\section{Formal quantizations of  Whitney functions}
Assume to be given a smooth manifold $M$, and let $X\subset M$ be a  closed subset.
Denote by $\mathcal{J}^\infty (X,M) \subset \mathcal{C}^\infty (M)$ the ideal of smooth functions on $M$ which are
flat  on $X$, i.e.~the space of all $f \in \mathcal{C}^\infty (M)$ such that for every differential
operator $D$ on $M$ the restricted function $Df_{|X}$ vanishes.
By Whitney's Extension Theorem, the quotient
$\mathcal{E}^\infty (X) := \mathcal{C}^\infty (M) / \mathcal{J}^\infty (X,M)$ naturally coincides
with the algebra of Whitney functions on $X$. This implies in particular that
  $\mathcal{E}^\infty (X) \subset \mathsf{J}^\infty (X)$, where $\mathsf{J}^\infty (X)$ denotes the space of
infinite jets over $X$.
Now consider the complex $\Omega (M)$ of differential
forms on $M$. Then the spaces $ \Omega^k_{\mathcal{J}^\infty}  (X,M) :=   \mathcal{J}^\infty (X,M) \cdot \Omega^k (M)$
are modules over $\mathcal{C}^\infty (M)$ preserved by the exterior derivative $d$, which means that
$d \big( \Omega^k_{\mathcal{J}^\infty}  (X,M) \big) \subset \Omega^{k+1}_{\mathcal{J}^\infty}  (X,M)$.
One thus obtains a subcomplex $\Omega^\bullet_{\mathcal{J}^\infty}  (X,M) \subset \Omega^\bullet (M)$
which we call the complex of differential forms on $M$ which are flat on $X$.
The quotient complex
$ \Omega^\bullet_{\mathcal{E}^\infty} (X) := \Omega^\bullet (M)/ \Omega^\bullet_{\mathcal{J}^\infty}  (X,M)$ will be called
the complex of Whitney-de Rham forms on $X$. According to \cite{BraPfl}, the cohomology of
$\Omega^\bullet_{\mathcal{E}^\infty} (X)$ coincides with the singular cohomology (with values in $\mathbb{R}$), if $M$
is an analytic manifold, and $X\subset M$ a subanalytic subset.

Let us now define what we understand by a deformation quantization of Whitney functions.
\begin{definition}
  Assume to be given a manifold $M$, a closed subset $X \subset M$ and a \emph{Whitney--Poisson} structure
  on $X$, i.e.~a bilinear map $\{ - , - \}$ on $\mathcal{E}^\infty (X)$ which satisfies  for all $F,G,H \in \mathcal{E}^\infty (X)$
  the relations
 \begin{enumerate}[(P1)]
  \item $\{F, GH\}=\{F,G\}H+G\{F,H\}$, and
  \item $\{\{F,G\}, H\}+\{\{H,F\},G\}+\{\{G,H\},F\}=0$.
\end{enumerate}

  By a \emph{formal deformation quantization} of the algebra $\mathcal{E}^\infty (X)$  or in other words
  a \emph{star product} on $\mathcal{E}^\infty (X)$ we understand an associative product
  \[
   \star : \mathcal{E}^\infty (X)[[\hbar]] \times \mathcal{E}^\infty (X)[[\hbar]] \rightarrow \mathcal{E}^\infty (X)[[\hbar]]
  \]
  on the space $\mathcal{E}^\infty (X)[[\hbar]]$ of formal power series in the variable $\hbar$ with coefficients
  in $\mathcal{E}^\infty (X)$ such that the following is satisfied:
  \begin{enumerate}[(DQ1)]
  \setcounter{enumi}{-1}
  \item
      The product $\star $ is $\mathbb{R}[[\hbar]]$-linear and $\hbar$-adically continuous in each argument.
  \item
     There exist $\mathbb{R}$-bilinear operators $c_k : \mathcal{E}^\infty (X) \times \mathcal{E}^\infty (X) \rightarrow \mathcal{E}^\infty (X)$, $k\in \mathbb{N}$
     such that $c_0 $ is the standard commutative product on $\mathcal{E}^\infty (X)$ and such that for all $F,G \in \mathcal{E}^\infty (X)$
     there is an expansion of the product $ F \star G $ of the form
     \begin{equation}
     \label{eq:starprodexp}
       F \star G = \sum_{k\in \mathbb{N}} c_k (F,G) \hbar^k .
     \end{equation}
  \item
     The constant function $1 \in \mathcal{E}^\infty$ satisfies $1 \star F = F \star 1 = F$  for all $F \in \mathcal{E}^\infty (X)$.
  \item
     The star commutator $[ F, G]_\star := F\star G - G \star F$ of two Whitney functions $F,G \in \mathcal{E}^\infty (X)$
     satisfies the  commutation relation
     \[
        [ F, G]_\star = - i \hbar \{ F, G\} + o(\hbar^2) .
     \]
  \end{enumerate}
    If in addition $\star$ is local in the sense that
   \begin{enumerate}[(DQ1)]
   \setcounter{enumi}{3}
  \item
    $\operatorname{supp} (F \star G) \subset \operatorname{supp} (F) \cap \operatorname{supp} (G)$ for all $F,G \in \mathcal{E}^\infty (X)$,
  \end{enumerate}
    then the star product is called \emph{local}.
\end{definition}
\begin{remark}
  If $(M,\Pi)$ is a Poisson manifold, the ideal $\mathcal{J}^\infty (X;M)$ is a even Poisson
  ideal in $\mathcal{C}^\infty (M)$.
  This implies that the Poisson bracket on $\mathcal{C}^\infty (M)$ factors to the
  quotient $\mathcal{E}^\infty (X)$.
  We denote the inherited Poisson bracket on $\mathcal{E}^\infty (X)$ also by $\{ - , - \}$, and call it
  \emph{global Whitney--Poisson structure}.
\end{remark}

