Hausdorff. This is needed to accommodate certain examples of interest,
such as groupoids arising from foliations. Dealing with the non-Hausdorff case
requires modifications to the definitions one would use in the Hausdorff case.
These kinds of issues are well understood, see [2], [3], [11]. We recall
Hausdorffness of the unit space automatically implies that of the source and
target fibers.

######  Lemma 4.1.

Let $G\rightrightarrows B$ be a possibly non-Hausdorff Lie groupoid. Then, for
any $x\in B$, the source and target fibers $G_{x}$ and $G^{x}$ are Hausdorff.

###### Proof.

See [18], Proposition 2.8. ∎

We also recall that the space of units always admits a Hausdorff open
neighbourhood.

######  Proposition 4.2.

Let $G\rightrightarrows B$ be a Lie groupoid. Then, there exists a Hausdorff
open set $W\subseteq G$ with $B\subseteq W$.

###### Proof.

This follows from [4], Lemma 4.18. See also the discussion in Section 7 of
[16]. ∎

###  4.2 The Lie algebroid of a Lie groupoid

######  Definition 4.3 (Definition 3.1, [13]).

The Lie algebroid of a Lie groupoid $G\rightrightarrows B$ is the vector
bundle $AG\to B$ obtained by restricting the source fiber tangent bundle
$\ker(ds)\subseteq TG$ to $B$.

There is a canonical bundle map $AG\to TB$ called the anchor map given by
restricting the differential of the target map $dt:TG\to TB$ to $AG$.
Significantly, there is also a natural bracket operation on the sections of
$AG$, but we will not need this.

######  Definition 4.4.

A vector field $X$ on a Lie groupoid $G$ is said to be right-invariant if it
is tangent to the source fibers of $G$ and, for all $\gamma\in G$, one has

$\displaystyle(R_{\gamma})_{*}X=X$ $\displaystyle\gamma\in G,$

where $R_{\gamma}$ denotes right-multiplication by $\gamma$. Note that
$R_{\gamma}$ is a diffeomorphism $G_{t(\gamma)}\to G_{s(\gamma)}$ and the
above equation is implicitly understood to refer to the restriction of $X$ to
the relevant source fibers.

It is easy to see that a right-invariant vector field on $G$ is completely
determined by its restriction to $B$, which is naturally a section of $AG$.
Conversely, every section of $AG$ extends uniquely to a right-invariant vector
field on $G$. We will freely denote a section of the Lie algebroid and its
extension to a right-invariant vector field by the same symbol. We note that
the bracket of two right-invariant vector fields is easily seen to be right-
invariant, leading to the bracket operation on sections of $AG$ (that we will
not be needing).

Because right-invariant vector fields are tangent to source fibers and source
fibers are Hausdorff (Lemma 4.1), we have the following.

######  Lemma 4.5.

Every right-invariant vector field on a possibly non-Hausdorff Lie groupoid is
non-branching, i.e. has a well-defined flow . ∎

It is common practice to denote the anchor map of a Lie algebroid by $\\#$,
but we will not do this here because it conflicts with the standard notation
for fundamental vector field in the context of smooth actions, which we are
also using. Instead, given a right-invariant vector field $X$ on
$G\rightrightarrows B$, we write $X^{B}$ for the corresponding vector field on
$B$ arising from the anchor map. To be more specific, the vector fields $X$
and $X^{B}$ are $t$-related.

In general, the flow of a right-invariant vector field on a Lie groupoid need
not be complete. A sufficient condition for the flow of a right-invariant
vector field to be complete is that the associated smooth section of the Lie
algebroid is compactly-supported. See, for instance, Proposition 3.6 in [9].
If $X$ is a complete, right-invariant vector field on $G\rightrightarrows M$,
then the induced vector field $X^{B}$ on the base is complete as well. Indeed,
its flow is determined by:

$\displaystyle e^{rX^{B}}b=t(e^{rX}b)$ $\displaystyle r\in\mathbb{R},b\in B.$

###  4.3 Smooth Haar systems

We now turn to Haar systems. Their general theory appears in [15] and their
routine adaptation to the smooth case is discussed in various sources such as
[1].

######  Definition 4.6.

Let $\pi:M\to N$ be a submersion of possibly non-Hausdorff manifolds. Assume
that the fibers of $\pi$ are Hausdorff and $\lambda_{y}$ is a smooth measure
on $\pi^{-1}(y)$ for each $y\in N$. We say that $\lambda=(\lambda_{y})_{y\in
N}$ is a smooth system of measures for $\pi$ if the following condition holds:
if $x\in M$, $U\subseteq M$ is a neighbourhood of $x$, $V\subseteq N$ is a
neighbourhood of $\pi(x)$, $\pi(U)=V$ and there are diffeomorphisms
$\chi_{U}:U\to\mathbb{R}^{d}\times\mathbb{R}^{k}$,
$\chi_{V}:V\to\mathbb{R}^{d}$ (so $\dim(M)=d+k$, $\dim(N)=d$) making the
following diagram commutative

${U}$${\mathbb{R}^{d}\times\mathbb{R}^{k}}$${V}$${\mathbb{R}^{d},}$$\scriptstyle{\chi_{U}}$$\scriptstyle{\pi}$$\scriptstyle{\mathrm{proj}_{2}}$$\scriptstyle{\chi_{V}}$

then the family of measures pushed forward through $\chi_{U}$ is the standard
volume measure of $\mathbb{R}^{k}$ copied on vertical fibers, multiplied by a
smooth, positive-valued function on $\mathbb{R}^{d}\times\mathbb{R}^{k}$.

Working locally, one sees that the above definition allows for integration
along fibers of bump functions, even in the non-Hausdorff case where
Definition 2.2 is in effect.

######  Lemma 4.7.

Let $\pi:M\to N$ be a submersion of possibly non-Hausdorff manifolds with
Hausdorff fibers that is equipped with a smooth system of measures
$\lambda=(\lambda_{y})_{y\in N}$. Then, integration along fibers with respect
to $\lambda$ defines a linear map $\pi_{!}:C_{c}^{\infty}(M)\to
C_{c}^{\infty}(N)$.

###### Proof.

Use Lemma 2.10 to reduce to charts on which $\pi$ is a projection. ∎

######  Definition 4.8.

Let $G\rightrightarrows B$ be a Lie groupoid. A (smooth, right) Haar system
for $G$ is a smooth system of measures $\lambda=(\lambda_{b})_{b\in B}$ for
the source submersion $s:G\to B$ which is furthermore right-invariant in the
sense that, for every $\gamma\in G$, the right multiplication map $R_{\gamma}$
is a measure preserving diffeomorphism $G_{t(\gamma)}\to G_{s(\gamma)}$.

Fixing a Haar system $\lambda$ for a Lie groupoid $G$ turns
$C_{c}^{\infty}(G)$ into a (generally noncommutative) algebra
$C_{c}^{\infty}(G,\lambda)$ with respect to the convolution product $*$
defined by either of the following equivalent integrals:

$\displaystyle(f*g)(\gamma_{0})=\int_{G_{t(\gamma_{0})}}f(\gamma^{-1})g(\gamma\gamma_{0})\
d\lambda_{t(\gamma_{0})}=\int_{G_{s(\gamma_{0})}}f(\gamma_{0}\gamma^{-1})g(\gamma)\
d\lambda_{s(\gamma_{0})}.$ (1)

The isomorphism class of this algebra does not depend on the choice of Haar
system. Indeed, if $\lambda$ and $\lambda^{\prime}$ are two Haar systems for
$G\rightrightarrows B$, then there is a unique smooth, positive-valued
function $\rho$ on $B$ such that $\lambda^{\prime}=(\rho\circ t)\lambda$.
Moreover $\rho$ can be used to define a canonical algebra isomorphism
$C_{c}^{\infty}(G,\lambda)\to C_{c}^{\infty}(G,\lambda^{\prime})$.
Specifically:

$f\mapsto\frac{1}{(\rho\circ s)^{1/2}(\rho\circ
t)^{1/2}}f:C_{c}^{\infty}(G,\lambda)\to C_{c}^{\infty}(G,\lambda^{\prime}).$

Since the convolution algebra of $G$ is independent of $\lambda$, we will tend
to write $C_{c}^{\infty}(G)$ instead of $C_{c}^{\infty}(G,\lambda)$. Note it
is also possible to skirt the discussion of Haar systems completely in
defining the convolution algebra of $G$ if one works with appropriate
densities instead of functions, see [3], Section 2.5.

###  4.4 Smooth groupoid actions

We now turn our attention to groupoid actions.

######  Definition 4.9.

Let $G\rightrightarrows M$ be a Lie groupoid. Let $M$ be a possibly non-
Hausdorff manifold together with a smooth map $\tau:M\to B$. Note the fiber
product

$G\ltimes M\colonequals G\times_{s,\tau}M$

is a closed submanifold of $G\times M$ because it is the preimage of the
diagonal under the submersion $s\times\tau:G\times M\to B\times B$. A left
action of $G$ on $M$ is a smooth product $G\ltimes
M\ni(\gamma,x)\mapsto\gamma\cdot x\in M$, such that $\tau(\gamma\cdot
x)=t(\gamma)$ for all $(\gamma,x)\in G\ltimes M$, $\tau(x)\cdot x=x$ for all
$x\in M$ and $(\gamma_{1}\gamma_{2})\cdot x=\gamma_{1}\cdot(\gamma_{2}\cdot
x)$ for all $\gamma_{1},\gamma_{2}\in G$, $x\in M$ with
$s(\gamma_{1})=t(\gamma_{2})$, $s(\gamma_{2})=\mu(x)$.

