Differential operators on $C^{*}$ -algebras and applications to smooth functional calculus and Schwartz functions on the tangent groupoid

Omar Mohsen

Abstract

We introduce the notion of a differential operator on $C^{*}$ -algebras. This is a noncommutative analogue of a differential operator on a smooth manifold. We show that the common closed domain of all differential operators is closed under smooth functional calculus. As a corollary, we show that Schwartz functions on Connes tangent groupoid are closed under smooth functional calculus.

Introduction

Let $M$ be a compact smooth manifold. In [Con94a, Chapter 2.5], Connes introduced the tangent groupoid

\mathbb{T}M:=M\times M\times\mathbb{R}_{+}^{\times}\sqcup TM\times\{0\}.

The space $\mathbb{T}M$ is equipped with a smooth manifold structure which is a special case of the deformation to the normal cone construction, see [Ful84a, Chapter 5]. The space $C^{\infty}_{c}(\mathbb{T}M)$ is equipped with a convolution law making it a $*$ -algebra. The convolution law is a deformation of the usual convolution law on $M\times M$ which is used in Schwartz kernels of operators acting on $L^{2}M$ and the convolution law on $T_{x}M$ for $x\in M$ which comes from the commutative group structure. As Connes shows, the space $\mathbb{T}M$ very naturally gives a canonical deformation from a differential operator to its principal symbol. Connes used this deformation to give a conceptual proof of the Atiyah-Singer index theorem [AS68a].

The tangent groupoid has been studied and generalised by many authors. In this article, we will be interested in the space of Schwartz functions $\mathcal{S}(\mathbb{T}M)$ on $\mathbb{T}M$ introduced by Carrillo-Rouse [Car08a]. The $*$ -algebra $C^{\infty}_{c}(\mathbb{T}M)$ can be naturally completed into a $C^{*}$ -algebra $C^{*}\mathbb{T}M$ . In [Car08a], Carrillo-Rouse proves that $\mathcal{S}(\mathbb{T}M)$ is a $*$ -subalgebra of $C^{*}\mathbb{T}M$ . In this article, we improve his result by showing that $\mathcal{S}(\mathbb{T}M)$ is closed under smooth functional calculus.

Theorem A.

1.

If $a\in\mathcal{S}(\mathbb{T}M)$ is a Schwartz function and $a$ is normal ( $a^{*}a=aa^{*}$ ) and $f$ is a smooth function on some open neighbourhood of $\mathrm{spec}(a)$ with $f(0)=0$ , then $f(a)\in\mathcal{S}(\mathbb{T}M)$ .
2.

If $a\in\mathcal{S}(\mathbb{T}M)$ is a Schwartz function and $f$ is a holomorphic function on some open neighbourhood of $\mathrm{spec}(a)$ with $f(0)=0$ , then $f(a)\in\mathcal{S}(\mathbb{T}M)$ .

Of course $f(a)$ in Theorem A means the functional calculus in $C^{*}\mathbb{T}M$ applied to $a$ . The main difficulty in Theorem A is the derivability of $f(a)$ in the $\mathbb{R}_{+}$ direction of $\mathbb{T}M$ because as $t\to 0^{+}$ , the structure of $C^{*}\mathbb{T}M$ changes from compact operators on $L^{2}M$ to $C_{0}(T^{*}M)$ . We remark that we suppose $f(0)=0$ because $C^{*}\mathbb{T}M$ isn’t unital.

To prove Theorem A we introduce differential operators on an arbitrary $C^{*}$ -algebra $A$ . Recall that a derivation on $A$ is a map $\delta:\mathcal{A}\to A$ defined on a $*$ -subalgebra $\mathcal{A}\subseteq A$ such that

\delta(ab)=\delta(a)b+a\delta(b),\quad\forall a,b\in\mathcal{A}.

Derivations usually come from the derivative of an $\mathbb{R}$ action on $A$ . If $A=C(N)$ and $\mathcal{A}=C^{\infty}(N)$ where $N$ is a smooth manifold $N$ , then a derivation is a vector field on $N$ . Therefore, one can think of derivations as differential operators of order $1$ on $A$ with domain $\mathcal{A}$ . We introduce the notion of a differential operator of higher order on an arbitrary $C^{*}$ -algebra $A$ . Our differential operators coincide with differential operators on $N$ if $A=C(N)$ and $\mathcal{A}=C^{\infty}(N)$ . We show that the derivative in the direction of $\mathbb{R}_{+}$ is a differential operator on $C^{*}\mathbb{T}M$ of order $2$ . The main theorem of this article is the following, see Definition 1.8 for the precise definition of $\mathcal{D}(A)\subseteq A$ . We remark that $\mathcal{D}(A)$ is a $*$ -subalgebra of $A$ which contains $\mathcal{A}$ .

Theorem B.

Let $\mathcal{A}\subseteq A$ be a dense $*$ -subalgebra and $\mathcal{D}(\mathcal{A})$ be the intersection of the domains of the closures of all differential operators defined on $\mathcal{A}$ . Then

1.

If $n\in\mathbb{N}$ , $a\in M_{n}(\mathcal{D}(\mathcal{A}))$ is normal ( $a^{*}a=aa^{*}$ ) and $f$ is a smooth function defined on some open neighbourhood of $\mathrm{spec}(a)$ with $f(0)=0$ , then $f(a)\in M_{n}(\mathcal{D}(\mathcal{A}))$ .
2.

If $n\in\mathbb{N}$ , $a\in M_{n}(\mathcal{D}(\mathcal{A}))$ and $f$ is a holomorphic function defined on some open neighbourhood of $\mathrm{spec}(a)$ with $f(0)=0$ , then $f(a)\in M_{n}(\mathcal{D}(\mathcal{A}))$ .

Here $\mathrm{spec}(a)$ is the spectrum of $a$ as an element of $M_{n}(A)$ , the $C^{*}$ -algebra of $n\times n$ matrices with coefficients in $A$ .

The methods we use to prove B are based on arguments by Bratteli and Robinson [BR75a, BR76a] and Pedersen [Ped00a]. We obtain Theorem A as a corollary of Theorem B because $\mathcal{D}(\mathcal{S}(TM))=\mathcal{S}(TM)$ .

The main interest in subalgebras closed under holomorphic calculus in the theory of operator algebras is that they give isomorphisms in $K$ -theory, thus allowing tools from cyclic cohomology to be used. Cyclic cocycles are rarely defined on the whole $C^{*}$ -algebra and only defined on a subalgebra, see for example [Con86a]

We remark that our approach to Schwartz functions is different from that of Carrillo-Rouse, and our algebra differs slightly from his algebra, see Remark 2.10. His approach is based on using local coordinates for $\mathbb{T}M$ and demanding growth conditions in such coordinates. He then shows that this is independent of local coordinates. Our approach is more global in nature. Recall that Schwartz functions on a vector space are smooth functions $f$ such that $D(f)$ is bounded whenever $D$ is a differential operator with polynomial coefficients. Analogously, on the space $\mathbb{T}M$ , we define differential operators which play the role of differential operators with polynomial coefficients. We then define Schwartz functions to be functions whose derivative by such operators is bounded.

Acknowledgments

We thank the referee for his valuable remarks which helped immensely with the exposition.

Organization of the article.

The article is organized as follows.

•

In Section 1, we define differential operators on $C^{*}$ -algebras and prove Theorem B.
•

In Section 2, we give a quick introduction to Connes’s tangent groupoid. We then prove Theorem A.

1 Differential operators on $C^{*}$ -algebras

Throughout the article $\mathbb{N}$ stands for $\{1,2,\cdots\}$ , $\mathbb{R}_{+}^{\times}$ for $\{x\in\mathbb{R}:x>0\}$ and $\mathbb{R}_{+}$ for $\{x\in\mathbb{R}:x\geq 0\}$ .

Example 1.1.

Let

\mathbb{H}:=\mathbb{R}^{3},\quad(x,y,z)\cdot(x^{\prime},y^{\prime},z^{\prime})=(x+x^{\prime},y+y^{\prime},z+z^{\prime}+xy^{\prime}-yx^{\prime})

be the Heisenberg group and

\displaystyle\delta_{x},\delta_{y},\delta_{z}:C^{\infty}_{c}(\mathbb{H})\to C^{\infty}_{c}(\mathbb{H}),\quad\delta_{x}(f)=xf,\quad\delta_{y}(f)=yf,\quad\delta_{z}(f)=zf.

(1)

The maps $\delta_{x}$ and $\delta_{y}$ satisfy the Leibniz rule

\displaystyle\delta_{x}(f\star g)=\delta_{x}(f)\star g+f\star\delta_{x}(g),\quad\delta_{y}(f\star g)=\delta_{y}(f)\star g+f\star\delta_{y}(g)

but $\delta_{z}$ satisfies

\displaystyle\delta_{z}(f\star g)=\delta_{z}(f)\star g+f\star\delta_{z}(g)+\delta_{x}(f)\star\delta_{y}(g)-\delta_{y}(f)\star\delta_{x}(g),

(2)

where $\star$ denotes the convolution. We define differential operators on noncommutative algebras to be maps which satisfy a Leibniz rule as in (2).

Definition 1.2.

Let $\mathcal{A}$ be a $\mathbb{C}$ -algebra equipped with an involution, $\mathcal{D}\subseteq\operatorname{End}(\mathcal{A})$ a $\mathbb{C}$ -subalgebra of $\mathbb{C}$ -linear endomorphisms of $\mathcal{A}$ which contains the identity.

1.

We say that $\delta\in\mathcal{D}$ is a $\mathcal{D}$ -differential operator of order $0$ if $\delta(ab)=\delta(a)b$ for all $a,b\in\mathcal{A}$ .

We define $\mathcal{D}$ -differential operators of higher order by recurrence. A map $\delta\in\mathcal{D}$ is called a $\mathcal{D}$ -differential operator of order $n\in\mathbb{N}$ , if there exists a finite family of linear maps

\delta_{10},\cdots,\delta_{1r},\delta_{20},\cdots,\delta_{2r}\in\mathcal{D}

which are $\mathcal{D}$ -differential operators of order $<n$ such that

\displaystyle\delta(ab)=\delta(a)\delta_{20}(b)+\delta_{10}(a)\delta(b)+\sum_{i=1}^{r}\delta_{1i}(a)\delta_{2i}(b).

(3)

Furthermore, we suppose that

\displaystyle\delta_{20}(x^{*})=\delta_{20}(x)^{*},\quad\forall x\in\mathcal{A}

(4)

We denote by $\operatorname{\mathrm{Diff}_{\mathcal{D}}}(\mathcal{A})\subseteq\mathcal{D}$ the space of $\mathcal{D}$ -differential operators of any order.

Remarks 1.3.

Let us explain Definition 1.2 in more details.

1.

We use a subspace $\mathcal{D}$ of linear maps instead of the space of all linear endomorphisms of $\mathcal{A}$ , because in applications, it seems unhelpful to consider all linear endomorphisms. Usually, one has a given subspace of linear endomorphisms of interest.
2.

Condition (4) is included because our main theorem fails without it. It plays an absolutely necessary role in the proof of Lemma 1.15. In all our applications (Theorem A and Corollary 1.18), the $\mathcal{D}$ -differential operators we consider have the property $\delta_{20}=c\mathrm{Id}$ with $c\in\{0,1\}$ and thus trivially satisfy (4). We include the general case of arbitrary $\delta_{20}$ satisfying (4) because the arguments in this section don’t change with its inclusion.
3.

It is worth nothing that if $\delta$ is a $\mathcal{D}$ -differential operator of order $n$ , then it is also a $\mathcal{D}$ -differential operator of order $m$ for any $m>n$ . It might be tempting to define the order of a $\mathcal{D}$ -differential operator $\delta$ to be the minimal $n$ such that $\delta$ is of order $n$ . We won’t do so because the order behaves in a rather unintuitive way. For example, the sum of a $\mathcal{D}$ -differential operator of order $n$ and another of order $m$ isn’t necessarily of order $\max(n,m)$ . It is always of order $\max(n,m)+1$ . See the proof of Proposition 1.5.

Remark 1.4.

The following observation will be used in the proof of Proposition 1.5. If $\delta\in\mathcal{D}$ such that there exists $\mathcal{D}$ -differential operators $\delta_{10},\cdots,\delta_{1r},\delta_{20},\cdots,\delta_{2r}\in\mathcal{D}$ such that (3) and (4) hold, then $\delta$ is a $\mathcal{D}$ -differential operator of order one plus the maximum of the orders of $\delta_{10},\cdots,\delta_{1r},\delta_{20},\cdots,\delta_{2r}$ . Whenever we apply this remark, to make it clearer how it is applied, we colored the term $\delta(a)\delta_{20}(b)$ by red, the term $\delta_{10}(a)\delta(b)$ by green, and the rest by blue.

Proposition 1.5.

The space $\operatorname{\mathrm{Diff}_{\mathcal{D}}}(\mathcal{A})$ is a unital subalgebra of $\mathcal{D}$ and thus of $\operatorname{End}(\mathcal{A})$ as well.

Proof.