Assume now to be given a Poisson manifold $(M,\Pi)$, a  closed subset $X\subset M$, and let $\star$ be a local
star product on $\mathcal{C}^\infty (M)$. By Peetre's Theorem one then knows that each of the operators
$c_k: \mathcal{C}^\infty (M) \times  \mathcal{C}^\infty (M) \rightarrow \mathcal{C}^\infty (M)$ in the expansion
Eq.~\eqref{eq:starprodexp} of the star product on  $\mathcal{C}^\infty (M)$ is locally bidifferential. But this implies that
for every $k\in \mathbb{N}$ the sets
$c_k \big( \mathcal{J}^\infty (X,M ) \times \mathcal{C}^\infty (M) \big) $ and $c_k \big( \mathcal{C}^\infty (M) \times \mathcal{J}^\infty (X,M ) \big) $
are contained in $\mathcal{J}^\infty (X,M )$. This immediately entails the following result.

\begin{proposition}
  Let $(M,\Pi)$ be a Poisson manifold and $\star$ a local star product on $\mathcal{C}^\infty (M)$. Then
  for each closed subset $X \subset M$ the subspace  $\mathcal{J}^\infty (X,M )[[\hbar]]$ is an ideal in
  $\big( \mathcal{C}^\infty (M) , \star \big)$ which gives rise to an exact sequence of deformed algebras
  \[
    0 \rightarrow \big( \mathcal{J}^\infty (X,M )[[\hbar]] , \star \big)
      \rightarrow \big( \mathcal{C}^\infty (M) , \star \big)  \rightarrow \big( \mathcal{E}^\infty (X) , \star \big)
      \rightarrow 0 ,
  \]
  where the induced star product on $\mathcal{E}^\infty (X)$ is denoted by $\star$ as well.
\end{proposition}

\begin{remark}
  One knows by the work of {\sc Fedosov} \cite{FedDQIT} that every symplectic manifold carries a local star product, and
  by {\sc Kontsevich} \cite{KonDQPM} that on every Poisson manifold there exists  a local star product.
  The proceeding proposition then entails that for every closed subset $X$ of a Poisson manifold  $(M,\Pi)$
  there exists a deformation quantization of $\mathcal{E}^\infty (X)$ with the induced global Whitney--Poisson structure.
\end{remark}