Similarly, if $M$ is equipped with a smooth map $\sigma:M\to G$, a right
action of $G$ on $M$ is a smooth product $M\rtimes G\ni(x,\gamma)\mapsto
x\cdot\gamma\in M$, where $M\rtimes G\colonequals M\times_{t,\sigma}G$, such
that $\sigma(x\cdot\gamma)=s(\gamma)$ for all $(x,\gamma)\in M\rtimes G$,
$x\cdot\sigma(x)=x$ for all $x\in M$ and
$x\cdot(\gamma_{1}\gamma_{2})=(x\cdot\gamma_{1})\cdot\gamma_{2}$ for all
$\gamma_{1},\gamma_{2}\in G$, $x\in M$ with $\sigma(x)=t(\gamma_{1})$,
$s(\gamma_{1})=t(\gamma_{2})$.

Supposing $G_{1}\rightrightarrows B_{1}$ acts on $M$ from the left with
respect to a map $\tau:M\to B_{1}$ and $G_{2}\rightrightarrows B_{2}$ acts on
$M$ from the right with respect to the map $\sigma:M\to B_{2}$, we say that
the left and right action commute with one another if
$\gamma_{1}\cdot(x\cdot\gamma_{2})=(\gamma_{1}\cdot x)\cdot\gamma_{2}$ for all
$\gamma_{1}\in G_{1}$, $\gamma_{2}\in G_{2}$, $x\in M$ satisfying
$s_{1}(\gamma_{1})=\tau(x)$, $\sigma(x)=t_{2}(\gamma_{2})$. In particular,
$\sigma(\gamma_{1}\cdot x)=\sigma(x)$, $\tau(x\cdot\gamma_{2})=\tau(x)$.

######  Remark 4.10.

Of course, we use the notation $G\ltimes M$ for $G\times_{s,\tau}M$ because it
is the _transformation groupoid_. There is a small snag here because we have
insisted that our groupoids have Hausdorff unit spaces and $M$, which is the
unit space of $G\ltimes M$, may be non-Hausdorff. In any event, we do not need
to view $G\rtimes M$ as a groupoid in its own right here.

The following diagram summarizes some of the data in Definition 4.9

${G_{1}}$${X}$${G_{2}}$${B_{1}}$${B_{2}}$$\scriptstyle{s_{1}}$$\scriptstyle{\tau}$$\scriptstyle{\sigma}$$\scriptstyle{t_{2}}$

One obvious example of commuting left and right actions are the left and right
multiplication action of a Lie groupoid on itself. Importantly, commuting
left/right actions also appear in one formulation of the notion of Morita
equivalence for Lie groupoids, see [21].

If $G\rightrightarrows B$ acts from the left on $\tau:M\to B$, then a right-
invariant vector field $X$ on $G$ determines a corresponding vector field
$X^{M}$ on $M$ (called the fundamental vector field) determined by

$\displaystyle X^{M}(m)=\tfrac{d}{dr}(e^{rX}\tau(m))\cdot m\Big{|}_{r=0}$
$\displaystyle m\in M.$

If $X$ is complete, then $X^{M}$ is complete and, indeed, the flow of $X^{M}$
is given by:

$\displaystyle e^{rX^{M}}m=(e^{rX}\tau(m))\cdot m$ $\displaystyle
r\in\mathbb{R},m\in M.$

If a Lie groupoid $G\rightrightarrows B$ with given right Haar system acts on
$\tau:M\to B$ from the left, then $C_{c}^{\infty}(M)$ becomes a left
$C_{c}^{\infty}(G)$-module with respect to the product

$C_{c}^{\infty}(G)\times C_{c}^{\infty}(M)\ni(f,\psi)\mapsto f*\psi\in
C_{c}^{\infty}(M)$

defined by

$\displaystyle(f*\psi)(x)=\int_{G_{\tau(x)}}f(\gamma^{-1})g(\gamma\cdot x)\
d\lambda_{\tau(x)}.$ (2)

To see this formula does indeed define a product (especially in the non-
Hausdorff case where things may be less clear), it is helpful to cast the
formula (2) in slightly more abstract form, as outlined below:

  1. 1.

Let $G\rightrightarrows B$ be a Lie groupoid acting smoothly from the left on
$\tau:M\to B$. Here, $G$ and $M$ are possibly non-Hausdorff. Let
$\pi,\alpha:G\ltimes M\to M$ be the (restrictions of) the second factor
projection and the action map, respectively. Let $\jmath:G\ltimes M\to
G\ltimes M$ be defined by $\jmath(\gamma,m)=(\gamma^{-1},\gamma\cdot m)$.
Then, $\jmath$ is an order-2 diffeomorphism satisfying
$\alpha=\pi\circ\jmath$. Indeed, although we do not make explicit use of it
because its unit space $M$ may be non-Hausdorff, $\alpha,\pi$ and $\jmath$ are
the target, source and inversion map for the transformation groupoid $G\ltimes
M$. The full set of structure maps are as follows:

source: $\displaystyle(\gamma,m)\mapsto m$ target:
$\displaystyle(\gamma,m)\mapsto\gamma\cdot m$ product:
$\displaystyle(\gamma^{\prime},\gamma\cdot
m)(\gamma,m)=(\gamma^{\prime}\gamma,m)$ inversion:
$\displaystyle(\gamma,m)\mapsto(\gamma^{-1},\gamma\cdot m)$

  2. 2.

Let $\lambda$ be the Haar system for $G$ and let $\widetilde{\lambda}$ be the
obvious smooth system of measures for $\pi$ determined by copying $\lambda$ on
fibers (one has $\pi^{-1}(m)=G_{\tau(m)}\times\\{m\\}$ for $m\in M$). In other
words, $\widetilde{\lambda}$ arises from the pullback square:

${G\ltimes
M}$${G}$${M}$${B}$$\scriptstyle{\mathrm{pr}_{1}}$$\scriptstyle{\pi,\widetilde{\lambda}}$$\scriptstyle{s,\lambda}$$\scriptstyle{\tau}$

  3. 3.

Using the map $\jmath$, we also endow $\alpha:G\ltimes M\to M$ with a smooth
system of measures (indeed, this is the left Haar system for the
transformation groupoid) so that there is a fiberwise integration map
$\alpha_{!}:C_{c}^{\infty}(G\ltimes M)\to C_{c}\infty(M)$ (Proposition 2.11
(3)). By construction, the following diagram is commutative:

${C_{c}^{\infty}(G\ltimes M)}$${C_{c}^{\infty}(G\ltimes
M)}$${C_{c}^{\infty}(M)}$${C_{c}^{\infty}(M)}$$\scriptstyle{\jmath_{*}}$$\scriptstyle{\alpha_{!}}$$\scriptstyle{\pi_{!}}$

  4. 4.

Given $f\in C_{c}^{\infty}(G)$ and $\varphi\in C_{c}^{\infty}(M)$, define
$f\ltimes\varphi$ to be the restriction of $f\otimes\varphi$ to $G\ltimes
M\subseteq G\times M$. Using Proposition 2.11 (1) and (2), one has that
$f\ltimes\varphi\in C_{c}^{\infty}(G\ltimes M)$.

  5. 5.

In terms of the above notations, the left $C_{c}^{\infty}(G)$-module structure
(2) of $C_{c}^{\infty}(M)$ may be expressed as follows:

$\displaystyle f*\psi\colonequals\alpha_{!}(f\ltimes\psi)$ $\displaystyle f\in
C_{c}^{\infty}(G),\psi\in C_{c}^{\infty}(M).$ (3)

Similarly, if $G\rightrightarrows B$ acts on $\sigma:M\to B$ from the right,
then $C_{c}^{\infty}(M)$ becomes a right $C_{c}^{\infty}(G)$-module with
respect to the product $C_{c}^{\infty}(M)\times
C_{c}^{\infty}(G)\ni(\psi,f)\mapsto\psi*f\in C_{c}^{\infty}(M)$ defined by

$\displaystyle(\psi*f)(x)=\int_{G_{\sigma(x)}}\psi(x\cdot\gamma^{-1})g(\gamma)\
d\lambda_{\sigma(x)}.$ (4)

Symmetrically to the case of left actions, there is a canonical way to equip
the right action map $\beta:M\rtimes G\to M$ with a smooth system of measures
in terms of which the right $C_{c}^{\infty}(G)$-module structure may be
alternatively expressed as follows:

$\displaystyle\psi*f\colonequals\beta_{!}(\psi\rtimes f)$ $\displaystyle f\in
C_{c}^{\infty}(G),\psi\in C_{c}^{\infty}(M).$ (5)

Again, although we do not make explicit use of it, we remark that the
structure maps of the right transformation groupoid $M\rtimes G$ are given by:

source: $\displaystyle(m,\gamma)\mapsto m\cdot\gamma$ target:
$\displaystyle(m,\gamma)\mapsto m$ product:
$\displaystyle(m,\gamma)(m\cdot\gamma,\gamma^{\prime})=(m,\gamma\gamma^{\prime})$
inversion: $\displaystyle(m,\gamma)\mapsto(m\cdot\gamma,\gamma^{-1})$

If $M$ carries commuting actions of $G_{1}$ from the left and $G_{2}$ from the
right, then $(f_{1}*\psi)*f_{2}=f_{1}*(\psi*f_{2})$ is satisfied for all
$f_{1}\in C_{c}^{\infty}(G_{1})$, $f_{2}\in C_{c}^{\infty}(G_{2})$, $\psi\in
C_{c}^{\infty}(M)$ so that $C_{c}^{\infty}(M)$ has the structure of a
$C_{c}^{\infty}(G_{1})$-$C_{c}^{\infty}(G_{2})$-bimodule. In the special case
of a groupoid acting on itself from the left and right, this recovers the
usual convolution product on $C_{c}^{\infty}(G)$.