It is clear that $\operatorname{Id}$ is a $\mathcal{D}$ -differential operator of order $0$ . Let $\delta$ and $\delta^{\prime}$ be $\mathcal{D}$ -differential operators of order $n$ and $m$ respectively with $\delta_{10},\cdots,\delta_{1r},\delta_{20},\cdots,\delta_{2r}$ and $\delta_{10}^{\prime},\cdots,\delta_{1r}^{\prime},\delta_{20}^{\prime},\cdots,\delta_{2r}^{\prime}$ as in Definition 1.2. Then,

	$\displaystyle\delta(ab)+\delta^{\prime}(ab)$	$\displaystyle=\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\delta(a)\delta_{20}(b)}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\delta_{10}(a)\delta(b)\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\sum_{i=1}^{r}\delta_{1i}(a)\delta_{2i}(b)$
		$\displaystyle\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\delta^{\prime}(a)\delta_{20}^{\prime}(b)\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\delta_{10}^{\prime}(a)\delta^{\prime}(b)\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\sum_{i=1}^{r^{\prime}}\delta^{\prime}_{1i}(a)\delta^{\prime}_{2i}(b).$

Hence, $\delta+\delta^{\prime}$ is a $\mathcal{D}$ -differential operator of order $\max(n,m)+1$ by Remark 1.4. Notice here that there are only blue terms. This is possible because, one already supposes that $\delta$ and $\delta^{\prime}$ are $\mathcal{D}$ -differential operators.

We prove that $\delta\circ\delta^{\prime}$ is a $\mathcal{D}$ -differential operator by induction on the order. The case $n=m=0$ is clear. We prove the case $m=0$ by induction on $n$ . We have

\displaystyle\delta(\delta^{\prime}(ab))=\delta(\delta^{\prime}(a)b)=\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\delta(\delta^{\prime}(a))\delta_{20}(b)\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\delta_{10}(\delta^{\prime}(a))\delta(b)\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\sum_{i=1}^{r}\delta_{1i}(\delta^{\prime}(a))\delta_{2i}(b)

(5)

All the terms $\delta_{10}\circ\delta^{\prime},\cdots,\delta_{1r}\circ\delta^{\prime}$ are $\mathcal{D}$ -differential operators by recurrence. Hence, by Remark 1.4, $\delta\circ\delta^{\prime}$ is also a $\mathcal{D}$ -differential operator.

The case $n=0$ is also proved by induction on $m$ . We have

\displaystyle\delta(\delta^{\prime}(ab))=\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\delta(\delta^{\prime}(a))\delta^{\prime}_{20}(b)\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\delta(\delta^{\prime}_{10}(a))\delta^{\prime}(b)\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\sum_{i=1}^{r^{\prime}}\delta(\delta^{\prime}_{1i}(a))\delta^{\prime}_{2i}(b).

(6)

All the terms $\delta\circ\delta^{\prime}_{10},\cdots,\delta\circ\delta^{\prime}_{1r^{\prime}}$ are $\mathcal{D}$ -differential operators by recurrence. Hence, by Remark 1.4, $\delta\circ\delta^{\prime}$ is again a $\mathcal{D}$ -differential operator.

We can now prove the general case of the composition of $\mathcal{D}$ -differential operators by induction on $(n,m)$ . We suppose that it is true for any $n^{\prime},m^{\prime}\in\mathbb{N}\cup\{0\}$ such that $n^{\prime}\leq n$ and $m^{\prime}\leq m$ and $n^{\prime}+m^{\prime}<n+m$ . One has

$\displaystyle\delta(\delta^{\prime}(ab))$	$\displaystyle=\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\delta(\delta^{\prime}(a))\delta_{20}(\delta_{20}^{\prime}(b))\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\delta_{10}(\delta^{\prime}(a))\delta(\delta_{20}^{\prime}(b))\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\sum_{i=1}^{r}\delta_{1i}(\delta^{\prime}(a))\delta_{2i}(\delta_{20}^{\prime}(b)).$	(7)
	$\displaystyle\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\delta(\delta_{10}^{\prime}(a))\delta_{20}(\delta^{\prime}(b))\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\delta_{10}(\delta_{10}^{\prime}(a))\delta(\delta^{\prime}(b))\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\sum_{i=1}^{r}\delta_{1i}(\delta_{10}^{\prime}(a))\delta_{2i}(\delta^{\prime}(b)).$
	$\displaystyle\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\sum_{j=1}^{r^{\prime}}\Big{(}\delta(\delta^{\prime}_{1j}(a))\delta_{20}(\delta^{\prime}_{2j}(b))\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\delta_{10}(\delta^{\prime}_{1j}(a))\delta(\delta^{\prime}_{2j}(b))\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\sum_{i=1}^{r}\delta_{1i}(\delta^{\prime}_{1j}(a))\delta_{2i}(\delta^{\prime}_{2j}(b))\Big{)}.$

Again by Remark 1.4, $\delta\circ\delta^{\prime}$ is a $\mathcal{D}$ -differential operator.

Finally, since the map $a\mapsto\lambda a$ is a differential operator of order $0$ for any $\lambda\in\mathbb{C}$ , it follows that $\lambda\delta$ , which is a composition of two $\mathcal{D}$ -differential operators, is also a $\mathcal{D}$ -differential operator. This finishes the proof that $\operatorname{\mathrm{Diff}_{\mathcal{D}}}(\mathcal{A})$ is a unital subalgebra of $\mathcal{D}$ .: ∎

Definition 1.6.

We say that a unital subalgebra $\mathcal{D}$ of $\operatorname{End}(\mathcal{A})$ is stable if $\mathcal{D}=\operatorname{\mathrm{Diff}_{\mathcal{D}}}(\mathcal{A})$ .

It is clear from Definition 1.2 that for any unital subalgebra $\mathcal{D}$ of $\operatorname{End}(\mathcal{A})$ , $\operatorname{\mathrm{Diff}_{\mathcal{D}}}(\mathcal{A})$ is stable.

We now give natural examples of stable subalgebras. Let $\mathfrak{g}$ be a nilpotent Lie algebra. The Baker-Campbell-Hausdorff formula is a finite sum because $\mathfrak{g}$ is nilpotent. Hence, we can view $\mathfrak{g}$ as a nilpotent Lie group. Let $\mathcal{A}=C^{\infty}_{c}(\mathfrak{g})$ . We equip $\mathcal{A}$ with the convolution product after fixing some Haar measure on $\mathfrak{g}$ . We remark that a Haar measure is equivalently a Lebesgue measure because $\mathfrak{g}$ is nilpotent.

Proposition 1.7.

For $D$ a differential operator on $\mathfrak{g}$ with polynomial coefficients, let

\delta_{D}:\mathcal{A}\to\mathcal{A},\quad\delta_{D}(f)=D(f).

The space $\mathcal{D}$ of all maps $\delta_{D}$ as $D$ varies is a stable unital subalgebra of $\operatorname{End}(\mathcal{A})$ , i.e., $\delta_{D}$ is a $\mathcal{D}$ -differential operator for every $D$ .

Proof.

It is clear that $\mathcal{D}$ is a unital subalgebra. We now show that it is stable. For $X\in\mathfrak{g}$ , let $X_{R}$ be the associated right invariant vector field. One easily sees that $\delta_{X_{R}}$ is a $\mathcal{D}$ -differential operator of order $0$ . We can write $D$ as a sum of terms of the form $PX_{1R}\cdots X_{kR}$ , where $P$ is a polynomial and $X_{1R},\cdots,X_{kR}$ are right-invariant vector fields. Hence, by Proposition 1.5, it suffices to prove that if $D(f)=Pf$ , then $\delta_{D}$ is a $\mathcal{D}$ -differential operator. Again, by Proposition 1.5, we can suppose that $P$ is linear. Let $\mathfrak{g}^{1}=\mathfrak{g}$ and $\mathfrak{g}^{n+1}=[\mathfrak{g}^{n},\mathfrak{g}]$ be the central series of $\mathfrak{g}$ . Since $\mathfrak{g}$ is nilpotent, there exists $k\in\mathbb{N}$ such that $\mathfrak{g}^{k+1}=0$ . We choose a linear decomposition $\mathfrak{g}=V^{1}\oplus\cdots\oplus V^{k}$ such that $\mathfrak{g}^{n}=V^{n}\oplus\cdots\oplus V^{k}$ . It is enough to consider $P\in V^{i*}$ for some $i$ . We proceed by induction on $i$ . For $i=1$ , $\delta_{P}$ is a $\mathcal{D}$ -differential operator of order $1$ . In fact

\displaystyle\delta_{P}(f\star g)=\delta_{P}(f)\star g+f\star\delta_{P}(g).

(8)

The induction case follows by using the Baker-Campbell-Hausdorff formula like in (2). ∎

From now on, we suppose that $\mathcal{A}$ is a dense $*$ -subalgebra of a $C^{*}$ -algebra $A$ .

Definition 1.8.

Let $\mathcal{D}$ be a stable unital subalgebra of $\operatorname{End}(\mathcal{A})$ . We denote by $\mathcal{D}(\mathcal{A})$ the set of $a\in A$ such that there exists a sequence $(x_{n})_{n\in\mathbb{N}}\in\mathcal{A}$ such that $x_{n}\to a$ and for every $\delta\in\mathcal{D}$ , $\delta(x_{n})$ and $\delta(x_{n}^{*})$ converge in $A$ .

By taking constant sequences, we see that $\mathcal{A}\subseteq\mathcal{D}(\mathcal{A})$ .

Remark 1.9.

Let $\delta\in\mathcal{D}$ . Since $\delta$ is generally an unbounded operator, it is desirable to take the closure of the graph of $\delta$ . Some care should be taken, as there exists derivations which aren’t closable. See [BR75a, BR76a] for examples and further discussion of this issue. This issue doesn’t concern us as we only care about the domain of the closure and not the linear map on the closure.

In the next theorem, $M_{n}(\mathcal{A})$ , $M_{n}(\mathcal{D}(\mathcal{A}))$ and $M_{n}(A)$ denotes the space of $n\times n$ matrices with coefficients in $\mathcal{A}$ , $\mathcal{D}(\mathcal{A})$ and $A$ respectively.

Theorem 1.10.

Let $\mathcal{D}$ be a stable unital subalgebra of $\operatorname{End}(\mathcal{A})$ . The set $\mathcal{D}(\mathcal{A})$ is a dense $*$ -subalgebra of $A$ . It is also closed under smooth functional calculus, i.e,

1.

If $n\in\mathbb{N}$ , $a\in M_{n}(\mathcal{D}(\mathcal{A}))$ is normal ( $a^{*}a=aa^{*}$ ) and $f$ is a smooth function defined on some neighborhood of $\mathrm{spec}(a)$ with $f(0)=0$ , then $f(a)\in M_{n}(\mathcal{D}(\mathcal{A}))$ .
2.

If $n\in\mathbb{N}$ , $a\in M_{n}(\mathcal{D}(\mathcal{A}))$ and $f$ is a holomorphic function defined on some neighborhood of $\mathrm{spec}(a)$ with $f(0)=0$ , then $f(a)\in M_{n}(\mathcal{D}(\mathcal{A}))$ .

Here $\mathrm{spec}(a)$ is the spectrum of $a$ as an element of $M_{n}(A)$ .

For clarity of the exposition, the proof of Theorem 1.10 will be divided into several lemmas

Lemma 1.11.

The space $\mathcal{D}(\mathcal{A})$ is a dense $*$ -subalgebra of $\mathcal{A}$ .

Proof.

The space $\mathcal{D}(\mathcal{A})$ is dense because $\mathcal{A}\subseteq\mathcal{D}(\mathcal{A})$ . Proving that $\mathcal{D}(\mathcal{A})$ is closed under addition, involution, multiplication by scalars is straightforward. Let $a,b\in\mathcal{D}(\mathcal{A})$ , $(x_{n})_{n\in\mathbb{N}},(y_{n})_{n\in\mathbb{N}}\in\mathcal{A}$ such that $x_{n}\to a$ and $y_{n}\to b$ and $\delta(x_{n})$ , $\delta(x_{n}^{*})$ , $\delta(y_{n})$ , $\delta(y_{n}^{*})$ converge for every $\delta\in\mathcal{D}$ . For any $\delta\in\mathcal{D}$ , we can find $\delta_{10},\cdots,\delta_{1r}$ , $\delta_{20},\cdots,\delta_{2r}\in\mathcal{D}$ such that

\displaystyle\delta(x_{n}y_{n})=\delta(x_{n})\delta_{20}(y_{n})+\delta_{10}(x_{n})\delta(y_{n})+\sum_{i=1}^{r}\delta_{1i}(x_{n})\delta_{2i}(y_{n}).

(9)

Since $\delta_{1i}(x_{n})$ and $\delta_{2i}(y_{n})$ converge for every $i\in\{0,\cdots,r\}$ , we obtain that $\delta(x_{n}y_{n})$ converges. By a similar argument, we deduce that $\delta(y_{n}^{*}x_{n}^{*})$ converges. Hence, $ab\in\mathcal{D}(\mathcal{A})$ . Therefore, $\mathcal{D}(\mathcal{A})$ is a dense $*$ -algebra. ∎

Let $n\in\mathbb{N}$ ,

\displaystyle M_{n}:\operatorname{End}(\mathcal{A})\to\operatorname{End}(M_{n}(\mathcal{A})),\quad M_{n}(\delta)((a)_{1\leq i,j\leq n})=(\delta(a_{ij}))_{1\leq i,j\leq n}.

(10)

Clearly $M_{n}$ is an algebra homomorphism.

Lemma 1.12.