In order to check the associativity of the bimodule structure, it is useful to
pass through the two-sided transformation groupoid:

$G_{1}\ltimes M\rtimes G_{2}\colonequals\\{(\gamma_{1},m,\gamma_{2})\in
G_{1}\times M\times
G_{2}:s_{1}(\gamma_{1})=\tau(m),\sigma(m)=t_{2}(\gamma_{2})\\}$

whose structure maps are listed below:

source: $\displaystyle(\gamma_{1},m,\gamma_{2})$ $\displaystyle\mapsto
m\cdot\gamma_{2}$ target: $\displaystyle(\gamma_{1},m,\gamma_{2})$
$\displaystyle\mapsto\gamma_{1}\cdot m$ inversion: $\displaystyle(m,\gamma)$
$\displaystyle\mapsto(m\cdot\gamma,\gamma^{-1})$ product:
$\displaystyle(\gamma_{1},m,\gamma_{2})(\gamma_{1}^{\prime},m^{\prime},\gamma_{2}^{\prime})$
$\displaystyle=(\gamma_{1}\gamma_{1}^{\prime},(\gamma_{1}^{\prime})^{-1}\cdot
m,\gamma_{2}\gamma_{2}^{\prime})$ (where
$m\cdot\gamma_{2}=\gamma_{1}^{\prime}=m^{\prime}$)
$\displaystyle=(\gamma_{1}\gamma_{1}^{\prime},m^{\prime}\cdot(\gamma_{2})^{-1},\gamma_{2}\gamma_{2}^{\prime}).$

Because the actions commute, we have the following commuting diagram:

${G_{1}\ltimes M\rtimes G_{2}}$${M\rtimes G_{2}}$${G_{1}\rtimes
M}$${M}$$\scriptstyle{\mu}$$\scriptstyle{\alpha\times\mathrm{id}}$$\scriptstyle{\mathrm{id}\times\beta}$$\scriptstyle{\beta}$$\scriptstyle{\alpha}$

where

$\mu(\gamma_{1},m,\gamma_{2})\colonequals\gamma_{1}\cdot m\cdot\gamma_{2}.$

Introducing fiberwise measures in the natural way, commutativity holds as well
at the level of bump functions and integration maps:

${C_{c}^{\infty}(G_{1}\ltimes M\rtimes G_{2})}$${C_{c}^{\infty}(M\rtimes
G_{2})}$${C_{c}^{\infty}(G_{1}\rtimes
M)}$${C_{c}^{\infty}(M)}$$\scriptstyle{\mu_{!}}$$\scriptstyle{(\alpha\times\mathrm{id})_{!}}$$\scriptstyle{(\mathrm{id}\times\beta)_{!}}$$\scriptstyle{\beta_{!}}$$\scriptstyle{\alpha_{!}}$

leading to the desired associativity property:

$\displaystyle f*(\psi*g)=(f*\psi)*g=\mu_{!}(f\ltimes\psi\rtimes g)$
$\displaystyle f\in C_{c}^{\infty}(G_{1}),\psi\in C_{c}^{\infty}(M),g\in
C_{c}^{\infty}(G_{2}).$

###  4.5 Dixmier-Malliavin for $\mathbb{R}$-actions

Suppose the additive group $\mathbb{R}$ acts smoothly on a possibly non-
Hausdorff smooth manifold $M$. Let $X$ be the (non-branching, complete) vector
field on $M$ generating the action. The _integrated form_ of the action is the
representation $\pi$ of $C_{c}^{\infty}(\mathbb{R})$ on $C_{c}^{\infty}(M)$
defined by:

$\displaystyle(\pi(f)\psi)(m)=\int_{\mathbb{R}}f(-t)\psi(e^{tX}m)\ dt.$ (6)

Of course, this is a very special case of the integrated forms of groupoid
actions just discussed. As in the more general case, especially when $M$ is
non-Hausdorff, in order to see $\pi(f)$ does indeed define a mapping
$C_{c}^{\infty}(M)\to C_{c}^{\infty}(M)$, it helps to express this action in
the more abstract form:

$\displaystyle\pi(f)\varphi=\alpha_{!}(f\otimes\varphi)$ $\displaystyle f\in
C_{c}^{\infty}(\mathbb{R}),\varphi\in C_{c}^{\infty}(M)$

Here, $\alpha:\mathbb{R}\times M\to M$ is the action map, made into a measured
submersion so as to render the following diagram commutative:

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where $\jmath(t,m)=(-t,e^{tX}m)$ and $\mathrm{pr}_{2}$ is a measured
submersion in the obvious way (copying Lebesgue measure).

As one expects, integration by parts enables one to move differentiation
across the convolution map $C_{c}^{\infty}(\mathbb{R})\times
C_{c}^{\infty}(M)\to C_{c}^{\infty}(M)$. That is,
$\pi(f^{\prime})\varphi=\pi(f)X\varphi$ holds, where $X$ is the vector field
generating the $\mathbb{R}$-action. We sketch a proof of this (quite routine)
fact mainly to indicate how it works in the formalism
$\pi(f)\varphi=\alpha_{!}(f\otimes\varphi)$ which is convenient for rigorous
treatment of the non-Hausdorff case.

######  Lemma 4.11.

Let $\mathbb{R}$ act smoothly on a possibly non-Hausdorff smooth manifold $M$
via the complete vector field $X$. Let $\pi$ be the associated representation
of $C_{c}^{\infty}(\mathbb{R})$ on $C_{c}^{\infty}(M)$. Then,
$\pi(f^{\prime})\varphi=\pi(f)X\varphi$ for all $f\in
C_{c}^{\infty}(\mathbb{R})$, $\varphi\in C_{c}^{\infty}(\mathbb{R})$.

###### Proof.

It is clear that $(\mathrm{pr}_{2})_{!}:C_{c}^{\infty}(\mathbb{R}\times M)\to
C_{c}^{\infty}(M)$ maps the image of the (non-branching) vector field
$\langle\frac{d}{dt},0\rangle$ on $\mathbb{R}\times M$ to zero. Conjugating by
$\jmath(t,m)=(-t,e^{tX}m)$, this gives that integration along the action map
$\alpha_{!}:C_{c}^{\infty}(\mathbb{R}\times M)\to C_{c}^{\infty}(M)$ maps the
image of $\langle-\frac{d}{dt},X\rangle$ to zero, giving the desired result. ∎

The above lemma leads to the following following preliminary factorization
result. This was also the essential ingredient in [9].

######  Theorem 4.12.

Let $\mathbb{R}$ act smoothly on a smooth manifold $M$ via a complete vector
field $X$ and let $\pi$ be the representation of $C_{c}^{\infty}(\mathbb{R})$
on $C_{c}^{\infty}(M)$ defined by (6). Suppose $f_{0},f_{1}\in
C_{c}^{\infty}(\mathbb{R})$ and $P(z)=\sum_{m\geq
0}a_{m}z^{m}\in\mathbb{C}[[z]]$ are such that $\delta=f_{0}+\sum_{m\geq
0}a_{m}f_{1}^{(m)}$, as in Lemma 1.2. Then, for any
$\varphi\in\operatorname{dom}(P(X))\subseteq C_{c}^{\infty}(M)$, one has:

$\varphi=\pi(f_{0})\varphi+\pi(f_{1})P(X)\varphi.$

###### Proof.

See Theorem 8.1 in [9]. ∎

It will be important for us to know that, if $M$ carries a (left)
$\mathbb{R}$-action generated by a vector field $X$ that furthermore commutes
with a given right action of a groupoid $G^{\prime}$, then the operators
$P(X):\operatorname{dom}(P(X))\to C_{c}^{\infty}(M)$ for
$P(z)\in\mathbb{C}[[z]]$ are right-linear for the
$C_{c}^{\infty}(G^{\prime})$-module structure of $C_{c}^{\infty}(M)$. The case
of interest for us is where $M$ carries commuting actions of groupoids $G$ and
$G^{\prime}$ from the left and the right, respectively, and the vector field
on $M$ is the fundamental vector field $X^{M}$ associated to some section $X$
of the Lie algebroid $AG$. The desired statement is Corollary 4.16 below,
which follows directly from the three lemmas preceding it. The proofs of the
these lemmas are straightforward, and we omit them.

######  Lemma 4.13.

Let $\pi:M\to N$ be a measured submersion of possibly non-Hausdorff smooth
manifolds. Let $X$ be a non-branching smooth vector field on $M$ and $Y$ a
non-branching smooth vector field on $N$ and suppose $X$ and $Y$ are
$\pi$-related. Let $P(z)\in\mathbb{C}[[z]]$. Then, $\pi_{!}$ maps
$\operatorname{dom}(P(X))$ into $\operatorname{dom}(P(Y))$ and the following
diagram is commutative:

${\operatorname{dom}(P(X))}$${C_{c}^{\infty}(M)}$${\operatorname{dom}(P(Y))}$${C_{c}^{\infty}(N)}$$\scriptstyle{P(X)}$$\scriptstyle{\pi_{!}}$$\scriptstyle{\pi_{!}}$$\scriptstyle{P(Y)}$

∎

Regarding the above, we note that, if $X$ is non-branching, it is not
automatic that $Y$ is non-branching, as may be seen in the case where $\pi$ is
the quotient map from two disjoint copies of $\mathbb{R}$ to the line with two
origins. Of course, if $Y$ is non-branching, it is also not automatic that $X$
is non-branching (one could take $N$ to be a single point).