For any $n\in\mathbb{N}$ , $M_{n}(\mathcal{D})$ is a stable subalgebra of $\operatorname{End}(M_{n}(\mathcal{A}))$ . Furthermore, $M_{n}(\mathcal{D})(M_{n}(\mathcal{A}))=M_{n}(\mathcal{D}(\mathcal{A}))$ .

Proof.

A straightforward computation shows that if $\delta$ is a left multiplier, then $M_{n}(\delta)$ is also a left multiplier. Similar computation shows that if (3) holds, then

\displaystyle M_{n}(\delta)(ab)=M_{n}(\delta)(a)M_{n}(\delta_{20})(b)+M_{n}(\delta_{10})(a)M_{n}(\delta)(b)+\sum_{i=1}^{r}M_{n}(\delta_{1i})(a)M_{n}(\delta_{2i})(b).

(11)

It is also clear that if $\delta_{20}$ satisfies (4), then $M_{n}(\delta_{20})$ satisfies (4). The lemma follows. ∎

By Lemma 1.12, it follows that to prove Theorem 1.10, it suffices to deal with the case $n=1$ .

Lemma 1.13.

Let $a\in\mathcal{D}(\mathcal{A})$ be self-adjoint and $\xi\in\mathbb{R}$ . Then, $ae^{i\xi a}\in\mathcal{D}(\mathcal{A})$ . Furthermore, we can find a family $(x_{n,\xi})_{n\in\mathbb{N},\xi\in\mathbb{R}}\in\mathcal{A}$ such that

1.

For every compact subset $K$ of $\mathbb{R}$ , the sequence $(x_{n,\xi})_{n\in\mathbb{N}}$ converges uniformly in $\xi\in K$ to $ae^{i\xi a}$ as $n\to+\infty$ . Moreover,

$\displaystyle\sup\{\left\lVert x_{n,\xi}\right\rVert:n\in\mathbb{N},\xi\in\mathbb{R}\}<+\infty$ (12)
2.

For every compact subset $K$ of $\mathbb{R}$ and for every $\delta\in\mathcal{D}$ , the sequence $\delta(x_{n,\xi})$ converges uniformly in $\xi\in K$ to an element in $A$ as $n\to+\infty$ . We denote the limit by $\delta(ae^{i\xi a})$ .
3.

For every $\delta\in\mathcal{D}$ , the function

$\displaystyle\mathbb{R}\to A,\quad\xi\mapsto\delta(ae^{i\xi a})$ (13)

is continuous. Furthermore, if $\delta$ is of order $l$ , then

$\displaystyle\sup\left\{\frac{\left\lVert\delta(x_{n,\xi})\right\rVert}{|\xi|^{l}+1}:n\in\mathbb{N},\xi\in\mathbb{R}\right\}<+\infty$ (14)
4.

For any $\xi\in\mathbb{R}$ , $n\in\mathbb{N}$ , $x_{n,\xi}^{*}=x_{n,-\xi}$ .

We remark that the notation $\delta(ae^{i\xi a})$ is an abuse of notation. In general $ae^{i\xi a}$ doesn’t belong to $\mathcal{A}$ , therefore the value of $\delta(ae^{i\xi a})$ usually depends on the choice of a sequence $x_{n}$ converging to $a$ as in Definition 1.8, see Remark 1.9.

We will delay the proof of Lemma 1.13 for the moment. Instead, we will finish the proof of Theorem 1.10.1.

Proof of Theorem 1.10.1.

To prove Theorem 1.10.1, we first deal with the self-adjoint case, then with the general case of a normal element.

Let $a\in\mathcal{D}(\mathcal{A})$ be self-adjoint, $f$ a $C^{\infty}$ function on a neighbourhood of $\operatorname{Spec}(a)\subseteq\mathbb{R}$ with $f(0)=0$ . Since we are only interested in $f(a)$ , without loss of generality, we can suppose $f\in C^{\infty}_{c}(\mathbb{R})$ . Since $f(0)=0$ , we can write $f(x)=xg(x)$ where $g\in C^{\infty}_{c}(\mathbb{R})$ . The Fourier transform $\hat{g}$ of $g$ is a Schwartz function. By the Fourier inversion formula

f(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}xe^{ix\xi}\hat{g}(\xi)d\xi.

Hence,

\displaystyle f(a)=\frac{1}{2\pi}\int_{-\infty}^{\infty}ae^{ia\xi}\hat{g}(\xi)d\xi

(15)

and the integral is absolutely convergent. We approximate the integral $\frac{1}{2\pi}\int_{-\infty}^{\infty}ae^{ia\xi}\hat{g}(\xi)d\xi$ by the finite Riemann sums

u_{n}:=\frac{1}{2\pi}\sum_{j=-n^{2}}^{n^{2}}\frac{1}{n}ae^{ia\frac{j}{n}}\hat{g}\left(\frac{j}{n}\right)

It is clear that $u_{n}\to f(a)$ as $n\to+\infty$ . Let $(x_{n,\xi})_{n\in\mathbb{N},\xi\in\mathbb{R}}$ be as in Lemma 1.13. We define

s_{n}=\frac{1}{2\pi}\sum_{j=-n^{2}}^{n^{2}}\frac{1}{n}x_{n,\frac{j}{n}}\hat{g}\left(\frac{j}{n}\right).

It is clear that $s_{n}\in\mathcal{A}$ . Since $x_{n,\xi}$ converges to $ae^{i\xi a}$ uniformly in $\xi$ on compact sets, and $\hat{g}$ is Schwartz, and by (12), we deduce that $\lim_{n\to+\infty}\left\lVert s_{n}-u_{n}\right\rVert=0$ . Hence, $s_{n}\to f(a)$ . Let $\delta\in\mathcal{D}$ . By Lemma 1.13.3, the integral

\displaystyle\frac{1}{2\pi}\int_{-\infty}^{\infty}\delta(ae^{i\xi a})\hat{g}(\xi)d\xi

(16)

is absolutely convergent. By using Riemann sums like we did with Integral (15), we deduce that

\lim_{n\to+\infty}\delta(s_{n})=\frac{1}{2\pi}\int_{-\infty}^{\infty}\delta(ae^{i\xi a})\hat{g}(\xi)d\xi.

Since $x_{n,\xi}^{*}=x_{n,-\xi}$ , we can reuse our argument to deduce that

\lim_{n\to+\infty}\delta(s_{n}^{*})=\frac{1}{2\pi}\int_{-\infty}^{\infty}\delta(ae^{i\xi a})\overline{\hat{g}(-\xi)}d\xi.

Hence, $f(a)\in\mathcal{D}(\mathcal{A})$ .

Let $a\in\mathcal{D}(\mathcal{A})$ be a normal element, $f$ a $C^{\infty}$ function on a neighbourhood of $\operatorname{Spec}(a)\subseteq\mathbb{C}$ with $f(0)=0$ . Like before, we can suppose that $f\in C^{\infty}_{c}(\mathbb{C})$ . We write $z=x+iy$ for the coordinates in $\mathbb{C}$ . Consider the function $g(z)=f(x)$ . Then, $g(a)=f_{|\mathbb{R}}(\frac{a+a^{*}}{2})$ . It follows from the self-adjoint case that $g(a)\in\mathcal{D}(\mathcal{A})$ . Hence, by replacing $f$ with $f-g$ , we can suppose that $f=0$ on $\mathbb{R}$ . By a similar argument, we can suppose that $f=0$ on both $\mathbb{R}$ and $i\mathbb{R}$ . Hence, there exists $g\in C^{\infty}_{c}(\mathbb{C})$ such that $f(z)=xyg(z)$ . Since $g$ is smooth compactly supported, it follows that $\hat{g}$ is a Schwartz function. Let $\Re(a)=\frac{a+a^{*}}{2}$ and $\Im(a)=\frac{a-a^{*}}{2i}$ . By the Fourier inversion formula

\displaystyle f(a)=\frac{1}{(2\pi)^{2}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\Re(a)e^{i\xi\Re(a)}\Im(a)e^{i\eta\Im(a)}\hat{g}(\xi+i\eta)d\xi d\eta.

(17)

Theorem 1.10.1 follows from Lemma 1.13 applied to $\Re(a)$ and $\Im(a)$ by writing Integral (17) as a Riemann sum like we did in the self adjoint case. This finishes the proof of Theorem 1.10.1 under the assumption of Lemma 1.13.∎

The proof of Lemma 1.13 requires a few lemmas. Let $a\in\mathcal{D}(\mathcal{A})$ be a self-adjoint element. We fix a sequence $(x_{n})_{n\in\mathbb{N}}$ such that $x_{n}\to a$ and $\delta(x_{n})$ and $\delta(x_{n}^{*})$ converge for every $\delta\in\mathcal{D}$ . By replacing $x_{n}$ with $\frac{x_{n}+x_{n}^{*}}{2}$ , we can suppose that $x_{n}=x_{n}^{*}$ .

Lemma 1.14.

For every $l\in\mathbb{N}$ and $\delta\in\mathcal{D}$ of order $l$ , there exists $C_{1},C_{2}>0$ such that

\left\lVert\delta(x_{n}^{m})\right\rVert\leq C_{1}m^{l}C_{2}^{m},\quad\forall m,n\in\mathbb{N}.

(18)

Proof.

We prove the lemma by induction on $l$ . For $l=0$ and $m\in\mathbb{N}$ , we have

\displaystyle\delta(x_{n}^{m+1})=\delta(x_{n})x_{n}^{m}

(19)

Since $\left\lVert\delta(x_{n})\right\rVert$ and $\left\lVert x_{n}\right\rVert$ converge, it follows that they are bounded. The case $l=0$ is finished.

Suppose the lemma holds for $\delta\in\mathcal{D}$ of order $<l$ . Let $\delta\in\mathcal{D}$ be of order $l$ . For each $i\in\{0,\cdots,r\}$ , the Lemma holds for $\delta_{1i}$ . Therefore, there exists constants $C_{1},C_{2}$ such that

\displaystyle\left\lVert\delta_{1i}(x_{n}^{m})\right\rVert\leq C_{1}m^{\text{order of }\delta_{1i}}C_{2}^{m}\leq C_{1}m^{l-1}C_{2}^{m},\quad\forall n,m\in\mathbb{N},i\in\{0,\cdots,r\}.

(20)

Let $C_{1}^{\prime}$ be a constant such that

\displaystyle\left\lVert\delta(x_{n})\right\rVert\leq C_{1}^{\prime},\quad\text{and}\quad\left\lVert\delta_{2i}(x_{n})\right\rVert\leq C_{1}^{\prime},\quad\text{and}\quad C_{1}\leq C_{1}^{\prime},\quad\forall n\in\mathbb{N},i\in\{1,\cdots,r\}.

(21)

Let $C_{2}^{\prime}$ be a constant such that

\displaystyle\left\lVert\delta_{20}(x_{n})\right\rVert\leq C_{2}^{\prime},\quad\text{and}\quad C_{2}\leq C_{2}^{\prime},\quad\forall n\in\mathbb{N}.

(22)

Some constants $C_{1}^{\prime},C_{2}^{\prime}$ exist because $\delta(x_{n})$ , $\delta_{20}(x_{n}),\cdots,\delta_{2r}(x_{n})$ converge in $A$ .

By (3), for $m\in\mathbb{N}$ , we have

\displaystyle\delta(x_{n}^{m+1})=\delta(x_{n}^{m})\delta_{20}(x_{n})+\delta_{10}(x_{n}^{m})\delta(x_{n})+\sum_{k=1}^{r}\delta_{1k}(x_{n}^{m})\delta_{2k}(x_{n}),\quad\forall n,m\in\mathbb{N}.

(23)

By recurrence, we deduce that

\displaystyle\delta(x^{m+1}_{n})=\delta(x_{n})\delta_{20}(x_{n})^{m}+\sum_{j=1}^{m}\delta_{10}(x^{j}_{n})\delta(x_{n})\delta_{20}(x_{n})^{m-j}+\sum_{j=1}^{m}\sum_{k=1}^{r}\delta_{1k}(x^{j}_{n})\delta_{2k}(x_{n})\delta_{20}(x_{n})^{m-j}

(24)

Hence

	$\displaystyle\left\lVert\delta(x_{n}^{m+1})\right\rVert$	$\displaystyle\leq C_{1}^{\prime}C_{2}^{\prime m}+\sum_{j=1}^{m}j^{l-1}C_{1}^{\prime 2}C_{2}^{\prime m}+\sum_{j=1}^{m}j^{l-1}rC_{1}^{\prime 2}C_{2}^{\prime m}$		(25)
		$\displaystyle\leq C_{1}^{\prime}C_{2}^{\prime m}+m^{l+1}C_{1}^{\prime 2}C_{2}^{\prime m}+m^{l+1}rC_{1}^{\prime 2}C_{2}^{\prime m}$		(25)

The result follows. ∎

Let $\xi\in\mathbb{R}$ , $\delta\in\mathcal{D}$ . By Lemma 1.14, the sum

\displaystyle\sum_{m=0}^{\infty}(i\xi)^{m}\frac{\delta(x_{n}^{m+1})}{m!}

(26)

converges in norm uniformly in $n$ . By an abuse of notation, we denote the sum by $\delta(x_{n}e^{i\xi x_{n}})$ . Since $\lim_{n\to+\infty}\delta(x_{n}^{m+1})$ exists for all $m\in\mathbb{N}$ and the convergence in (26) is uniform, it follows that

\displaystyle\delta(ae^{i\xi a}):=\lim_{n\to+\infty}\delta(x_{n}e^{i\xi x_{n}})=\sum_{m=0}^{\infty}(i\xi)^{m}\frac{\lim_{n\to+\infty}\delta(x_{n}^{m+1})}{m!}

(27)

exists.

Lemma 1.15.

For every $l\in\mathbb{N}$ , $\delta\in\mathcal{D}$ of order $l$ , there exists constant $C>0$ such that

\left\lVert\delta(x_{n}e^{i\xi x_{n}})\right\rVert\leq C(|\xi|^{l}+1),\quad\forall n\in\mathbb{N},\xi\in\mathbb{R}.

Proof.

We prove the lemma by induction on $l$ . If $l=0$ , then by (19)

\displaystyle\delta(x_{n}e^{i\xi x_{n}})=\sum_{m=0}^{\infty}(i\xi)^{m}\frac{\delta(x_{n}^{m+1})}{m!}=\delta(x_{n})\sum_{m=0}^{\infty}(i\xi)^{m}\frac{x_{n}^{m}}{m!}=\delta(x_{n})e^{i\xi x_{n}}

(28)

Since $x_{n}$ are self-adjoint, it follows that $\left\lVert e^{i\xi x_{n}}\right\rVert=1$ . The case $l=0$ follows.

Let $\delta\in\mathcal{D}$ of order $l$ . We suppose the lemma holds for $\mathcal{D}$ -differential operators of order $<l$ . By (24), and the identity

\displaystyle\frac{1}{(j-1)!(m-j)!}\int_{0}^{1}s^{j-1}(1-s)^{m-j}ds=\frac{1}{m!},\quad 1\leq j\leq m,

(29)

it follows that

	$\displaystyle\delta(x_{n}e^{i\xi x_{n}})=\delta(x_{n})e^{i\xi\delta_{20}(x_{n})}$	$\displaystyle+i\xi\int_{0}^{1}\delta_{10}(x_{n}e^{is\xi x_{n}})\delta(x_{n})e^{i(1-s)\xi\delta_{20}(x_{n})}ds$
		$\displaystyle+i\xi\sum_{k=1}^{r}\int_{0}^{1}\delta_{1k}(x_{n}e^{is\xi x_{n}})\delta_{2k}(x_{n})e^{i(1-s)\xi\delta_{20}(x_{n})}ds$

Now we use crucially (4), to deduce that $\delta_{20}(x_{n})$ is self-adjoint. Therefore,

\left\lVert e^{i\xi\delta_{20}(x_{n})}\right\rVert=\left\lVert e^{i(1-s)\xi\delta_{20}(x_{n})}\right\rVert=1.

The lemma follows from the induction hypothesis on $\delta_{1k}(x_{n}e^{is\xi x_{n}})$ for $k\in\{0,\cdots,r\}$ . ∎

Proof of Lemma 1.13.

Let $\phi:\mathbb{R}_{+}\to\mathbb{N}$ be any function such which satisfies

\displaystyle\phi(\xi)<m\implies\frac{\xi^{m}}{m!}\leq\frac{1}{\sqrt{m!}},\quad\forall\xi\in\mathbb{R}_{+},m\in\mathbb{N}.

(30)

We take

\displaystyle x_{n,\xi}:=\sum_{m=0}^{\phi(|\xi|)+n}(i\xi)^{m}\frac{x_{n}^{m+1}}{m!}.

(31)

The element $x_{n,\xi}$ is a finite sum of elements of $\mathcal{A}$ . Hence, it belongs to $\mathcal{A}$ . We now check that $x_{n,\xi}$ satisfies the required properties.

It is rather straightforward to show that $x_{n,\xi}$ converges to $ae^{i\xi a}$ uniformly on each compact subset of $\mathbb{R}$ . Let us show (12). Let $C=\sup\{\left\lVert x_{n}\right\rVert:n\in\mathbb{N}\}$ . It is finite because $x_{n}\to a$ . Since

\sum_{m=0}^{\infty}(i\xi)^{m}\frac{x_{n}^{m+1}}{m!}=x_{n}e^{i\xi x_{n}},

and $x_{n}$ is self-adjoint, we deduce that

\displaystyle\left\lVert\sum_{m=0}^{\infty}(i\xi)^{m}\frac{x_{n}^{m+1}}{m!}\right\rVert\leq\left\lVert x_{n}\right\rVert\leq C.

(32)

$\displaystyle\left\lVert x_{n,\xi}-\sum_{m=0}^{\infty}(i\xi)^{m}\frac{x_{n}^{m+1}}{m!}\right\rVert$	$\displaystyle=\left\lVert\sum_{m=\phi(\|\xi\|)+n+1}^{\infty}(i\xi)^{m}\frac{x_{n}^{m+1}}{m!}\right\rVert$	(33)
	$\displaystyle\leq\sum_{m=\phi(\|\xi\|)+n+1}^{\infty}\|\xi\|^{m}\frac{C^{m+1}}{m!}$
	$\displaystyle\leq\sum_{m=\phi(\|\xi\|)+n+1}^{\infty}\frac{C^{m+1}}{\sqrt{m!}}$
	$\displaystyle\leq\sum_{m=0}^{\infty}\frac{C^{m+1}}{\sqrt{m!}}<+\infty$

we deduce (12).

2.

By Lemma 1.14, it is again straightforward to show that $\delta(x_{n,\xi})$ converges to $\delta(ae^{i\xi a})$ uniformly on each compact subset of $\mathbb{R}$ .

The function $\xi\mapsto\delta(ae^{i\xi a})$ is a continuous function because it is the uniform limit (over compact subsets of $\mathbb{R}$ ) of continuous function by (27). Let us show (14). Let $\delta\in\mathcal{D}$ of order $l$ . By Lemma 1.14, there exists $C_{1},C_{2}$ such that $\left\lVert\delta(x_{n}^{m})\right\rVert\leq C_{1}m^{l}C_{2}^{m}$ . Therefore, we have

$\displaystyle\left\lVert\delta(x_{n,\xi})-\delta(x_{n}e^{i\xi x_{n}})\right\rVert$	$\displaystyle=\left\lVert\sum_{m=\phi(\|\xi\|)+n+1}^{\infty}(i\xi)^{m}\frac{\delta(x_{n}^{m+1})}{m!}\right\rVert$	(34)
	$\displaystyle\leq\sum_{m=\phi(\|\xi\|)+n+1}^{\infty}\|\xi\|^{m}\frac{C_{1}(m+1)^{l}C_{2}^{m+1}}{m!}$
	$\displaystyle\leq\sum_{m=\phi(\|\xi\|)+n+1}^{\infty}\frac{C_{1}(m+1)^{l}C_{2}^{m+1}}{\sqrt{m!}}$
	$\displaystyle\leq\sum_{m=0}^{\infty}\frac{C_{1}(m+1)^{l}C_{2}^{m+1}}{\sqrt{m!}}<+\infty$

By Lemma 1.15, we deduce (12).

4.

The identity $x_{n,\xi}^{*}=x_{n,-\xi}$ is trivial to verify.

This finishes the proof of Lemma 1.13. ∎

We now proceed to prove Theorem 1.10.2. The proof of Theorem 1.10.2 relies on the fact that Theorem 1.10.1 is true even in a parametrized setting. By this, we mean the following

Lemma 1.16.

Let $K$ be a compact topological space, $(a_{w})_{w\in K}\subseteq\mathcal{D}(\mathcal{A})$ , $(x_{n,w})_{n\in\mathbb{N},w\in K}\subseteq\mathcal{A}$ a family of elements which satisfies the following:

1.

For every $n\in\mathbb{N},w\in K$ , $x_{n,w}^{*}=x_{n,w}$ .
2.

For every $n\in\mathbb{N}$ , the function $n\mapsto x_{n,w}$ is continuous in $w\in K$ .
3.

The limit of $x_{n,w}$ as $n\to+\infty$ is equal to $a_{n,w}$ uniformly in $w\in K$ .
4.

For every $\delta\in\mathcal{D}$ , the limit $\delta(x_{n,w})$ as $n\to+\infty$ exists in $A$ uniformly in $w\in K$
5.

For every $\delta\in\mathcal{D}$ , the limit $\delta(x_{n,w}^{*})$ as $n\to+\infty$ exists in $A$ uniformly in $w\in K$

Let $f$ be a continuous complex-valued function defined in an open neighbourhood of

\{(w,\lambda)\in K\times\mathbb{C}:\lambda\in\operatorname{Spec}(a_{w})\}

which is smooth in the $\lambda$ variable, and which satisfies $f(w,0)=0$ . Then, there exists a family $(y_{n,w})_{n\in\mathbb{N},w\in K}\subseteq\mathcal{A}$ which satisfies the following:

1.

For every $n\in\mathbb{N}$ , the function $n\mapsto y_{n,w}$ is continuous in $w\in K$ .
2.

The limit of $y_{n,w}$ as $n\to+\infty$ is equal to $f(w,a_{w})$ uniformly in $w\in K$ .
3.

For every $\delta\in\mathcal{D}$ , the limit $\delta(y_{n,w})$ as $n\to+\infty$ exists in $A$ uniformly in $w\in K$
4.

For every $\delta\in\mathcal{D}$ , the limit $\delta(y_{n,w}^{*})$ as $n\to+\infty$ exists in $A$ uniformly in $w\in K$

To prove Lemma 1.16, one follows the proof of Theorem 1.10.1 carefully making sure that adding a parameter doesn’t cause any problems. We leave this verification to the reader.

The following is a counterpart to Lemma 1.13 for non self-adjoint elements.

Lemma 1.17.

Let $a\in\mathcal{D}(\mathcal{A})$ . For every $w\notin\mathrm{spec(a)}\cup\{0\}$ , $\frac{a}{w-a}\in\mathcal{D}(\mathcal{A})$ . Furthermore, if $K\subseteq\mathbb{C}\backslash(\mathrm{spec(a)}\cup\{0\})$ is a compact subset, then we can find a family $(x_{n,w})_{n\in\mathbb{N},w\in K}\in\mathcal{A}$ such that the following is satisfied:

•

The sequence $(x_{n,w})_{n\in\mathbb{N}}$ converges uniformly in $w\in K$ to $\frac{a}{w-a}$ .
•

For every $\delta\in\mathcal{D}$ , $\delta(x_{n,w})$ converges uniformly in $w\in K$ to an element denoted by $\delta(\frac{a}{w-a})$ .
•

For every $\delta\in\mathcal{D}$ , $\delta(x_{n,w}^{*})$ converges uniformly in $w\in K$ to an element denoted by $\delta(\frac{a^{*}}{\bar{w}-a^{*}})$ .
•

For every $\delta\in\mathcal{D}$ , the maps $w\mapsto\delta(\frac{a}{w-a})$ and $w\mapsto\delta(\frac{a^{*}}{\bar{w}-a^{*}})$ are continuous.

Proof.

Let $w\notin\mathrm{spec(a)}\cup\{0\}$ . The element

a_{w}:=(w-a)^{*}(w-a)-|w|^{2}=-wa^{*}-\bar{w}a+a^{*}a\in\mathcal{D}(\mathcal{A})

is self-adjoint whose spectrum is contained in $]-|w|^{2},+\infty[$ . By Theorem 1.10.1, applied to $f(x)=\frac{1}{x+|w|^{2}}-\frac{1}{|w|^{2}}$ , we deduce that $f(a_{w})\in\mathcal{D}(\mathcal{A})$ . It follows that

\frac{a}{w-a}=\left(f(a_{w})+\frac{1}{|w|^{2}}\right)(w-a)^{*}a\in\mathcal{D}(\mathcal{A}).

Lemma 1.17 now follows from Lemma 1.16. ∎

Proof of Theorem 1.10.2.

Let $a\in\mathcal{D}(\mathcal{A})$ , and $f$ a holomorphic function on a neighbourhood of $\operatorname{Spec}(a)$ with $f(0)=0$ . Since $f(0)=0$ , $f(z)=zg(z)$ for some holomorphic function $g$ . Let $\gamma$ be a contour around $\mathrm{spec}(a)$ such that $\gamma$ is in the domain of $f$ and the Cauchy integral formula applies

f(z)=\frac{z}{2\pi i}\int_{\gamma}\frac{g(w)}{w-z}dw.

We thus have

f(a)=\frac{1}{2\pi i}\int_{\gamma}g(w)\frac{a}{w-a}dw.

Without loss of generality, we can suppose that $0$ doesn’t belong to the image of $\gamma$ .

Let $x_{n,w}$ be as in Lemma 1.17 applied to $K$ equal to the image of $\gamma$ . We approximate $f(a)$ by a Riemann sum, then approximate each term $\frac{a}{w-a}$ by $x_{n,w}$ . This way we obtain a sequence $s_{n}\in\mathcal{A}$ which converges to $f(a)$ and such that $\delta(s_{n})$ converges to $\frac{1}{2\pi i}\int_{\gamma}g(w)\delta(\frac{a}{w-a})dw$ and $\delta(s_{n}^{*})$ converges to $\frac{1}{2\pi i}\int_{\gamma}\bar{g}(w)\delta(\frac{a^{*}}{\bar{w}-a^{*}})dw$ . The result follows. ∎

Corollary 1.18.

Let $\mathfrak{g}$ , $A$ be as in Proposition 1.7, $\mathcal{S}(\mathfrak{g})\subseteq A$ be the $*$ -subalgebra of Schwartz functions. Then

1.