######  Lemma 4.14.

Let $M$ be a possibly non-Hausdorff smooth manifold and let $N$ be a closed
submanifold of $M$. Let $X$ be a non-branching smooth vector field on $M$ that
is tangent to $N$ and denote the restriction of $X$ to $N$ by $Y$. Let
$P(z)\in\mathbb{C}[[z]]$. Then, the restriction map $C_{c}^{\infty}(M)\to
C_{c}^{\infty}(N)$ maps $\operatorname{dom}(P(X))$ into
$\operatorname{dom}(P(Y))$ and the following diagram is commutative:

${\operatorname{dom}(P(X))}$${C_{c}^{\infty}(M)}$${\operatorname{dom}(P(Y))}$${C_{c}^{\infty}(N)}$$\scriptstyle{P(X)}$$\scriptstyle{\mathrm{restr}}$$\scriptstyle{\mathrm{restr}}$$\scriptstyle{P(Y)}$

∎

######  Lemma 4.15.

Let $M$ and $N$ be possibly non-Hausdorff smooth manifolds. Let $X$ be a non-
branching smooth vector field on $M$ and let $Y=\langle X,0\rangle$, a non-
branching smooth vector field on $M\times N$. Let $P(z)\in\mathbb{C}[[z]]$.
Then, if $f\in\operatorname{dom}(P(X))\subseteq C_{c}^{\infty}(M)$ and $g\in
C_{c}^{\infty}(N)$, we have $f\otimes g\in\operatorname{dom}(P(Y))$ and
$Y(f\otimes g)=(Xf)\otimes g$. ∎

######  Corollary 4.16.

Let $G^{\prime}\rightrightarrows B$ be a possibly non-Hausdorff Lie groupoid
acting smoothly from the right on a possibly non-Hausdorff smooth manifold $M$
with respect to a smooth map $\sigma:M\to B$. Let $X$ be a smooth vector field
on $M$ which commutes with the right action of $G^{\prime}$ on $M$. Let
$P(z)\in\mathbb{C}[[z]]$. Then, $\operatorname{dom}(P(X))$ is right
$C_{c}^{\infty}(G^{\prime})$-submodule of $C_{c}^{\infty}(M)$, and
$P(X)(\varphi*f)=(P(X)\varphi)*f$ for all $\varphi\in\operatorname{dom}(P(X))$
and $f\in C_{c}^{\infty}(M)$.

###### Proof.

Let $\beta:M\rtimes G^{\prime}\to M$ be the action map. Put
$Y\colonequals\langle X,0\rangle$. Observe that $Y$ is a vector field on
$M\times G^{\prime}$ which is tangent to $M\rtimes G^{\prime}$. Put
$Z=Y|_{M\rtimes G^{\prime}}$. Then we have

$\displaystyle(P(X)\varphi)*f$ $\displaystyle=\beta_{!}((P(X)\varphi)\rtimes
f)$ $\displaystyle=\beta_{!}((P(Y)(\varphi\otimes f))|_{M\rtimes G^{\prime}})$
by Lemma 4.14 $\displaystyle=\beta_{!}(P(Z)(\varphi\rtimes f))$ by Lemma 4.13
$\displaystyle=P(X)\beta_{!}(\varphi\rtimes f)$ by Lemma 4.15
$\displaystyle=P(X)(\varphi*f),$

as required. ∎

###  4.6 Ideals associated to invariant subsets

Next, we turn our attention to ideals in smooth groupoid algebras. A subset
$Z$ of the unit space of a groupoid is called invariant if every arrow with
source in $Z$ also has target in $Z$. Given a Lie groupoid $G\rightrightarrows
B$ with Haar system, a closed, invariant set $Z\subseteq B$ and
$p\in\mathbb{N}\cup\\{\infty\\}$, one may consider the $p$th order vanishing
ideal $J_{Z}^{p}\subseteq C_{c}^{\infty}(G)$ consisting of the functions
vanishing to (at least) $p$th order on $G_{Z}\colonequals
s^{-1}(Z)=t^{-1}(Z)$. We discuss these ideals in greater detail below.

######  Definition 4.17.

Let $M$ be a possibly non-Hausdorff smooth manifold, let $Z\subseteq M$ be a
closed set, and let $p\in\mathbb{N}\cup\\{\infty\\}$. We say that $\varphi\in
C_{c}^{\infty}(M)$ vanishes to pth order on $Z$ if it can be decomposed as
$\varphi=\sum_{i=1}^{N}(f_{i}\circ\chi_{i})_{0}$ where $U_{i}\subseteq M$ are
chart neighbourhoods, $\chi_{i}:U_{i}\to\mathbb{R}^{d}$ are diffeomorphisms,
and $f_{i}\in C_{c}^{\infty}(\mathbb{R}^{d})$ are such that $f_{i}$ and all
its partial derivatives of order $<p$ vanish on $\chi_{i}(U_{i}\cap Z)$. The
subscript $0$ above indicates the operation of extension by zero.

######  Proposition 4.18.

Let $G\rightrightarrows B$ be a Lie groupoid with given Haar system acting
from the left on a smooth manifold $M$ with respect to a smooth map $\tau:M\to
B$ so that $C_{c}^{\infty}(M)$ is a left $C_{c}^{\infty}(G)$-module. Let
$Z\subseteq B$ be an invariant, closed subset and let
$p,q\in\mathbb{N}\cup\\{\infty\\}$. If $f\in C_{c}^{\infty}(G)$ vanishes to
$p$th order on $G_{Z}\colonequals s^{-1}(Z)=t^{-1}(Z)$ and $\varphi\in
C_{c}^{\infty}(M)$ vanishes to $q$th order on $M^{Z}\colonequals\tau^{-1}(Z)$,
then $f*\varphi$ vanishes to $(p+q)$th order on $M^{Z}$.

###### Proof.

Let $\alpha:G\ltimes M\to M$ be the action map. It is simple to check in
charts that $f\otimes\varphi\in C_{c}^{\infty}(G\times M)$ vanishes to order
$p+q$ on $G_{Z}\times M^{Z}$. Restricting to $G\ltimes M\subseteq G\times M$,
one has that $f\ltimes\varphi$ vanishes to order $p+q$ on $(G_{Z}\times
M^{Z})\cap(G\ltimes M)=\alpha^{-1}(M^{Z})$ (the latter equality uses the
invariance of $Z$). The conclusion follows by expressing the convolution of
$f$ and $\varphi$ in the form $f*\varphi=\alpha_{!}(f\ltimes\varphi)$, as
explained above. ∎

As a corollary of the above proposition, one has that the $p$th order
vanishing ideals

$J^{p}_{Z}\colonequals\\{f\in C_{c}^{\infty}(G):\text{ $f$ vanishes to $p$th
order on $G_{Z}$}\\}$

are indeed ideals for $Z\subseteq B$ closed and invariant and
$p\in\mathbb{N}\cup\\{\infty\\}$. Moreover, $J^{p}_{Z}*J^{q}_{Z}\subseteq
J^{p+q}_{Z}$. In [9], it was shown that this containment is an equality in the
case where $Z\subseteq B$ is a closed, invariant submanifold. In particular,
$J_{Z}^{\infty}*J_{Z}^{\infty}=J_{Z}^{\infty}$. One of our goals in the
present article is to show, more generally, that $J_{Z}^{\infty}$ is H-unital
for $Z\subseteq B$ any closed, invariant subset.

We give one final definition/notation. As a point of clarification, what
follows are straightforward groupoid analogs of the fact that matrices of the
form $\left(\begin{smallmatrix}*&*\\\ 0&0\end{smallmatrix}\right)$,
respectively $\left(\begin{smallmatrix}*&0\\\ *&0\end{smallmatrix}\right)$,
constitute a right ideal, respectively left ideal, in the the algebra of
$2$-by-$2$ matrices. Assume as above that $M$ carries commuting actions of
$G_{1}$ from the left and $G_{2}$ from the right. Then, given a closed set
$K\subseteq B_{1}$, we define

$\displaystyle M^{K}$ $\displaystyle\colonequals\tau^{-1}(K)$ $\displaystyle
C_{c}^{\infty}(M)^{K}$ $\displaystyle\colonequals\\{\psi\in
C_{c}^{\infty}(M):\mathrm{supp}(\psi)\subseteq M^{K}\\}.$

It is straightforward to check that $C_{c}^{\infty}(M)^{K}$ is a right
$C_{c}^{\infty}(G_{2})$-submodule of $C_{c}^{\infty}(M)$. Symmetrically, if
$K\subseteq B_{2}$ is a closed set, one may define

$\displaystyle M_{K}$ $\displaystyle\colonequals\sigma^{-1}(K)$ $\displaystyle
C_{c}^{\infty}(M)_{K}$ $\displaystyle\colonequals\\{\psi\in
C_{c}^{\infty}(M):\mathrm{supp}(\psi)\subseteq M_{K}\\}$

and check that $C_{c}^{\infty}(M)_{K}$ is a left
$C_{c}^{\infty}(G_{1})$-submodule of $C_{c}^{\infty}(M)$.

##  5 H-unitality of the convolution algebra of a Lie groupoid

In this section, we prove the first of our main results, the H-unitality of
$C_{c}^{\infty}(G)$ for any Lie groupoid $G$ (Theorem 1.9 from the
introduction). Indeed we prove a slightly stronger statement involving smooth
actions of groupoids. The latter results are applicable, for instance, in
context of Morita equivalences of Lie groupoids.

Throughout this section, $G\rightrightarrows B$ and
$G^{\prime}\rightrightarrows B^{\prime}$ are Lie groupoids with given Haar
systems and $M$ is a smooth manifold carrying commuting actions of $G$ from
left and $G^{\prime}$ from the right with respect to smooth maps $\tau:M\to B$
and $\sigma:M\to B^{\prime}$. Thus, $C_{c}^{\infty}(M)$ has the structure of a
$C_{c}^{\infty}(G)$-$C_{c}^{\infty}(G^{\prime})$-bimodule. Recall as well the
notations:

$\displaystyle M^{K}$ $\displaystyle\colonequals\tau^{-1}(K)$ $\displaystyle
C_{c}^{\infty}(M)^{K}$ $\displaystyle\colonequals\\{\psi\in
C_{c}^{\infty}(M):\mathrm{supp}(\psi)\subseteq M^{K}\\}.$

and the fact that $C_{c}^{\infty}(M)^{K}$ is a right
$C_{c}^{\infty}(G^{\prime})$-submodule of $C_{c}^{\infty}(M)$.