If $a\in\mathcal{S}(\mathfrak{g})$ is normal and $f$ is a smooth function defined on some neighbourhood of $\mathrm{spec}(a)$ with $f(0)=0$ , then $f(a)\in\mathcal{S}(\mathfrak{g})$ .
2.

If $a\in\mathcal{S}(\mathfrak{g})$ and $f$ is a holomorphic function defined on some neighbourhood of $\mathrm{spec}(a)$ with $f(0)=0$ , then $f(a)\in\mathcal{S}(\mathfrak{g})$ .

Corollary 1.18 is well known, see for example [HJ84a]. A well known argument using holomorphic functional calculus (see for example [Bla98a]) implies the following corollary.

Corollary 1.19.

The inclusion $i:\mathcal{D}(\mathcal{A})\to A$ induces an isomorphism in $K$ -theory $K_{0}(\mathcal{D}(\mathcal{A}))\simeq K_{0}(A)$ .

2 Schwartz functions on Connes’s tangent groupoid

Let $M$ be a compact smooth manifold. The set

\mathbb{T}M=M\times M\times\mathbb{R}_{+}^{\times}\sqcup TM\times\{0\}

can be equipped with the structure of a smooth manifold with boundary $TM\times\{0\}$ as follows. The subset $M\times M\times\mathbb{R}_{+}^{\times}$ is declared an open subset of $\mathbb{T}M$ with its usual smooth structure. If $U\subseteq M$ is an open set, $\phi:U\to\mathbb{R}^{\dim(M)}$ is a diffeomorphism, then $U\times U\times\mathbb{R}_{+}^{\times}\sqcup TU\times\{0\}$ is declared an open subset of $\mathbb{T}M$ and the following bijection is declared a diffeomorphism

$\displaystyle U\times U\times\mathbb{R}_{+}^{\times}\sqcup TU\times\{0\}$	$\displaystyle\to\mathbb{R}^{\dim(M)}\times\mathbb{R}^{\dim(M)}\times\mathbb{R}_{+}$	(35)
$\displaystyle(y,x,t)$	$\displaystyle\mapsto\left(\frac{\phi(y)-\phi(x)}{t},\phi(x),t\right),\quad x,y\in U,t\in\mathbb{R}_{+}^{\times}$
$\displaystyle(v,x,0)$	$\displaystyle\mapsto(d\phi_{x}(v),\phi(x),0),\quad x\in U,v\in T_{x}M$

One can check that this defines a smooth structure on $\mathbb{T}M$ . A more conceptual way to define the smooth structure is to notice that the smooth structure (and the topology) is uniquely determined by requiring that the maps

\displaystyle\mathbb{T}M\to M\times M\times\mathbb{R}_{+},\quad(y,x,t)\mapsto(y,x,t),\quad(v,x,0)\mapsto(x,x,0)

(36)

and

\displaystyle\operatorname{DNC}(f):\mathbb{T}M\to\mathbb{C},\quad(y,x,t)\mapsto\frac{f(y)-f(x)}{t},\quad(v,x,0)\mapsto df_{x}(v),

(37)

be smooth, where $f\in C^{\infty}(M)$ .

Differential operators.

Let $X\in\mathcal{X}(M)$ be a vector field. We can define a vector field $\mathbb{X}$ on $\mathbb{T}M$ as follows. If $f\in C^{\infty}(\mathbb{T}M)$ is a smooth function, then we define $\mathbb{X}\cdot f\in C^{\infty}_{c}(\mathbb{T}M)$ by

\mathbb{X}\cdot f(y,x,t)=tX_{y}f(y,x,t),\quad\mathbb{X}\cdot f(v,x,0)=X(x)\cdot f(v,x,0),

where $X_{y}$ means that $X$ acts on the $y$ variable, and $X(x)\cdot f(v,x,0)$ means that we view $X(x)$ as a constant vector field on $T_{x}M$ which acts on the smooth function $f(\cdot,x,0)$ . By the description of charts above, one checks that $\mathbb{X}$ is indeed a vector field.

More generally, let $D$ be a differential operator of order $d$ acting on $M$ . Then we can define a differential operator denoted $\mathbb{D}$ of order $d$ acting on $\mathbb{T}M$ by the formula

\mathbb{D}\cdot f(y,x,t)=t^{d}D_{y}f(y,x,t),\quad\mathbb{D}\cdot f(v,x,0)=D(x)\cdot f(v,x,0),

(38)

where $D_{y}$ means that $D$ acts on the $y$ variable, and $D(x)\cdot f(v,x,0)$ means that we view the principal part of $D$ as a constant coefficient differential operator on $T_{x}M$ acting on the smooth function $f(\cdot,x,0)$ . Locally by the principal part, we mean that if $D=\sum_{|I|\leq d}g_{I}\frac{\partial}{\partial x_{I}}$ , then $D(x)=\sum_{|I|=d}g_{I}(x)\frac{\partial}{\partial x_{I}}$ seen as a constant coefficient differential operator on $T_{x}M$ . Using the local charts of $\mathbb{T}M$ , one can check that $\mathbb{D}$ is indeed a differential operator.

Convolution and adjoint.

From now on, we suppose that $M$ is equipped with a Riemannian metric. If $f,g\in\mathbb{C}^{\infty}_{c}(\mathbb{T}M)$ , then we define their convolution $f\star g\in C^{\infty}_{c}(\mathbb{T}M)$ by the formula

	$\displaystyle f\star g(y,x,t)$	$\displaystyle=t^{-\dim(M)}\int_{M}f(y,z,t)g(z,x,t)dz,\quad t\neq 0$		(39)
	$\displaystyle f\star g(v,x,0)$	$\displaystyle=\int_{T_{x}M}f(v-w,x,0)g(w,x,0)dw,$		(39)

where the integral over $T_{x}M$ is with respect to the constant Riemannian metric on $T_{x}M$ induced from the Riemannian metric on $M$ . We also define the adjoint by

f^{*}(y,x,t)=\bar{f}(x,y,t),\quad f^{*}(v,x,0)=\bar{f}(-v,x,0)

We leave it to the reader to check that $C^{\infty}_{c}(\mathbb{T}M)$ with the operations defined above is a $*$ -algebra. We refer the reader to [Con94a] and [DL10a] for more details on the convolution algebra of the tangent groupoid.

Adjoint of Differential operators on $\mathbb{T}M$ .

Let $D$ be a differential operator on $M$ of order $d$ , $\mathbb{D}$ the associated differential operator on $\mathbb{T}M$ . We sometimes denote $\mathbb{D}(f)$ by $\mathbb{D}\star f$ . We also define $f\star\mathbb{D}$ by the formula

f\star\mathbb{D}:=(\mathbb{D}^{*}\star f^{*})^{*},

where $\mathbb{D}^{*}$ is the differential operator on $\mathbb{T}M$ associated to the formal adjoint $D^{*}$ of $D$ . Equivalently

f\star\mathbb{D}(y,x,t)=t^{d}D^{\prime}_{x}f(y,x,t),\quad f\star\mathbb{D}(v,x,0)=D(x)\cdot f(v,x,0),

where $D^{\prime}$ is the formal transpose given by

\int_{M}Df(x)g(x)dx=\int_{M}f(x)D^{\prime}g(x)dx,\quad f,g\in C^{\infty}(M).

One can check that

(\mathbb{D}\star f)\star g=\mathbb{D}\star(f\star g),\quad(f\star\mathbb{D})\star g=f\star(\mathbb{D}\star g)\quad f,g\in C^{\infty}_{c}(\mathbb{T}M).

(40)

This justifies our notation $f\star\mathbb{D}$ and $\mathbb{D}\star f$ .

Derivations in direction of $M$ .

The functions $\mathbb{D}\star f$ and $f\star\mathbb{D}$ agree at $t=0$ . It follows that the map

\displaystyle\delta_{D}:C^{\infty}_{c}(\mathbb{T}M)\to C^{\infty}_{c}(\mathbb{T}M),\quad f\mapsto\frac{1}{t}(\mathbb{D}\star f-f\star\mathbb{D})

is a well-defined differential operator on $\mathbb{T}M$ of order $d$ . If $D$ is an order $0$ differential operator given by multiplication by $g\in C^{\infty}(M)$ , then $\delta_{g}$ is an order $0$ differential operator given by multiplication by $\operatorname{DNC}(g)$ , see (37). For $X\in\mathcal{X}(M)$ , one can describe $\delta_{X}$ as follows. The flow of $X$ gives an $\mathbb{R}_{+}^{\times}$ action on $\mathbb{T}M$ by

\beta_{\lambda}(y,x,t)=(\exp(\lambda X)\cdot y,\exp(\lambda X)\cdot x,t),\quad\beta_{\lambda}(v,x,0)=(d\exp(\lambda X)(x)(v),\exp(\lambda X)\cdot x,0).

The differential operator $\delta_{X}$ is the derivative of the action $\beta$ (plus multiplication by the divergence of $X$ ). Since $\delta_{D}$ is given by a commutator, it satisfies the Leibniz rule

\delta_{D}(f\star g)=\delta_{D}(f)\star g+f\star\delta_{D}(g),\quad f,g\in C^{\infty}_{c}(\mathbb{T}M)

(41)

and

\delta_{D_{1}D_{2}}(f)=\delta_{D_{1}}(f)\star\mathbb{D}_{2}+\mathbb{D}_{1}\star\delta_{D_{2}}(f),\quad f\in C^{\infty}_{c}(\mathbb{T}M)

(42)

Derivation in direction of $t$ .

The group $\mathbb{R}_{+}^{\times}$ acts smoothly on the left on $\mathbb{T}M$ by the formula

\alpha_{\lambda}:\mathbb{T}M\to\mathbb{T}M,\quad\alpha_{\lambda}(y,x,t)=(y,x,\lambda^{-1}t),\quad\alpha_{\lambda}(v,x,0)=(\lambda v,x,0),\quad\lambda\in\mathbb{R}_{+}^{\times}.

(43)

We also let $\mathbb{R}_{+}^{\times}$ acts on $C^{\infty}$ functions by the formula

\displaystyle\alpha_{\lambda}(f)=\lambda^{-\dim(M)}f\circ\alpha_{\lambda^{-1}},\quad f\in C^{\infty}_{c}(\mathbb{T}M),\lambda\in\mathbb{R}_{+}^{\times}.

One has

\alpha_{\lambda}(f\star g)=\alpha_{\lambda}(f)\star\alpha_{\lambda}(g),\quad\alpha_{\lambda}(f)^{*}=\alpha_{\lambda}(f^{*}).

(44)

We define $\delta_{\alpha}$ to be the derivative of the $\mathbb{R}_{+}^{\times}$ -action at $\lambda=1$ . One can check that

	$\displaystyle\delta_{\alpha}(f)(y,x,t)$	$\displaystyle=-\dim(M)f(y,x,t)+t\frac{\partial}{\partial t}f(y,x,t)$
	$\displaystyle\delta_{\alpha}(f)(v,x,0)$	$\displaystyle=-\dim(M)f(v,x,0)-\sum_{i}v_{i}\frac{\partial}{\partial v_{i}}f(v,x,0),$

where $v_{i}$ are any local coordinates for $T_{x}M$ . It follows from (44) that $\delta_{\alpha}$ satisfies the Leibniz rule

\delta_{\alpha}(f\star g)=\delta_{\alpha}(f)\star g+f\star\delta_{\alpha}(g),\quad f,g\in C^{\infty}_{c}(\mathbb{T}M).

(45)

Lemma 2.1.

One can find a finite family of smooth functions $f_{1},\cdots,f_{k}\in C^{\infty}(M)$ and vector fields $X_{1},\cdots,X_{k}\in\mathcal{X}(M)$ such that for any $X\in\mathcal{X}(M)$ , $X=\sum_{i=1}^{k}X(f_{i})X_{i}$ .

Proof.

Let $f_{1},\cdots,f_{k}:M\to\mathbb{R}$ be smooth functions such that for all $x\in M$ , $df_{1,x},\cdots,df_{k,x}$ span $T_{x}^{*}M$ . Let $\theta=M\times\mathbb{R}^{k}$ be the trivial vector bundle on $M$ of rank $k$ . The forms $df_{i}$ define a surjective bundle map $\theta\to T^{*}M$ . Its dual is an injective bundle map $i:TM\to\theta$ . Let $p:\theta\to TM$ be any bundle map such that $p\circ i=\mathrm{Id}$ . The map $p$ determines the vector fields $X_{1},\cdots,X_{k}$ . ∎

We fix a choice of $f_{i},X_{i}$ for the rest of this section. By Lemma 2.1, if $x\in M$ , then $\sum_{i=1}^{k}df_{i}(x)X_{i}(x)$ is equal to the identity $T_{x}M\to T_{x}M$ . By taking the trace of $\sum_{i=1}^{k}df_{i}(x)X_{i}(x)$ , we deduce that

\sum_{i=1}^{k}X_{i}(f_{i})(x)=\dim(M),\quad\forall x\in M

(46)

For $f\in C^{\infty}_{c}(\mathbb{T}M)$ , the functions $\delta_{\alpha}(f)$ and $-\dim(M)f-\sum_{i=1}^{k}\delta_{f_{i}}(\mathbb{X}_{i}\star f)$ agree at $t=0$ . Hence, we can define

	$\displaystyle\hat{\delta}(f):$	$\displaystyle=\frac{1}{t}\left(\delta_{\alpha}(f)+\dim(M)f+\sum_{i=1}^{k}\delta_{f_{i}}(\mathbb{X}_{i}\star f)\right)$		(47)
		$\displaystyle=\frac{1}{t}\left(\delta_{\alpha}(f)+\sum_{i=1}^{k}\mathbb{X}_{i}\star\delta_{f_{i}}(f)\right),$		(47)

where in the second equality we used (46).