######  Lemma 5.1.

Let $G\rightrightarrows B$ be a Lie groupoid with given Haar system. Let
$X_{1},\ldots,X_{k}\in C_{c}^{\infty}(B,AG)$, viewed as complete, right-
invariant vector fields on $G$. Define $u:\mathbb{R}^{k}\times B\to G$ by

$u(t_{1},\ldots,t_{k},b)=e^{t_{1}X_{1}}\cdots e^{t_{k}X_{k}}b$

Suppose that $\widetilde{W}\subseteq\mathbb{R}^{k}\times B$ is an open set
which is mapped diffeomorphically by $u$ onto an open set $W\subseteq G$. Then
there is a linear bijection $\theta$ from
$C_{c}^{\infty}(\widetilde{W})\subseteq C_{c}^{\infty}(\mathbb{R}^{k}\times
B)$ to $C_{c}^{\infty}(W)\subseteq C_{c}^{\infty}(G)$ given as pushforward by
$u$, followed by multiplication by a suitable Jacobian factor with the
property described below.

Let $G$ act smoothly from the left on $\tau:M\to B$ and define

$\widetilde{\pi}(f)\psi(m)=f(-t_{k},\ldots,-t_{1},\tau(e^{t_{1}X_{1}^{M}}\cdots
e^{t_{k}X_{k}^{M}}m))\psi(e^{t_{1}X_{1}^{M}}\cdots e^{t_{k}X_{k}^{M}}m).$

Then, one has:

$\widetilde{\pi}(f)\psi=\theta(f)*\psi$

for all $f\in C_{c}^{\infty}(\widetilde{W})$ and all $\psi\in
C_{c}^{\infty}(M)$.

We will generalize the following result from [9].

######  Theorem 5.2 (Theorem 5.1, [9]).

Suppose $G\rightrightarrows B$ is a Lie groupoid with a given Haar system and
$M$ is a smooth manifold equipped with a left action of $G$. Thus,
$C_{c}^{\infty}(M)$ has the structure of a $C_{c}^{\infty}(G)$-module. Then,
for every $\varphi\in C_{c}^{\infty}(M)$, there exist $f_{1},\ldots,f_{N}\in
C_{c}^{\infty}(G)$ and $\psi_{1},\ldots,\psi_{N}\in C_{c}^{\infty}(M)$ such
that

$\varphi=f_{1}*\psi_{1}+\ldots+f_{N}*\psi_{N}.$

Moreover, this factorization can be taken such that, for all $i$,
$\mathrm{supp}(\psi_{i})\subseteq\mathrm{supp}(\varphi)$ and
$\mathrm{supp}(f_{i})\subseteq W$, where $W$ is a prescribed open subset of
$G$ containing $\tau(\mathrm{supp}(\varphi))$.

As in the result above, we shall consider the more general situation of a
smooth groupoid action. This is done with an eye to certain applications to
smooth Morita equivalences which may be explored elsewhere. The case of
interest here in this article is that of a groupoid acting on itself.

######  Theorem 5.3.

Let $\mathscr{P}$ be any finite subset of $C_{c}^{\infty}(M)$. Define
$K=\bigcup_{\varphi\in\mathscr{P}}\tau(\operatorname{supp}(\varphi))$ and let
$W$ be any open subset of $G$ containing $K$. Then, there exist:

  1. 1.

$f_{1},\ldots,f_{N}\in C_{c}^{\infty}(W)\subseteq C_{c}^{\infty}(G)$.

  2. 2.

a right $C_{c}^{\infty}(G^{\prime})$-submodule $A_{0}$ with
$\mathscr{P}\subseteq A_{0}\subseteq C_{c}^{\infty}(M)^{K}$,

  3. 3.

right $C_{c}^{\infty}(G^{\prime})$-linear maps
$\Psi_{1},\ldots,\Psi_{N}:A_{0}\to C_{c}^{\infty}(M)$ that do not increase
supports

such that, for all $\varphi\in A_{0}$, we have
$\varphi=f_{1}*\Psi_{1}(\varphi)+\ldots+f_{N}*\Psi_{N}(\varphi)$.

We remark that, in view of Example 2.6, (1) can in general be a stronger
assertion than: $f_{i}\in C_{c}^{\infty}(G)$ and
$\operatorname{supp}f_{i}\subseteq W$. That being said, since the unit space
of $G$ admits a Hausdorff neighbourhood (Proposition 4.2), this distinction is
not important here as $W$ may without loss of generality be assumed to be
Hausdorff.

###### Proof.

Use the map $\tau:M\to B$ to endow $C_{c}^{\infty}(M)$ with the structure of a
left $C^{\infty}(B)$-module as follows:

$\displaystyle(\rho\cdot\varphi)(x)=\rho(\tau(x))\varphi(x)$
$\displaystyle\rho\in C^{\infty}(B),\varphi\in C_{c}^{\infty}(M),x\in M.$

It is simple to confirm that the left $C^{\infty}(B)$-module structure on
$C_{c}^{\infty}(M)$ commutes with the right
$C_{c}^{\infty}(G^{\prime})$-module structure.

Note that $K\subseteq B$ is compact by Proposition 2.4. First we argue that it
suffices to prove this theorem under the additional hypothesis that the Lie
algebroid $AG$ is trivial (as a vector bundle) over some open neighbourhood of
$K$. Indeed, assume this special case is already proven, and choose smooth,
compactly-supported functions $\rho_{1},\ldots,\rho_{N}:B\to[0,1]$ such that
$\sum_{i=1}^{N}\rho_{i}=1$ holds on $K$ and $AG$ is trivial on a neighbourhood
of $\operatorname{supp}(\rho_{i})$ for each $i$. Define
$\mathscr{P}^{(i)}=\\{\rho_{i}\cdot\varphi:\varphi\in\mathscr{P}\\}$ for
$i=1,\ldots,N$. By hypothesis, for $1\leq i\leq N$, there exist

  1. 1.

$f_{1}^{(i)},\ldots,f^{(i)}_{N_{i}}\in C_{c}^{\infty}(G)$ with supports
contained in $W$,

  2. 2.

A right $C_{c}^{\infty}(G^{\prime})$-submodule $A_{0}^{(i)}\subseteq
C_{c}^{\infty}(M)$ with $\mathscr{P}^{(i)}\subseteq A_{0}^{(i)}\subseteq
C_{c}^{\infty}(M)^{K}$,

  3. 3.

Right $C_{c}^{\infty}(G^{\prime})$-linear maps
$\Psi_{1}^{(i)},\ldots,\Psi_{N_{i}}^{(i)}:A_{0}^{(i)}\to C_{c}^{\infty}(M)$
that do not increase supports

such that, for all $\varphi\in A_{0}^{(i)}$,

$\varphi=\sum_{j=1}^{N_{i}}f_{j}^{(i)}*\Psi_{j}(\varphi)$

Define:

$\displaystyle A_{0}$ $\displaystyle=\\{\varphi\in
C_{c}^{\infty}(M)^{K}:\rho_{i}\cdot\varphi\in A_{0}^{(i)}\text{ for
}i=1,\ldots,N\\}.$

By definition, $\mathscr{P}\subseteq A_{0}\subseteq C_{c}^{\infty}(M)^{K}$,
and it is easy to check that $A_{0}$ is a right
$C_{c}^{\infty}(G^{\prime})$-submodule.

For $1\leq i\leq N,1\leq j\leq N_{i}$, define maps $\Psi_{ij}:A_{0}\to
C_{c}^{\infty}(M)$ by

$\Psi_{ij}(\varphi)=\Psi_{j}^{(i)}(\rho_{i}\cdot\varphi).$

It is easy to check these are linear maps which do not increase supports and
are right $C_{c}^{\infty}(G^{\prime})$-linear. Moreover, if $\varphi\in A_{0}$

$\varphi=\sum_{i=1}^{N}\rho_{i}\cdot\varphi=\sum_{i=1}^{N}\sum_{j=1}^{N_{i}}f^{(i)}_{j}*\Psi_{j}(\rho_{i}\cdot\varphi)=\sum_{i=1}^{N}\sum_{j=1}^{N_{i}}f_{j}^{(i)}*\Psi_{ij}(\varphi).$

It remains to check the case where $AG$ is trivial over a neighbourhood of
$K$. Then, by inverse function theorem, there exists an open set $U\subseteq
B$ with $K\subseteq U$ and (complete by Lemma 4.5) right-invariant vector
fields $X_{1},\ldots,X_{k}\in C_{c}^{\infty}(B,AG)$ such that

$\displaystyle u:\mathbb{R}^{k}\times B\to G$ $\displaystyle
u(t_{1},\ldots,t_{k},b)=e^{t_{1}X_{1}}\cdots e^{t_{k}X_{k}}b$

maps $\widetilde{W}\colonequals(-1,1)^{k}\times U$ diffeomorphically onto an
open subset of $G$ contained in $W$. Indeed by shrinking $W$ we can, and do,
assume that $W=u(\widetilde{W})$.

Let $X_{1}^{M},\ldots,X_{k}^{M}$ denote the corresponding complete vector
fields on $M$ and let $\pi_{1},\ldots,\pi_{k}$ denote the corresponding
representations of $C_{c}^{\infty}(\mathbb{R})$ on $C_{c}^{\infty}(M)$ given
by (6). Because the action of $G$ on $M$ commutes with the action of
$G^{\prime}$ on $M$, the $\mathbb{R}$ actions determined by the fundamental
vector fields $X_{1}^{M},\ldots,X_{k}^{M}$ commute with the action of
$G^{\prime}$ as well.

By Lemma 4.2 in [9], there is a linear bijection $\theta_{W}$ from
$C_{c}^{\infty}(\widetilde{W})\subseteq C_{c}^{\infty}(\mathbb{R}^{k}\times
B)$ to $C_{c}^{\infty}(W)\subseteq C_{c}^{\infty}(G)$ given as pushforward by
$u$, followed by multiplication by a suitable smooth Jacobian factor such that

$\widetilde{\pi}(f)\psi=\theta_{W}(f)*\psi$

for all $f\in C_{c}^{\infty}(\widetilde{W})$ and all $\psi\in
C_{c}^{\infty}(M)$, where

$\widetilde{\pi}(f)\psi(m)\colonequals
f(-t_{k},\ldots,-t_{1},\tau(e^{t_{1}X_{1}^{M}}\cdots
e^{t_{k}X_{k}^{M}}m))\psi(e^{t_{1}X_{1}^{M}}\cdots e^{t_{k}X_{k}^{M}}m).$

In particular, fixing $\rho\in C_{c}^{\infty}(B)$ such that $\rho=1$ on $K$
(so that $\rho\cdot\varphi=\varphi$ for all $\varphi\in
C_{c}^{\infty}(M)^{K}$) and $\operatorname{supp}(\rho)\subseteq U$, we get
that

$\displaystyle\pi_{1}(f_{1})\cdots\pi_{k}(f_{k})\psi=\theta_{W}(f_{k}\otimes\ldots\otimes
f_{1}\otimes\rho)*\psi$ (7)

for all $f_{1},\ldots,f_{k}\in C_{c}^{\infty}(-1,1)\subseteq
C_{c}^{\infty}(\mathbb{R})$ and all $\psi\in C_{c}^{\infty}(M)^{K}$ (see also
Lemmas 4.1 and 4.3 in [9]).

By Lemma 3.6, there exists a sequence of positive reals $(c_{m})_{m\geq 0}$
such that, for any formal series $P(z)=\sum_{m\geq 0}a_{m}z^{m}$ with
$|a_{m}|\leq c_{m}$, we have

$\mathscr{P}\subseteq\mathrm{dom}(P(X_{i_{1}}^{M})\cdots P(X_{i_{n}}^{M}))$

for all $n,i_{1},\ldots,i_{n}\geq 1$. Using Lemma 1.2, write
$\delta_{0}=f_{0}+\sum_{m\geq 0}f_{1}^{(m)}$ where $f_{0},f_{1}\in
C_{c}^{\infty}(-1,1)\subseteq C_{c}^{\infty}(\mathbb{R})$ and $|a_{m}|\leq
c_{m}$. By repeated application of Theorem 4.12, we obtain

$\displaystyle\varphi=$
$\displaystyle\pi_{1}(f_{0})\varphi+\pi_{1}(f_{1})P(X^{M}_{1})\varphi$
$\displaystyle=$
$\displaystyle\pi_{1}(f_{0})\pi_{2}(f_{0})\varphi+\pi_{1}(f_{0})\pi_{2}(f_{1})P(X^{M}_{2})\varphi$
$\displaystyle+\pi_{1}(f_{1})\pi_{2}(f_{0})P(X_{M}^{1})\varphi+\pi_{1}(f_{1})\pi_{2}(f_{1})P(X^{M}_{2})P(X^{M}_{1})\varphi$
$\displaystyle\vdots$ $\displaystyle=$
$\displaystyle\sum_{i_{1},\ldots,i_{k}\in\\{0,1\\}}\pi_{1}(f_{i_{1}})\cdots\pi_{k}(f_{i_{k}})P(X^{M}_{k})^{i_{k}}\cdots
P(X^{M}_{1})^{i_{1}}\varphi$

for all $\varphi\in\mathscr{P}$, with the understanding that
$P(X_{i})^{0}=\mathrm{id}$. Thus, using (7),

$\varphi=\sum_{i_{1},\ldots,i_{k}\in\\{0,1\\}}f_{i_{1},\ldots,i_{k}}*\Psi_{i_{1},\ldots,i_{k}}(\varphi)$

where

$\displaystyle f_{i_{1},\ldots,i_{k}}$
$\displaystyle\colonequals\theta_{W}(f_{1}\otimes\cdots\otimes
f_{k}\otimes\rho)\subseteq C_{c}^{\infty}(W)\subseteq C_{c}^{\infty}(G)$
$\displaystyle A_{0}$
$\displaystyle\colonequals\bigcap_{i_{1},\ldots,i_{k}\in\\{0,1\\}}\operatorname{dom}\left(P(X^{M}_{k})^{i_{k}}\cdots
P(X^{M}_{1})^{i_{1}}\right)\cap C_{c}^{\infty}(M)^{K}$
$\displaystyle\Psi_{i_{1},\ldots,i_{k}}$ $\displaystyle\colonequals
P(X^{M}_{k})^{i_{k}}\cdots P(X^{M}_{1})^{i_{1}}|_{A_{0}}$

By Corollary 4.16, we have that $A_{0}$ is a right
$C_{c}^{\infty}(G^{\prime})$-submodule of $C_{c}^{\infty}(M)^{K}$ and that the
maps $\Psi_{i_{1},\ldots,i_{k}}$ are right
$C_{c}^{\infty}(G^{\prime})$-linear, as required. They do not increase
supports by Proposition 2.9. ∎

Specializing to the case where $G$ acts on itself from the right and left,
Theorem 5.3 amounts to the following.

######  Corollary 5.4.

Let $G\rightrightarrows B$ be a Lie groupoid with a given Haar system. Let
$\mathscr{P}$ be a finite subset of $C_{c}^{\infty}(G)$ and put
$K\colonequals\bigcup_{\varphi\in\mathscr{P}}t(\operatorname{supp}\varphi)$
(by Proposition 2.4, $K$ is a compact subset of $B$) and let $W\subseteq G$ be
an open set with $K\subseteq W$ (by Lemma 4.2, we may assume $W$ is
Hausdorff). Then, there exist:

  1. 1.

$f_{1},\ldots,f_{N}\in C_{c}^{\infty}(W)\subseteq C_{c}^{\infty}(G)$,

  2. 2.

a right ideal $A_{0}\subseteq C_{c}^{\infty}(G)$ with $\mathscr{P}\subseteq
A_{0}\subseteq C_{c}^{\infty}(G)^{K}$,

  3. 3.

right $C_{c}^{\infty}(G)$-linear maps $\Psi_{1},\ldots,\Psi_{N}:A_{0}\to
C_{c}^{\infty}(G)$ that do not increase supports

such that, for all $\varphi\in A_{0}$, we have
$\varphi=f_{1}*\Psi_{1}(\varphi)+\ldots+f_{N}*\Psi_{N}(\varphi)$. ∎

The desired result follows.

######  Corollary 5.5.

For any Lie groupoid $G$ with given Haar system, the smooth convolution
algebra $C_{c}^{\infty}(G)$ is $H$-unital.

###### Proof.

This follows from Corollary 5.4 and Proposition 1.13, taking $\phi$ to be the
map

$\varphi\mapsto
f_{1}\otimes\Psi_{1}(\varphi)+\ldots+f_{N}\otimes\Psi_{N}(\varphi).$

∎

##  6 Factorization of flat functions relative to a submersion

In this section, we extend the following known factorization result for
functions that are “flat” on a given closed set so as to allow factorization
to be performed relative to a submersion.

######  Theorem 6.1.

Suppose $M$ is a Hausdorff smooth manifold and $Z\subseteq M$ is closed. Let
$I_{Z}^{\infty}$ denote the ideal in $C^{\infty}_{c}(M)$ consisting of
functions that vanish to infinite order (i.e. vanish with all derivatives) on
$Z$. Then, given $\varphi_{1},\ldots,\varphi_{N}\in I_{Z}^{\infty}$, there
exists $\rho,\psi_{1},\ldots,\psi_{N}\in I_{Z}^{\infty}$ such that
$\varphi_{i}=\rho\psi_{i}$ for $i=1,\ldots,N$.

###### Proof.

This follows from Theorem 3.2 of [19]. See also the proof of Theorem 6.2 in
[20] (“property [F]” is defined on pp. 612). ∎

Our extended factorization result is Theorem 6.8 below, essentially a “with
parameters” version of the preceding result. It seems likely that Theorem 6.8
is known to experts but, as a reference could not be located, we provide a
proof.

Let us first record the following elementary extension principle.

######  Lemma 6.2.

Let $Z\subseteq\mathbb{R}^{d}$ be a closed set and let $f\in C^{\infty}(W)$,
$W\colonequals\mathbb{R}^{d}\setminus Z$. If $f$ and all its partial
derivatives vanish at the boundary of $W$, then $f$ extends by zero to a
smooth function on $\mathbb{R}^{d}$ vanishing to infinite order on $Z$.

###### Proof.

Let $g$ be the extension by zero of $f$. Using induction and the standard
result that a function on $\mathbb{R}^{d}$ is $C^{1}$ if and only each of its
first order partials exists and is continuous, one only needs to check that
the first order partials of $g$ exists and vanish on $Z$. One may therefore
reduce to the case $d=1$ where the desired conclusion may be deduced from the
mean value theorem. ∎

The preceding lemma leads to directly to a condition under which one may form
the quotient of two smooth functions vanishing to infinite order on a given
closed set.

######  Lemma 6.3.

Let $Z\subseteq\mathbb{R}^{d}$ be a closed set. Suppose $f,g\in
C^{\infty}(\mathbb{R}^{d})$ vanish to infinite order on $Z$ and $g>0$ on
$W\colonequals\mathbb{R}^{n}\setminus Z$. If, for every
$\alpha\in\mathbb{N}^{d}$ and $m\in\mathbb{N}$, the function
$\frac{\partial^{\alpha}f}{g^{m}}\in C^{\infty}(W)$ vanishes at the boundary
of $W$, then the extension by zero of $\frac{f}{g}$ is a smooth function on
$\mathbb{R}^{d}$ vanishing to infinite order on $Z$.