Lemma 2.2.

The map $\hat{\delta}$ satisfies

\hat{\delta}(f\star g)=\hat{\delta}(f)\star g+f\star\hat{\delta}(g)+\sum_{i=1}^{k}\delta_{X_{i}}(f)\star\delta_{f_{i}}(g),\quad f,g\in C^{\infty}_{c}(\mathbb{T}M)

and hence a derivation of order $2$ on $C^{\infty}_{c}(\mathbb{T}M)$ .

Proof.

By (41) and (45)

\hat{\delta}(f\star g)-\hat{\delta}(f)\star g-f\star\hat{\delta}(g)=\frac{1}{t}\left(\sum_{i=1}^{k}\mathbb{X}_{i}\star f\star\delta_{f_{i}}(g)-f\star\mathbb{X}_{i}\star\delta_{f_{i}}(g)\right)=\sum_{i=1}^{k}\delta_{X_{i}}(f)\star\delta_{f_{i}}(g)\qed

The reader is invited to see the similarity between (47) and [HH18a, The formula for $T$ on Page 4].

Schwartz functions.

Definition 2.3.

We define $\mathcal{S}(\mathbb{T}M)$ to be the space of smooth function $f\in C^{\infty}(\mathbb{T}M)$ such that $f$ is bounded and all iterated applications of the following differential operators give bounded functions on $\mathbb{T}M$

•

the operator $f\mapsto tf$ , where $t:\mathbb{T}M\to\mathbb{R}_{+}$ is the natural projection.
•

the operator $f\mapsto\mathbb{D}\star f$ , where $D$ is any differential operator on $M$
•

the operator $f\mapsto\delta_{D}(f)$ , where $D$ is any differential operator on $M$
•

the operator $f\mapsto\hat{\delta}(f)$ .

We can add the application $f\mapsto\delta_{\alpha}(f)$ to the above list, but this is redundant as it follows from the others using (47).

Proposition 2.4.

Let $f\in\mathcal{S}(\mathbb{T}M)$ . Then for every $x\in M$ , the function $v\mapsto f(v,x,0)$ is a Schwartz function on the vector space $T_{x}M$ in the classical sense.

Proof.

If $X\in\mathcal{X}(M)$ is a vector field, then $v\mapsto(\mathbb{X}\star f)(v,x,0)$ is the application of the constant vector field $X(x)$ to $v\mapsto f(v,x,0)$ . While if $g\in C^{\infty}(M)$ , then the map $v\mapsto\delta_{g}(f)(v,x,0)$ is the pointwise product of the map $v\mapsto f(v,x,0)$ with the linear map $dg_{x}:T_{x}M\to\mathbb{C}$ . By iterated application of the previous two operations, it follows that $v\mapsto f(v,x,0)$ is Schwartz. ∎

Example 2.5.

Let $g_{1},\cdots,g_{l}:M\to\mathbb{R}$ be smooth functions such that the map

M\mapsto\mathbb{R}^{l},\quad x\mapsto(g_{1}(x),\cdots,g_{l}(x))

is an embedding. Then, the function

e^{-\sum_{i=1}^{l}\operatorname{DNC}(g_{i})^{2}-t^{2}}\in C^{\infty}(\mathbb{T}M)

is Schwartz, where $\operatorname{DNC}(g_{i})$ is defined in (37).

The following proposition summarizes the main properties of Schwartz functions. In its proof, we remark that by (42), in Definition 2.3 the second and third conditions can be replaced by $\mathbb{D}\star f$ and $\delta_{D}(f)$ are bounded where $D$ is either a vector field or a smooth function on $M$ .

Proposition 2.6.

The following holds

1.

The definition of Schwartz functions doesn’t depend on the choice of $f_{i},X_{i}$ in Lemma 2.1.
2.

If $f\in\mathcal{S}(\mathbb{T}M)$ , then $f\in C_{0}(\mathbb{T}M)$ , i.e. $f$ vanishes at infinity.
3.

If $f,g\in\mathcal{S}(\mathbb{T}M)$ , then the integral in $f\star g$ is absolutely convergent and $f\star g\in\mathcal{S}(\mathbb{T}M)$
4.

If $f\in\mathcal{S}(\mathbb{T}M)$ , then $f^{*}\in\mathcal{S}(\mathbb{T}M)$

Proof.

For the first part, we first need some lemmas.

Lemma 2.7.

If $f\in C^{\infty}_{c}(M\times M\times\mathbb{R}_{+})$ and $g\in\mathcal{S}(\mathbb{T}M)$ , then the pointwise product $(f\circ\pi)g\in\mathcal{S}(\mathbb{T}M)$ , where $\pi:\mathbb{T}M\to M\times M\times\mathbb{R}_{+}$ is the map in (36)

Proof.

$(f\circ\pi)g$ is bounded because $g$ and $f$ are bounded. We argue that each of the applications in Definition 2.3 gives functions of the form $(f^{\prime}\circ\pi)g^{\prime}$ for some $f^{\prime}\in C^{\infty}_{c}(M\times M\times\mathbb{R}_{+})$ and $g^{\prime}\in\mathcal{S}(\mathbb{T}M)$ . The result then follows by induction. We thus have

•

the function $t(f\circ\pi)g=(f\circ\pi)(tg)$
•

If $D$ is a function on $M$ , then $\mathbb{D}\star((f\circ\pi)g)=(f\circ\pi)(\mathbb{D}\star g)$ . If $D$ is a vector field, then we have

$\mathbb{D}\star((f\circ\pi)g)=(D_{y}(f)\circ\pi)(tg)+(f\circ\pi)(\mathbb{D}\star g),$ (48)

where $D_{y}(f)$ is the action of $D$ on the $y$ variable in $f(y,x,t)$ . The equality can be checked directly on $M\times M\times\mathbb{R}_{+}^{\times}$ and by density equality follows on $\mathbb{T}M$ .
•

If $D$ is a function, then $\delta_{D}((f\circ\pi)g)=(f\circ\pi)\delta_{D}(g)$ . If $D$ is a vector field, then one can as in (48) show that

$\delta_{D}((f\circ\pi)g)=(D_{y}(f)\circ\pi-D_{x}^{\prime}(f)\circ\pi)(g)+(f\circ\pi)\delta_{D}(g)$

•

one has

\delta_{\alpha}((f\circ\pi)g)=(f\circ\pi)\delta_{\alpha}(g)+(\frac{\partial}{\partial t}(f)\circ\pi)(tg)

and so

\hat{\delta}((f\circ\pi)g)=(f\circ\pi)\hat{\delta}(g)+(\frac{\partial}{\partial t}(f)\circ\pi)(g)+\sum_{i=1}^{k}(X_{iy}(f)\circ\pi)(\delta_{f_{i}}(g))

This finishes the proof. ∎

Lemma 2.8.

Let $f\in C^{\infty}(M\times M)$ be a smooth function that vanishes to the order $1$ on the diagonal, and let $\tilde{f}\in C^{\infty}(\mathbb{T}M)$ be the smooth function given by

(y,x,t)\mapsto\frac{f(y,x)}{t^{2}},\quad(v,x,0)\mapsto\frac{1}{2}d^{2}f_{x}(v),

where $d^{2}f$ is the Hessian of $f$ . Then if $g\in\mathcal{S}(\mathbb{T}M)$ then the pointwise product $\tilde{f}g\in\mathcal{S}(\mathbb{T}M)$ .

Proof.

Any function $f$ which vanishes to the order $1$ can be written as a finite sum of functions of the form $h(y,x)(h_{1}(y)-h_{1}(x))(h_{2}(y)-h_{2}(x))$ where $h\in C^{\infty}(M\times M),h_{1},h_{2}\in C^{\infty}(M)$ . It follows that the pointwise product $\tilde{f}g$ is sum of pointwise product of $h\circ\pi$ with $\delta_{h_{1}}(\delta_{h_{2}}(g))$ where $\pi:\mathbb{T}M\to M\times M$ is the natural map. The result then follows from Lemma 2.7. ∎

Lemma 2.9.

Let $g_{1},\cdots,g_{k^{\prime}}$ and $Y_{1},\cdots,Y_{k^{\prime}}$ be another family satisfying Lemma 2.1. Then there exists a family of smooth functions $h_{ij}:M\to\mathbb{R}$ for $1\leq i\leq k$ and $1\leq j\leq k^{\prime}$ such that

df_{i}=\sum_{j=1}^{k^{\prime}}h_{ij}dg_{j},\quad Y_{j}=\sum_{i=1}^{k}h_{ij}X_{i}.

(49)

Proof.

Let $x_{1},\cdots,x_{\dim(M)}$ be local coordinates in $M$ . Then $df_{i}=\sum_{l=1}^{\dim(M)}\phi_{il}dx_{l}$ and $X_{i}=\sum_{l=1}^{\dim(M)}\phi^{\prime}_{il}\frac{\partial}{\partial x_{l}}$ for some functions $\phi_{il}$ and $\phi_{il}^{\prime}$ . Similarly, $dg_{j}=\sum_{l=1}^{{\dim(M)}}\psi_{jl}dx_{l}$ and $Y_{j}=\sum_{l=1}^{{\dim(M)}}\psi^{\prime}_{jl}\frac{\partial}{\partial x_{l}}$ . Then one can take $h_{ij}=\sum_{l=1}^{{\dim(M)}}\phi_{il}\psi_{jl}^{\prime}$ . To define $h$ globally one takes a partition of unity. ∎

We can now prove Proposition 2.6.1. Let $\hat{\delta}^{\prime}$ be the operator associated to $g_{1},\cdots,g_{k^{\prime}}$ and $Y_{1},\cdots,Y_{k^{\prime}}$ . We have

	$\displaystyle\hat{\delta}(f)-\hat{\delta}^{\prime}(f)$	$\displaystyle=\frac{1}{t}\left(\sum_{i=1}^{k}\delta_{f_{i}}(\mathbb{X}_{i}\star f)-\sum_{j=1}^{k^{\prime}}\delta_{g_{j}}(\mathbb{Y}_{j}\star f)\right)$
		$\displaystyle=\sum_{i=1}^{k}\frac{1}{t^{2}}\left(\left(f_{i}(y)-f_{i}(x)-\sum_{j=1}^{k^{\prime}}(g_{j}(y)-g_{j}(x))h_{ij}(y)\right)(\mathbb{X}_{i}\star f)\right).$

By (49), the function $f_{i}(y)-f_{i}(x)-\sum_{j=1}^{k^{\prime}}(g_{j}(y)-g_{j}(x))h_{ij}(y)$ vanishes to order $1$ on the diagonal. By Lemma 2.8, $\hat{\delta}^{\prime}(f)\in\mathcal{S}(\mathbb{T}M)$ . Proposition 2.6.1 follows.

For Proposition 2.6.2, suppose that $f\notin C_{0}(\mathbb{T}M)$ . Then there exists $\epsilon>0$ a sequence in $\mathbb{T}M$ and which goes to infinity yet $|f|\geq\epsilon$ for every element in the sequence. By passing to a subsequence we can suppose that the sequence is either of the form $(y_{n},x_{n},t_{n})$ or of the form $(v_{n},x_{n},0)$ . Suppose we have the first case. Then by taking a subsequence, we can suppose that $x_{n}\to x$ and $y_{n}\to y$ and $t_{n}\to t\in[0,+\infty]$ . If $t=+\infty$ , we get a contradiction to the fact that $tf$ is bounded. If $t\in]0,+\infty[$ , then the sequence $(y_{n},x_{n},t_{n})$ converges in $\mathbb{T}M$ to $(y,x,t)$ , again a contradiction. If $x\neq y$ then we get a contradiction to the fact that $\delta_{g}(f)$ is bounded where $g\in C^{\infty}(M)$ is any smooth function with $g(x)=g(y)$ . So we have $t=0$ and $x=y$ . The sequence $(y_{n},x_{n},t_{n})$ converging to infinity implies that there exists $g\in C^{\infty}(M)$ such that $\left|\frac{g(y_{n})-g(x_{n})}{t_{n}}\right|\to+\infty$ . We get then a contradiction to the fact that $\delta_{g}(f)$ is bounded. The case of a sequence $(v_{n},x_{n},0)$ is similar.

For Proposition 2.6.3, first the integral in $f\star g$ is absolutely convergent by Proposition 2.4. Hence, $f\star g$ is a well-defined function on $\mathbb{T}M$ . We now show its continuity. Let $g_{i}$ be as in Example 2.5. Then consider the function

\phi=(1+\operatorname{DNC}(g_{1})^{2}+\cdots+\operatorname{DNC}(g_{l})^{2})^{\frac{\dim(M)}{2}+1}\in C^{\infty}(\mathbb{T}M).

(50)

The function $f$ being Schwartz implies that $\phi f$ is bounded. One can easily show by looking at local coordinates of $\mathbb{T}M$ that there exists $C>0$ (only depends on $\phi$ ) such that

\left\lVert f\star g\right\rVert_{\infty}\leq C\left\lVert\phi f\right\rVert_{\infty}\left\lVert g\right\rVert_{\infty}.