###### Proof.

Let $\mathscr{F}\subseteq C^{\infty}(W)$ denote the
$C^{\infty}(\mathbb{R}^{d})$-linear span of the functions
$\frac{\partial^{\alpha}f}{g^{m}}$ for $\alpha\in\mathbb{N}^{d}$,
$m\in\mathbb{N}$. By assumption, the functions in $\mathscr{F}$ all vanish at
the boundary of $W$. Observe that $\mathscr{F}$ is closed under taking partial
derivatives. Indeed, if $\alpha\in\mathbb{N}^{d}$ is a multi-index,
$m\in\mathbb{N}$, $h\in C^{\infty}(\mathbb{R}^{d})$ and $\partial$ is one of
the first-order partials, then

$\partial(h\frac{\partial^{\alpha}f}{g^{m}})=(\partial
h)\frac{\partial^{\alpha}f}{g^{m}}+h\frac{(\partial\circ\partial^{\alpha})f}{g^{m}}-m(\partial
g)h\frac{\partial^{\alpha}f}{g^{m+1}}.$

Thus, thinking of $\frac{f}{g}$ as a smooth function on $W$, we have by
induction that all of its higher order partial derivatives vanish at the
boundary of $W$. Thus, $\frac{f}{g}$ extends to a smooth function on
$\mathbb{R}^{d}$ that vanishes to infinite order on $Z$ by Lemma 6.2. ∎

We use the existence of a regularized distance function for an arbitrary
closed subset of Euclidean space. Such distance functions appear in the proof
of the Whitney extension theorem given in [17] and can be explicitly
constructed using cubical meshes. Note that Wodzicki [20] also uses these
functions by way of an appeal to [19].

######  Theorem 6.4.

Let $Z\subseteq\mathbb{R}^{d}$ be closed,
$W\colonequals\mathbb{R}^{d}\setminus Z$ and let $\delta(x)$ denote the
distance from $x$ to $Z$. Then, there is a smooth function
$\Delta:W\to(0,\infty)$ such that:

  1. (i)

there exist constants $c_{2}>c_{1}>0$ such that
$c_{1}\delta(x)\leq\Delta(x)\leq c_{2}\delta(x)$, $x\in W$,

  2. (ii)

for each $\alpha\in\mathbb{N}^{d}$, there is a constant $B_{\alpha}\geq 0$
such that $|\partial^{\alpha}\Delta(x)|\leq B_{\alpha}\delta(x)^{1-|\alpha|}$,
$x\in W$.

The constants $c_{1},c_{2},B_{\alpha}$ are independent of $Z$.

###### Proof.

This is Theorem 2 on pp. 171 of [17]. ∎

Next, we apply Theorem 6.4 to construct smooth defining functions $\rho\in
C^{\infty}(\mathbb{R}^{d})$ for arbitrary closed sets
$Z\subseteq\mathbb{R}^{d}$ with rate of vanishing governed by an arbitrary
smooth function $g:(0,\infty)\to(0,\infty)$ vanishing together with all
derivatives at $0$.

######  Lemma 6.5.

Let $Z\subseteq\mathbb{R}^{d}$ be closed and put
$W\colonequals\mathbb{R}^{d}\setminus Z$. Let $\Delta:W\to(0,\infty)$ be a
regularized distance function for $Z$, as in Theorem 6.4. Then, given any
$g\in C^{\infty}(0,\infty)$ vanishing together with all derivatives at $0$,
the extension by zero of $g\circ\Delta$ is a smooth function $\rho$ on
$\mathbb{R}^{d}$ vanishing to infinite order on $Z$.

###### Proof.

In view of Lemma 6.2, we just need to show that all of the partial derivatives
of $g\circ\Delta$ vanish at the boundary of $W$. For nonzero
$\alpha\in\mathbb{N}^{d}$, we have the following formula for the $\alpha$th
partial derivative of a composition:

$\partial^{\alpha}(g\circ\Delta)=\sum_{k=1}^{|\alpha|}(g^{(k)}\circ\Delta)\sum_{\beta\in
J(\alpha,k)}C(\alpha,k,\beta)\prod_{j=1}^{k}\partial^{\beta(j)}\Delta$

where $C(\alpha,k,\beta)\in\mathbb{N}$ and

$J(\alpha,k)\colonequals\\{\beta=(\beta(1),\ldots,\beta(k))\in(\mathbb{N}^{d})^{k}:\beta(j)\neq
0,j=1,\ldots,k\text{ and }{\textstyle\sum_{j=1}^{k}}\beta(j)=\alpha\\}.$

See the proof of Lemma 3.1 in [19] as well as [8]. Applying the estimates of
Theorem 6.4 to the above expression for $\partial^{\alpha}(g\circ\Delta)$, it
is straightforward to derive an estimate of the form

$|\partial^{\alpha}(g\circ\Delta)|\leq\sum_{k=1}^{|\alpha|}C(k)(g^{(k)}\circ\Delta)\Delta^{k-|\alpha|},$

where $C(k)\geq 0$. Since $g(t)t^{-p}\to 0$ as $t\to 0^{+}$ for all
$p\in\mathbb{N}$ and $\Delta$ vanishes at the boundary of $W$, the estimate
above shows that $\partial^{\alpha}(g\ \circ\Delta)$ vanishes at the boundary
of $W$ as needed. ∎

Lemma 6.5, together with the following result, allows one to construct smooth
defining functions for a closed set $Z\subseteq\mathbb{R}^{d}$ which vanish to
infinite order on $Z$ “as slowly as desired”.

######  Lemma 6.6.

Let $f_{k}$ be a sequence of functions on $(0,\infty)$ such that $\lim_{t\to
0^{+}}f_{k}(t)t^{-p}=0$ for all $p\in\mathbb{N}$. Then, there exists a smooth,
positive-valued function $g$ on $(0,\infty)$ that vanishes together with all
its derivatives at $0$ such that $\lim_{t\to 0^{+}}\frac{f_{k}(t)}{g(t)}=0$
for all $k$.

###### Proof.

See Lemma 6.7 in [9] for a complete proof and references to the literature. ∎

######  Theorem 6.7.

Let $Z\subseteq\mathbb{R}^{k}$ be closed,
$W\colonequals\mathbb{R}^{k}\setminus Z$. Suppose $f\in
C^{\infty}(\mathbb{R}^{k}\times\mathbb{R}^{\ell})$ vanishes to infinite order
on $Z\times\mathbb{R}^{\ell}$. Then, there exists $\rho\in
C^{\infty}(\mathbb{R}^{k})$ vanishing to infinite order on $Z$ and strictly
positive on $W$ and $g\in C^{\infty}(\mathbb{R}^{k}\times\mathbb{R}^{\ell})$
vanishing to infinite order on $Z\times\mathbb{R}^{k}$ such that
$f(x,y)=\rho(x)g(x,y)$ for all
$(x,y)\in\mathbb{R}^{k}\times\mathbb{R}^{\ell}$.

###### Proof.

Because $f$ vanishes with all its derivatives on $Z\times\mathbb{R}^{\ell}$,
it follows (e.g. from Taylor’s theorem) that $(x,y)\mapsto
f(x,y)\delta(x)^{-p}$ vanishes at the boundary of $W\times\mathbb{R}^{\ell}$
for any $p\in\mathbb{N}$, where $\delta(x)$ denotes the distance from $x$ to
$Z$. The same is true if $\delta$ is replaced by a regularized distance
function $\Delta:W\to(0,\infty)$ (Theorem 6.4). Given
$\alpha\in\mathbb{N}^{k}\times\mathbb{N}^{\ell}$, $m\in\mathbb{N}$, $r>0$,
define $f_{\alpha,m,r}:(0,\infty)\to(0,\infty)$ by

$f_{\alpha,m,r}(t)=\sup\\{|\partial^{\alpha}f(x,y)|^{1/m}:|x|,|y|\leq r\text{
and }\Delta(x)\leq t\\}.$

By design, $f_{\alpha,m,r}$ is an increasing, continuous function satisfying

$\lim_{t\to 0^{+}}f_{\alpha,m,r}(t)t^{-p}=0$

for all $p\in\mathbb{N}$. Thus, by Lemma 6.6, there exists a smooth function
$g:(0,\infty)\to(0,\infty)$ that vanishes with all its derivatives at $0$ such
that

$\lim_{t\to 0^{+}}\frac{f_{\alpha,m,r}(t)}{g(t)}=0$

for all $\alpha\in\mathbb{N}^{k}\times\mathbb{N}^{\ell}$, $m\in\mathbb{N}$,
$r>0$. By Lemma 6.5, $g\circ\Delta$ extends by zero to a smooth function
$\rho:\mathbb{R}^{d}\to[0,\infty)$ which vanishes to infinite order on $Z$.
The estimate

$\left|\frac{\partial^{\alpha}f(x,y)}{\rho(x)^{m}}\right|\leq\left(\frac{f_{\alpha,m,r}(\Delta(x))}{g(\Delta(x))}\right)^{m}$

(valid for $|x|,|y|\leq r$) shows that
$(x,y)\mapsto\frac{\partial^{\alpha}f(x,y)}{\rho(x)^{m}}$ vanishes at the
boundary of $W\times\mathbb{R}^{\ell}$ for all
$\alpha\in\mathbb{N}^{k}\times\mathbb{N}^{\ell}$, $m\in\mathbb{N}$ and so, by
Theorem 6.3, $(x,y)\mapsto\frac{f(x,y)}{\rho(x)}$ extends by zero to a smooth
function on $\mathbb{R}^{k}\times\mathbb{R}^{\ell}$ vanishing to infinite
order on $Z\times\mathbb{R}^{k}$. ∎

We are now in a position to state and prove the main result of this section.

######  Theorem 6.8.

Let $\pi:M\to B$ be a submersion where $B$ is Hausdorff and $M$ is possibly
non-Hausdorff. View $C_{c}^{\infty}(M)$ as a $C^{\infty}(B)$-module with
module structure given by $f\cdot\varphi\colonequals(f\circ\pi)\varphi$ for
$f\in C^{\infty}(B)$, $\varphi\in C_{c}^{\infty}(M)$. Let $Z\subseteq B$ be a
closed set and let $\varphi_{1},\ldots,\varphi_{N}\in C_{c}^{\infty}(M)$
vanish to infinite order on $\pi^{-1}(Z)$ (Definition 4.17). Then, there
exists $\rho\in C^{\infty}(B)$ vanishing to infinite order on $Z$ and strictly
positive on $W\colonequals B\setminus Z$ and $\psi_{1},\ldots,\psi_{N}\in
C_{c}^{\infty}(M)$ vanishing to infinite order on $\pi^{-1}(Z)$ such that
$\varphi_{i}=\rho\cdot\psi_{i}$ for $i=1,\ldots,N$.