Since $\phi f$ and $g$ are Schwartz functions, $\phi f,g\in C_{0}(\mathbb{T}M)$ . We can thus approximate them uniformly with compactly supported functions. It follows that $f\star g\in C_{0}(\mathbb{T}M)$ . Smoothness of $f\star g$ as well as the fact that $f\star g\in\mathcal{S}(\mathbb{T}M)$ follow easily from (40), (41), (44) and Lemma 2.2.

For Proposition 2.6.4, since $f^{*}$ is bounded if $f$ is bounded, the result follows from the following identities

\displaystyle tf^{*}=(tf)^{*},\quad\mathbb{D}\star f^{*}=(-t\delta_{D^{*}}(f)+\mathbb{D}^{*}\star f)^{*},\quad\delta_{D}(f^{*})=-\delta_{D^{*}}(f)^{*},\quad\delta_{\alpha}(f^{*})=\delta_{\alpha}(f)^{*},

and the identity

	$\displaystyle\hat{\delta}(f^{})^{}-\hat{\delta}(f)$	$\displaystyle=\frac{1}{t}\left(\sum_{i=1}^{k}-\delta_{f_{i}}(f\star\mathbb{X}_{i}^{*})-\delta_{f_{i}}(\mathbb{X}_{i}\star f)\right)$
		$\displaystyle=\frac{1}{t}\left(\sum_{i=1}^{k}\delta_{f_{i}}(f\star\mathbb{X}_{i})-\delta_{f_{i}}(\mathbb{X}_{i}\star f)\right)+\sum_{i=1}^{k}\delta_{f_{i}}(f\star\mathrm{div}(X_{i}))$
		$\displaystyle=\sum_{i=1}^{k}-\delta_{f_{i}}(\delta_{X_{i}}(f))+\delta_{f_{i}}(f\star\mathrm{div}(X_{i}))\qed$

Remark 2.10.

There are different definitions in the literature of Schwartz functions. Our definition doesn’t precisely agree with [Car08a]. In [Car08a], the author adds a conical support condition which we don’t need. Our definition agrees with the one proposed by Debord and Skandalis [DS14a]. We refer the reader to [DS14a, Section 1.6] for a treatment of the Schwartz functions defined here using classical semi-norm estimates.

We give here a definition of Schwartz functions which is equivalent to ours by looking in local coordinates. A function $f\in\mathcal{S}(\mathbb{T}M)$ is Schwartz if and only if it satisfies the following:

1.

For every $k,l\in\mathbb{N}$ , $D$ a differential operator on $M\times M$ ,

$\sup\left\{\left|t^{k}\frac{d}{dt^{l}}Df(y,x,t)\right|:(y,x,t)\in M\times M\times[1,+\infty[\right\}<+\infty$
2.

For every $k,l\in\mathbb{N}$ , $D$ a differential operator on $M\times M$ , $K\subseteq M\times M\backslash M$ a compact subset outside the diagonal,

$\sup\left\{\left|t^{-k}\frac{d}{dt^{l}}Df(y,x,t)\right|:(y,x,t)\in K\times]0,1]\right\}<+\infty$

For every $U\subseteq M$ open subset diffeomorphic to $\mathbb{R}^{\dim(M)}$ by a map $\phi:U\to\mathbb{R}^{\dim(M)}$ , $\Phi$ the local chart of $\mathbb{T}M$ associated to $\phi$ defined in (35), $K\subseteq\mathbb{R}^{\dim(M)}$ a compact subset, $k,l\in\mathbb{N}$ , $\alpha,\beta\in\mathbb{N}^{\dim(M)}$ , one has

\sup\left\{\left|\left\lVert v\right\rVert^{k}\frac{d}{dt^{l}}\frac{d}{dv^{\alpha}}\frac{d}{dx^{\beta}}f\circ\Phi^{-1}(v,x,t)\right|:(v,x,t)\in\hat{K}\right\}<+\infty,

where $\hat{K}=\{(v,x,t)\in\mathbb{R}^{2\dim(M)+1}:x,x+tv\in K\}$ .

$C^{*}$ -algebra of the tangent groupoid.

Let $f\in C^{\infty}_{c}(\mathbb{T}M)$ , $t\neq 0$ . We define the operator

	$\displaystyle\pi_{t}(f):L^{2}M$	$\displaystyle\to L^{2}M$
	$\displaystyle g$	$\displaystyle\mapsto(y\mapsto t^{-\dim(M)}\int_{M}f(y,x,t)g(x)dx),\quad g\in L^{2}(M)$

For each $x\in M$ , we also define the operator

	$\displaystyle\pi_{x}(f):L^{2}T_{x}M$	$\displaystyle\to L^{2}T_{x}M$
	$\displaystyle g$	$\displaystyle\mapsto(v\mapsto\int_{T_{x}M}f(v-w,x,0)g(w)dw),\quad g\in L^{2}(T_{x}M)$

We then define

\left\lVert f\right\rVert:=\max{\left(\sup_{t\in\mathbb{R}_{+}^{\times}}\left\lVert\pi_{t}(f)\right\rVert,\sup_{x\in M}\left\lVert\pi_{x}(f)\right\rVert\right)}.

We define the $C^{*}$ -algebra $C^{*}\mathbb{T}M$ to be the completion of $C^{\infty}_{c}(\mathbb{T}M)$ with respect to $\left\lVert\cdot\right\rVert$ . The $C^{*}$ -algebra $C^{*}\mathbb{T}M$ lies in a short exact sequence

0\to\mathcal{K}(L^{2}M)\otimes C_{0}(\mathbb{R}_{+}^{\times})\to C^{*}\mathbb{T}M\to C_{0}(T^{*}M)\to 0,

where $\mathcal{K}(L^{2}M)$ is the $C^{*}$ -algebra of compact operators on $L^{2}M$ , see [Con94a, Proposition 5 Page 108]

Proposition 2.11.

The space $\mathcal{S}(\mathbb{T}M)$ is a $*$ -subalgebra of $C^{*}\mathbb{T}M$ .

Proof.

It is well known that

\left\lVert\pi_{t}(f)\right\rVert\leq\sup_{x\in M}\max\left(t^{-\dim(M)}\int_{M}|f(x,y,t)|dy,t^{-\dim(M)}\int_{M}|f(y,x,t)|dy\right),\ t\in\mathbb{R}_{+}^{\times},f\in C^{\infty}_{c}(\mathbb{T}M)

and

\left\lVert\pi_{x}(f)\right\rVert\leq\sup_{x\in M}\left(\int_{T_{x}M}|f(v,x,0)|dv\right),\quad x\in M,f\in C^{\infty}_{c}(\mathbb{T}M).

Hence,

\left\lVert f\right\rVert\leq\sup_{x\in M,t\in\mathbb{R}_{+}^{\times}}\max\left(t^{-\dim(M)}\int_{M}|f(x,y,t)|dy,t^{-\dim(M)}\int|f(y,x,t)|dy,\int_{T_{x}M}|f(v,x,0)|dv\right)

(51)

It follows that $C^{*}\mathbb{T}M$ contains measurable functions on $\mathbb{T}M$ for which the right-hand side of (51) is finite (usually denoted $L^{1}(\mathbb{T}M)$ ). Let $\phi$ be as in (50). If $f\in\mathcal{S}(\mathbb{T}M)$ , then the right-hand side of (51) is bounded by $\left\lVert\phi f\right\rVert_{\infty}$ . Hence, $f\in C^{*}\mathbb{T}M$ . Finally, $\mathcal{S}(\mathbb{T}M)$ is a $*$ -subalgebra by Proposition 2.6. ∎

Relation between uniform norm and $C^{*}$ -norm.

The following theorem which is essentially just the Sobolev embedding theorem will be very useful in allowing us to replace the uniform norm with the $C^{*}$ -norm, which is more convenient to use.

Theorem 2.12.

Let $\Delta$ be the positive Laplace-Beltrami operator on $M$ , $\Delta\!\!\!\!\Delta$ the corresponding differential operator on $\mathbb{T}M$ as in (38), and $k\in\mathbb{N}$ with $2k>\frac{\dim(M)}{2}$ . There exists $C>0$ , such that

\left\lVert f\right\rVert_{\infty}\leq C\left\lVert(1+t^{\dim(M)})(1+\Delta\!\!\!\!\Delta)^{k}\star f\star(1+\Delta\!\!\!\!\Delta)^{k}\right\rVert,\quad\forall f\in C^{\infty}_{c}(\mathbb{T}M)

Lemma 2.13.

Let $k\in\mathbb{N}$ with $2k>\frac{\dim(M)}{2}$ . There exists a constant $C>0$ such that for all $t>0$ and $x\in M$ , if $u_{x}$ denotes the Dirac delta distribution on $M$ at $x$ , then

\left\lVert(1+t^{2}\Delta)^{-k}(u_{x})\right\rVert_{L^{2}M}\leq C\max(t^{-\frac{\dim(M)}{2}},1).

Proof.

The Sobolev embedding lemma implies that $u_{x}\in H^{-2k}(M)$ where $H^{s}(M)$ denotes the $s$ Sobolev space. Hence, $(1+t^{2}\Delta)^{-k}(u_{x})\in L^{2}M$ . For $t>1$ we have and use the inequality

	$\displaystyle\left\lVert(1+t^{2}\Delta)^{-k}(u_{x})\right\rVert_{L^{2}M}$	$\displaystyle=\left\lVert(1+t^{2}\Delta)^{-k}(1+\Delta)^{k}(1+\Delta)^{-k}(u_{x})\right\rVert_{L^{2}M}$
		$\displaystyle\leq\left\lVert(1+t^{2}\Delta)^{-k}(1+\Delta)^{k}\right\rVert_{B(L^{2}M)}\left\lVert(1+\Delta)^{-k}(u_{x})\right\rVert_{L^{2}M}$
		$\displaystyle\leq\left\lVert(1+\Delta)^{-k}(u_{x})\right\rVert_{L^{2}M},$

where in the last inequality we used the fact that $\frac{(1+x)^{k}}{(1+t^{2}x)^{k}}\leq 1$ for all $x\in[0,+\infty[$ . For $t<1$ , we proceed differently. If $g\in L^{2}M$ , then

\langle(1+t^{2}\Delta)^{-k}(u_{x}),g\rangle_{L^{2}M}=|(1+t^{2}\Delta)^{-k}(g)(x)|.

We need to maximize $\frac{|(1+t^{2}\Delta)^{-k}(g)(x)|}{\left\lVert g\right\rVert_{L^{2}M}}$ as $g$ varies in $L^{2}M$ . The function $g$ needs to vanish outside any given neighbourhood of $x$ to reach the supremum. We can thus assume that $M=\mathbb{R}^{\dim(M)}$ and $x=0$ and the metric on $M$ is Euclidean outside the unit ball. The inequality follows from a change of variable $x\to\frac{x}{t}$ . ∎

Proof of Theorem 2.12.

Let $t>0$ . We apply

\pi_{t}((1+\Delta\!\!\!\!\Delta)^{k}\star f\star(1+\Delta\!\!\!\!\Delta)^{k})=t^{-\dim(M)}(1+t^{2}\Delta)^{k}\star f\star(1+t^{2}\Delta)^{k}

to $(1+t^{2}\Delta)^{-k}(u_{x})$ to deduce that

\displaystyle t^{-\dim(M)}\left\lVert(1+t^{2}\Delta)^{k}(f(\cdot,x,t))\right\rVert_{L^{2}M}

\displaystyle\leq C\max(t^{-\frac{\dim(M)}{2}},1)\left\lVert(1+\Delta\!\!\!\!\Delta)^{k}\star f\star(1+\Delta\!\!\!\!\Delta)^{k}\right\rVert

One has

	$\displaystyle\|f(y,x,t)\|$	$\displaystyle=\|\left\langle u_{y},f(\cdot,x,t)\right\rangle_{H^{-2k}(M)\times H^{2k}(M)}\|$
		$\displaystyle=\|\langle(1+t^{2}\Delta)^{-k}(u_{y}),(1+t^{2}\Delta)^{k}(f(\cdot,x,t))\rangle_{L^{2}(M)\times L^{2}(M)}\|$
		$\displaystyle\leq\left\lVert(1+t^{2}\Delta)^{-k}(u_{y})\right\rVert_{L^{2}M}\left\lVert(1+t^{2}\Delta)^{k}(f(\cdot,x,t))\right\rVert_{L^{2}M}$
		$\displaystyle\leq C\max(t^{-\frac{\dim(M)}{2}},1)\left\lVert(1+t^{2}\Delta)^{k}(f(\cdot,x,t))\right\rVert_{L^{2}M}$

It follows that

	$\displaystyle\|f(y,x,t)\|$	$\displaystyle\leq C^{2}t^{\dim(M)}\max(t^{-\dim(M)},1)\left\lVert(1+\Delta\!\!\!\!\Delta)^{k}\star f\star(1+\Delta\!\!\!\!\Delta)^{k}\right\rVert$
		$\displaystyle=C^{2}\max(1,t^{\dim(M)})\left\lVert(1+\Delta\!\!\!\!\Delta)^{k}\star f\star(1+\Delta\!\!\!\!\Delta)^{k}\right\rVert.$

By taking the limit as $t\to 0^{+}$ , one obtains a bound of $f$ on $TM\times\{0\}$ . The theorem follows. ∎

Schwartz functions are closed under smooth calculus.

Proof of Theorem A.

Let $\mathcal{D}\subseteq\operatorname{End}(\mathcal{S}(\mathbb{T}M))$ be the subalgebra of $\operatorname{End}(\mathcal{S}(\mathbb{T}M))$ generated by the maps in Definition 2.3. We will prove that $\mathcal{D}(\mathcal{S}(\mathbb{T}M))=\mathcal{S}(\mathbb{T}M)$ . Theorem A would then follow from Theorem 1.10. Let $a\in\mathcal{D}(\mathcal{S}(\mathbb{T}M))$ , $x_{n}\in\mathcal{S}(\mathbb{T}M)$ as in Definition 1.8. Since

x\mapsto(1+t^{\dim(M)})(1+\Delta\!\!\!\!\Delta)^{k}\star x\star(1+\Delta\!\!\!\!\Delta)^{k}

is a differential operator on $\mathcal{S}(\mathbb{T}M)$ which belongs to $\mathcal{D}$ , it follows that $(1+t^{\dim(M)})(1+\Delta\!\!\!\!\Delta)^{k}\star x_{n}\star(1+\Delta\!\!\!\!\Delta)^{k}$ converges in $C^{*}\mathbb{T}M$ . By Theorem 2.12, it follows that $a$ is a bounded continuous function and $x_{n}\to a$ in uniform norm. Let $\delta\in\mathcal{D}$ . Since the map $x\mapsto(1+t^{\dim(M)})(1+\Delta\!\!\!\!\Delta)^{k}\star\delta(x)\star(1+\Delta\!\!\!\!\Delta)^{k}$ belongs to $\mathcal{D}$ , again by Theorem 2.12, $\delta(x_{n})$ converges in the uniform norm to a bounded continuous function on $\mathbb{T}M$ . It follows that $a$ is smooth and $\delta(a)$ bounded for every $\delta$ . Hence, $a\in\mathcal{S}(\mathbb{T}M)$ . ∎

Remark 2.14.

In [Ewe21a], Ewert defined an algebra of Schwartz functions for the inhomogeneous tangent groupoid defined in [EY17a, Moh21a, HH18a, CP19a]. Does Theorem A hold for this Schwartz algebra, or a similarly defined one?

{refcontext}

[sorting=nyt]

References

[AS68] M.. Atiyah and I.. Singer “The index of elliptic operators. I” In Ann. of Math. (2) 87, 1968, pp. 484–530 DOI: 10.2307/1970715
[Bla98] B. Blackadar “ $K$ -theory for operator algebras” 5, Mathematical Sciences Research Institute Publications Cambridge University Press, Cambridge, 1998, pp. xx+300
[BR75] O. Bratteli and D.. Robinson “Unbounded derivations of $C^{\ast}$ -algebras” In Comm. Math. Phys. 42, 1975, pp. 253–268 URL: http://projecteuclid.org.ezproxy.universite-paris-saclay.fr/euclid.cmp/1103899048
[BR76] O. Bratteli and D.. Robinson “Unbounded derivations of $C^{\ast}$ -algebras. II” In Comm. Math. Phys. 46.1, 1976, pp. 11–30 URL: http://projecteuclid.org.ezproxy.universite-paris-saclay.fr/euclid.cmp/1103899543
[Car08] P. Carrillo Rouse “A Schwartz type algebra for the tangent groupoid” In $K$ -theory and noncommutative geometry, EMS Ser. Congr. Rep. Eur. Math. Soc., Zürich, 2008, pp. 181–199 DOI: 10.4171/060-1/7
[CP19] W. Choi and R. Ponge “Tangent maps and tangent groupoid for Carnot manifolds” In Differential Geom. Appl. 62, 2019, pp. 136–183 DOI: 10.1016/j.difgeo.2018.11.002
[Con86] A. Connes “Cyclic cohomology and the transverse fundamental class of a foliation” In Geometric methods in operator algebras (Kyoto, 1983) 123, Pitman Res. Notes Math. Ser. Longman Sci. Tech., Harlow, 1986, pp. 52–144
[Con94] A. Connes “Noncommutative geometry” Academic Press, Inc., San Diego, CA, 1994, pp. xiv+661
[DL10] C. Debord and J.-M. Lescure “Index theory and groupoids” In Geometric and topological methods for quantum field theory Cambridge Univ. Press, Cambridge, 2010, pp. 86–158 DOI: 10.1017/CBO9780511712135.004
[DS14] C. Debord and G. Skandalis “Adiabatic groupoid, crossed product by $\mathbb{R}_{+}^{\ast}$ and pseudodifferential calculus” In Adv. Math. 257, 2014, pp. 66–91 DOI: 10.1016/j.aim.2014.02.012
[EY17] E. Erp and R. Yuncken “On the tangent groupoid of a filtered manifold” In Bull. Lond. Math. Soc. 49.6, 2017, pp. 1000–1012 DOI: 10.1112/blms.12096
[Ewe21] E. Ewert “Pseudodifferential operators on filtered manifolds as generalized fixed points” In To appear in Journal of noncommutative Geometry arXiv, 2021 DOI: 10.48550/ARXIV.2110.03548
[Ful84] W. Fulton “Intersection theory” 2, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] Springer-Verlag, Berlin, 1984, pp. xi+470 DOI: 10.1007/978-3-662-02421-8
[HH18] A. Haj Saeedi Sadegh and N. Higson “Euler-like vector fields, deformation spaces and manifolds with filtered structure” In Doc. Math. 23, 2018, pp. 293–325
[HJ84] A. Hulanicki and J.. Jenkins “Nilpotent Lie groups and summability of eigenfunction expansions of Schrödinger operators” In Studia Math. 80.3, 1984, pp. 235–244 DOI: 10.4064/sm-80-3-235-244
[Moh21] O. Mohsen “On the deformation groupoid of the inhomogeneous pseudo-differential calculus” In Bull. Lond. Math. Soc. 53.2, 2021, pp. 575–592 DOI: 10.1112/blms.12443
[Ped00] G.. Pedersen “Operator differentiable functions” In Publ. Res. Inst. Math. Sci. 36.1, 2000, pp. 139–157 DOI: 10.2977/prims/1195143229

References

[AS68a] M.. Atiyah and I.. Singer “The index of elliptic operators. I” In Ann. of Math. (2) 87, 1968, pp. 484–530 DOI: 10.2307/1970715
[BR75a] O. Bratteli and D.. Robinson “Unbounded derivations of $C^{\ast}$ -algebras” In Comm. Math. Phys. 42, 1975, pp. 253–268 URL: http://projecteuclid.org.ezproxy.universite-paris-saclay.fr/euclid.cmp/1103899048
[BR76a] O. Bratteli and D.. Robinson “Unbounded derivations of $C^{\ast}$ -algebras. II” In Comm. Math. Phys. 46.1, 1976, pp. 11–30 URL: http://projecteuclid.org.ezproxy.universite-paris-saclay.fr/euclid.cmp/1103899543
[Ful84a] W. Fulton “Intersection theory” 2, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] Springer-Verlag, Berlin, 1984, pp. xi+470 DOI: 10.1007/978-3-662-02421-8
[HJ84a] A. Hulanicki and J.. Jenkins “Nilpotent Lie groups and summability of eigenfunction expansions of Schrödinger operators” In Studia Math. 80.3, 1984, pp. 235–244 DOI: 10.4064/sm-80-3-235-244
[Con86a] A. Connes “Cyclic cohomology and the transverse fundamental class of a foliation” In Geometric methods in operator algebras (Kyoto, 1983) 123, Pitman Res. Notes Math. Ser. Longman Sci. Tech., Harlow, 1986, pp. 52–144
[Con94a] A. Connes “Noncommutative geometry” Academic Press, Inc., San Diego, CA, 1994, pp. xiv+661
[Bla98a] B. Blackadar “ $K$ -theory for operator algebras” 5, Mathematical Sciences Research Institute Publications Cambridge University Press, Cambridge, 1998, pp. xx+300
[Ped00a] G.. Pedersen “Operator differentiable functions” In Publ. Res. Inst. Math. Sci. 36.1, 2000, pp. 139–157 DOI: 10.2977/prims/1195143229
[Car08a] P. Carrillo Rouse “A Schwartz type algebra for the tangent groupoid” In $K$ -theory and noncommutative geometry, EMS Ser. Congr. Rep. Eur. Math. Soc., Zürich, 2008, pp. 181–199 DOI: 10.4171/060-1/7
[DL10a] C. Debord and J.-M. Lescure “Index theory and groupoids” In Geometric and topological methods for quantum field theory Cambridge Univ. Press, Cambridge, 2010, pp. 86–158 DOI: 10.1017/CBO9780511712135.004
[DS14a] C. Debord and G. Skandalis “Adiabatic groupoid, crossed product by $\mathbb{R}_{+}^{\ast}$ and pseudodifferential calculus” In Adv. Math. 257, 2014, pp. 66–91 DOI: 10.1016/j.aim.2014.02.012
[EY17a] E. Erp and R. Yuncken “On the tangent groupoid of a filtered manifold” In Bull. Lond. Math. Soc. 49.6, 2017, pp. 1000–1012 DOI: 10.1112/blms.12096
[HH18a] A. Haj Saeedi Sadegh and N. Higson “Euler-like vector fields, deformation spaces and manifolds with filtered structure” In Doc. Math. 23, 2018, pp. 293–325
[CP19a] W. Choi and R. Ponge “Tangent maps and tangent groupoid for Carnot manifolds” In Differential Geom. Appl. 62, 2019, pp. 136–183 DOI: 10.1016/j.difgeo.2018.11.002
[Ewe21a] E. Ewert “Pseudodifferential operators on filtered manifolds as generalized fixed points” In To appear in Journal of noncommutative Geometry arXiv, 2021 DOI: 10.48550/ARXIV.2110.03548
[Moh21a] O. Mohsen “On the deformation groupoid of the inhomogeneous pseudo-differential calculus” In Bull. Lond. Math. Soc. 53.2, 2021, pp. 575–592 DOI: 10.1112/blms.12443

(Omar Mohsen) Paris-Saclay University, Paris, France e-mail: [email protected]

	$\displaystyle\|f(y,x,t)\|$	$\displaystyle=\|\left\langle u_{y},f(\cdot,x,t)\right\rangle_{H^{-2k}(M)\times H^{2k}(M)}\|$
		$\displaystyle=\|\langle(1+t^{2}\Delta)^{-k}(u_{y}),(1+t^{2}\Delta)^{k}(f(\cdot,x,t))\rangle_{L^{2}(M)\times L^{2}(M)}\|$
		$\displaystyle\leq\left\lVert(1+t^{2}\Delta)^{-k}(u_{y})\right\rVert_{L^{2}M}\left\lVert(1+t^{2}\Delta)^{k}(f(\cdot,x,t))\right\rVert_{L^{2}M}$
		$\displaystyle\leq C\max(t^{-\frac{\dim(M)}{2}},1)\left\lVert(1+t^{2}\Delta)^{k}(f(\cdot,x,t))\right\rVert_{L^{2}M}$

Differential operators on C∗C^{*}-algebras and applications to smooth functional calculus and Schwartz functions on the tangent groupoid

Abstract

Introduction

Theorem A.

Theorem B.

Acknowledgments

Organization of the article.

1 Differential operators on C∗C^{*}-algebras

Example 1.1.

Definition 1.2.

Remarks 1.3.

Remark 1.4.

Proposition 1.5.

Proof.

Definition 1.6.

Proposition 1.7.

Proof.

Definition 1.8.

Remark 1.9.

Theorem 1.10.

Lemma 1.11.

Proof.

Lemma 1.12.

Proof.

Lemma 1.13.

Proof of Theorem 1.10.1.

Lemma 1.14.

Proof.

Lemma 1.15.

Proof.

Proof of Lemma 1.13.

Lemma 1.16.

Lemma 1.17.

Proof.

Proof of Theorem 1.10.2.

Corollary 1.18.

Corollary 1.19.

2 Schwartz functions on Connes’s tangent groupoid

Differential operators.

Convolution and adjoint.

Adjoint of Differential operators on 𝕋​M\mathbb{T}M.

Derivations in direction of MM.

Derivation in direction of tt.

Lemma 2.1.

Proof.

Lemma 2.2.

Proof.

Schwartz functions.

Definition 2.3.

Proposition 2.4.

Proof.

Example 2.5.

Proposition 2.6.

Proof.

Lemma 2.7.

Proof.

Lemma 2.8.

Proof.

Lemma 2.9.

Proof.

Remark 2.10.

C∗C^{*}-algebra of the tangent groupoid.

Proposition 2.11.

Proof.

Relation between uniform norm and C∗C^{*}-norm.

Theorem 2.12.

Lemma 2.13.

Proof.

Proof of Theorem 2.12.

Schwartz functions are closed under smooth calculus.

Proof of Theorem A.

Remark 2.14.

References

References

Differential operators on $C^{*}$ -algebras and applications to smooth functional calculus and Schwartz functions on the tangent groupoid

1 Differential operators on $C^{*}$ -algebras

Adjoint of Differential operators on $\mathbb{T}M$ .

Derivations in direction of $M$ .

Derivation in direction of $t$ .

$C^{*}$ -algebra of the tangent groupoid.

Relation between uniform norm and $C^{*}$ -norm.