###### Proof.

Suppose $f,\rho_{1},\rho_{2}$ are smooth functions on
$\mathbb{R}^{k}\times\mathbb{R}^{\ell}$ vanishing to infinite order on
$\\{0\\}\times\mathbb{R}^{\ell}$ and that $0<\rho_{1}<\rho_{2}$ on the
complement of $\\{0\\}\times\mathbb{R}^{\ell}$. We remark that, if
$f/\rho_{1}$ extends to a smooth function on
$\mathbb{R}^{k}\times\mathbb{R}^{\ell}$ vanishing to infinite order on
$\\{0\\}\times\mathbb{R}^{\ell}$, then $f/\rho_{2}$ also extends to a smooth
function on $\mathbb{R}^{k}\times\mathbb{R}^{\ell}$ vanishing to infinite
order on $\\{0\\}\times\mathbb{R}^{\ell}$ (see Lemma 6.9 [9]). This remark,
together with the fact that $B$ is Hausdorff and therefore admits smooth
partitions of unity allow one to (i) consider only the local problem, where
$M$ and $N$ are Euclidean spaces and $\pi$ is projection, and (ii) consider
the case of only a single function $\varphi\in C_{c}^{\infty}(M)$. The local
case of a single function is given by Theorem 6.7. ∎

##  7 H-unitality of ideals arising from invariant subsets

In this final section, we prove the second of our main results, the
H-unitality of infinite order vanishing ideals in smooth groupoid algebras
(Theorem 1.11 from the introduction). This leads directly to an excision
principle for invariant subsets. Applications of this excision result will be
considered elsewhere.

Recall from Wodzicki’s seminal paper on H-unitality, if $Z$ is a closed subset
of a Hausdorff smooth manifold $M$, then the ideal in $C^{\infty}(M)$
consisting of functions that vanish together all derivatives on $Z$ is
$H$-unital (Theorem 6.1, [20]). The goal here is to obtain the noncommutative
analogue.

The main ingredient of the proof is the following direct corollary of Theorem
6.8 (take the submersion $\pi$ to be the target submersion).

######  Corollary 7.1.

If $G\rightrightarrows B$ is a Lie groupoid, $Z\subseteq B$ is an invariant,
closed subset and $\varphi_{1},\ldots,\varphi_{N}$ belong to the infinite
order vanishing ideal $J_{Z}^{\infty}\subseteq C_{c}^{\infty}(G)$ (see Section
4.6), then there exists a smooth function $\rho\in C^{\infty}(B)$, vanishing
to infinite order on $Z$ and positive on $B\setminus Z$ and
$\psi_{1},\ldots,\psi_{N}\in J_{Z}^{\infty}$ such that
$\varphi_{i}=\rho\cdot\psi_{i}$ for $i=1,\ldots,N$. ∎

Note that, if $\rho\in C^{\infty}(B)$ is nonvanishing on $B\setminus Z$, then
$\varphi\mapsto\rho\cdot\varphi:J^{\infty}_{Z}\to J^{\infty}_{Z}$ is clearly
injective. Indeed, if $G$ is Hausdorff and $Z$ is a closed submanifold, then
$\varphi\mapsto\rho\cdot\varphi$ is injective on all of $C_{c}^{\infty}(G)$.
When $G$ is non-Hausdorff and $Z\subseteq B$ is a closed submanifold (or more
generally has empty interior), then injectivity of
$\varphi\mapsto\rho\cdot\varphi$ on all of $C_{c}^{\infty}(G)$ may fail, as
the following example shows.

######  Example 7.2.

Let $B=\mathbb{R}$ and let $G=\mathbb{R}^{\times}\sqcup\mathbb{Z}$, the “line
with infinitely many origins” with its obvious non-Hausdorff smooth manifold
structure. Then, $G$ is a Lie groupoid over $B$ where $s=t$ is the obvious
projection to $G\to B$ and multiplication
$G^{(2)}=\mathbb{R}^{\times}\sqcup\mathbb{Z}^{2}\to G$ is
$(\mathrm{id}_{\mathbb{R}^{\times}}\sqcup\mathrm{addition})$. Define
$\varphi=0\sqcup f$ where $f:\mathbb{Z}\to\\{0,1\\}$ is given by $f(1)=1$,
$f(-1)=-1$, $f(n)=0$ for $n\neq\pm 1$. It is easy to see that $\varphi$ is a
(nonzero) element of $C_{c}^{\infty}(G)$. However, $\rho\cdot\varphi=0$ for
any $\rho\in C^{\infty}(\mathbb{R})$ vanishing at $0$.

As in Section 5, we deduce H-unitality from a technical result designed to be
used with Proposition 1.13.

######  Theorem 7.3.

Let $G\rightrightarrows B$ be a Lie groupoid with a given Haar system and let
$Z\subseteq B$ be a $G$-invariant, closed subset. Let $\mathscr{P}$ be a
finite subset of $J_{Z}^{\infty}$. Put
$K\colonequals\bigcup_{\varphi\in\mathscr{P}}t(\operatorname{supp}\varphi)$
and let $W\subseteq G$ be an open set with $K\subseteq W$. Then, there exist:

  1. 1.

$f_{1},\ldots,f_{N}\in J_{Z}^{\infty}\subseteq C_{c}^{\infty}(G)$,

  2. 2.

a right ideal $A_{0}\subseteq J_{Z}^{\infty}$ with $\mathscr{P}\subseteq
A_{0}\subseteq C_{c}^{\infty}(G)^{K}$,

  3. 3.

right $C_{c}^{\infty}(G)$-linear maps $\Psi_{1},\ldots,\Psi_{N}:A_{0}\to
J_{Z}^{\infty}$ that do not increase supports

such that, for all $\varphi\in A_{0}$, we have
$\varphi=f_{1}*\Psi_{1}(\varphi)+\ldots+f_{N}*\Psi(\varphi)$.

###### Proof.

Let $\mathscr{P}=\\{\varphi_{1},\ldots,\varphi_{n}\\}$. From Corollary 7.1
above, there exists $\rho\in C^{\infty}(B)$, vanishing to infinite order on
$Z$ and positive on $B\setminus Z$, and $\psi_{1},\ldots,\psi_{n}\in
J^{\infty}_{Z}$ such that $\varphi_{i}=\rho\cdot\psi_{i}$ for $i=1,\ldots,n$.
We have that $\varphi\mapsto\rho\cdot\varphi$ is a right
$C_{c}^{\infty}(G)$-linear bijection of $J_{Z}^{\infty}$ onto the right ideal
$\rho\cdot J_{Z}^{\infty}\subseteq J_{Z}^{\infty}$. Let
$M_{\frac{1}{\rho}}:\rho\cdot J_{Z}^{\infty}\to J_{Z}^{\infty}$ denote the
inverse isomorphism of right $C_{c}^{\infty}(G)$-modules. By design,
$M_{\frac{1}{\rho}}(\psi_{i})=\varphi_{i}$ for $i=1,\ldots,n$.

From Corollary 5.4, there exist:

  1. 1.

$g_{1},\ldots,g_{N}\in C_{c}^{\infty}(W)\subseteq C_{c}^{\infty}(G)$,

  2. 2.

a right ideal $B_{0}\subseteq C_{c}^{\infty}(G)$ with $\mathscr{P}\subseteq
B_{0}\subseteq C_{c}^{\infty}(G)^{K}$,

  3. 3.

right $C_{c}^{\infty}(G)$-linear maps $\Phi_{1},\ldots,\Phi_{N}:B_{0}\to
C_{c}^{\infty}(G)$ that do not increase supports

such that, for all $\varphi\in B_{0}$, we have
$\varphi=g_{1}*\Phi_{1}(\varphi)+\ldots+g_{N}*\Phi(\varphi)$. Define:

$\displaystyle f_{i}=g_{i}\cdot\rho$ $\displaystyle A_{0}\colonequals
B_{0}\cap\rho\cdot J_{Z}^{\infty}$ $\displaystyle\Psi_{i}\colonequals
M_{\frac{1}{\rho}}\circ\Phi_{i}|_{A_{0}}$ $\displaystyle i=1,\ldots,N$

so that

$\displaystyle\varphi$
$\displaystyle=g_{1}*\Phi_{1}(\varphi)+\ldots+g_{N}*\Phi(\varphi)$
$\displaystyle=g_{i}*(\rho\cdot\Psi_{1}(\varphi))+\ldots+g_{N}*(\rho\cdot\Psi_{N}(\varphi))$
$\displaystyle=f_{1}*\Psi_{1}(\varphi)+\ldots+f_{N}*\Psi(\varphi)$

and the $f_{i}$, $A_{0}$ and $\Psi_{i}$ are as needed. ∎

As a corollary, we obtain the desired H-unitality result and its consequence
for excision.

######  Corollary 7.4.

For any Lie groupoid $G\rightrightarrows B$ with a given Haar system and any
$G$-invariant, closed subset $Z\subseteq B$, the associated ideal
$J_{Z}^{\infty}\subseteq C_{c}^{\infty}(G)$ is $H$-unital. Consequently, the
short exact sequence

$0\to J_{Z}^{\infty}\to C_{c}^{\infty}(G)\to
C_{c}^{\infty}(G)/J_{Z}^{\infty}\to 0$

induces a corresponding long exact sequence in cyclic/Hochschild homology.

###### Proof.

Follows from Proposition 1.13 and Corollary 1.8. ∎

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