Let us briefly recall Fedosov's approach \cite{FedDQIT} for the construction of  a deformation quantization over a symplectic
manifold $(M,\omega)$ and use this to describe the induced star product on $\mathcal{E}^\infty (X)$ with $X\subset M$ closed
in more detail.
To this end, observe first that each of the tangent spaces $T_pM$ is a linear symplectic space,
hence  gives rise to the formal Weyl algebra $\mathbb W  (T_p M ) $. As a vector space, $\mathbb W  (T_p M ) $
coincides with $\operatorname{\widehat{Sym}} (T^*_pM)[[\hbar]]$, the space of formal power series in $\hbar$ with coefficients
in the space of  Taylor expansions at the origin of smooth functions on $T_pM$. Note that  $\operatorname{\widehat{Sym}} (T^*_pM)$
coincides with the $\mathfrak{m}$-adic completion of the space $\operatorname{Sym} (T^*_pM)$ of polynomial functions on $T_pM$,
where $\mathfrak{m} $ denotes the maximal ideal in $\operatorname{Sym} (T^*_pM)$. In other words this means that $\operatorname{\widehat{Sym}} (T^*_pM)$
can be identified with the product $\prod_{s\in \mathbb{N}} \operatorname{Sym}^s (T^*_pM)$, where $\operatorname{Sym}^s (T^*_pM)$ denotes the space
of $s$-homogenous polynomial functions on $T_pM$. Hence every element $a$ of $\mathbb W  (T_p M ) $ can be uniquely
expressed in the form
\begin{equation}
\label{eq:fWeylrep}
  a = \sum_{s\in \mathbb{N} , \, k\in \mathbb{N}} a_{s,k} \hbar^k ,
\end{equation}
where the $a_{s,k} \in \operatorname{Sym}^s (T^*_pM)$ are uniquely defined by $a$. For later purposes note that
$\mathbb W  (T_p M ) $ is filtered by the \emph{Fedosov-degree}
\[
  \deg_\textup{F} (a) := \min\{s +2k \mid a_{sk}\neq 0 \}, \quad
  a \in \mathbb W  (T_p M ).
\]
Next observe that the Poisson bivector $\Pi$ on $T_pM$ is linear and can be
written in the form
\begin{equation}
\label{eq:Poissonrep}
	  \Pi = \sum_{i=1}^{\frac{\dim T_pM}{2}} \Pi_{i1} \otimes \Pi_{i2} \quad \text{with $\Pi_{i1},\Pi_{i2} \in T_pM$,
            $i=1,\cdots , \frac{\dim T_pM}{2}$}.
\end{equation}
Since the elements of $T_pM$ act as derivations on $\operatorname{Sym} (T_pM)$ one obtains an operator
\begin{equation}
\begin{split}
  \widehat{\Pi} : \: & \operatorname{Sym} (T_pM) \otimes \operatorname{Sym} (T_pM) \rightarrow \operatorname{Sym} (T_pM) \otimes \operatorname{Sym} (T_pM), \\
  & a \otimes b \mapsto \sum_{i=1}^{\frac{\dim T_pM}{2}} \Pi_{i1} a \otimes \Pi_{i2} b ,
\end{split}
\end{equation}
which does not depend on the particular representation \eqref{eq:Poissonrep}. Note that by $\mathbb{C}[[\hbar]]$-linearity
and $\mathfrak{m}$-adic continuity, $\widehat{\Pi}$ uniquely extends to an operator
\[
 \widehat{\Pi} :  \operatorname{\widehat{Sym}} (T_pM)[[\hbar]] \otimes \operatorname{\widehat{Sym}} (T_pM)[[\hbar]]
  \rightarrow  \operatorname{\widehat{Sym}} (T_pM)[[\hbar]] \otimes \operatorname{\widehat{Sym}} (T_pM)[[\hbar]] .
\]
The product of two elements $a, b \in \mathbb W  (T_p M ) $ can now be written
down. It is the so-called \emph{Moyal--Weyl} product of $a$ and $b$ and is
given  by
\begin{equation}
  a \circ b := \sum \frac{ (- i \hbar)^k}{k!} \mu \big( \widehat{\Pi}
  ( a \otimes b ) \big) .
\end{equation}
One checks easily that $\circ$ is a star product on $\mathbb W  (T_p M )$.

Denote by $\mathbb W (M) $ the bundle of formal Weyl algebras over $M$, which is
the  (profinite dimensional) vector bundle over $M$ having fibers
$ \mathbb W  (T_p M ) $, $p\in M$.
Furthermore, let $\Omega^\bullet \mathbb W  $
be the sheaf of smooth differential forms with  values in the bundle $\mathbb W (M)$.
Note that both the space ${\mathcal{W}} (M)$ of smooth sections of  $\mathbb W (M) $
and the space $\Omega^\bullet \mathbb W (M)$ are filtered by the Fedosov-degree. More precisely,
the Fedosov filtration $\big( \mathcal{F}^k {\mathcal{W}} (M) \big)_{k\in \mathbb{N}}$ of ${\mathcal{W}} (M)$ is given by
\[
   \mathcal{F}^k {\mathcal{W}} (M) :=
   \{ a \in {\mathcal{W}} (M) \mid \deg_\textup{F} (a(p)) \geq k \text{ for all $p \in M$}  \} ,
\]
and similarly for $\Omega^\bullet \mathbb W (M)$.
Note also that an element $a \in {\mathcal{W}} (M)$ can be uniquely written
in the form \eqref{eq:fWeylrep}, where the $a_{s,k}$ with $s,k\in\mathbb{N}$ then are
smooth sections of the symmetric powers $\operatorname{Sym}^s (T^*M)$. This representation
allows us to define the \emph{symbol map}
$\sigma: {\mathcal{W}} \rightarrow \mathcal{C}^\infty (M)[[\hbar]]$ by
\[
  \sigma (a ) = \sum_{k\in \mathbb{N}} a_{0,k} \hbar^k \quad \text{for $a\in {\mathcal{W}}$}.
\]

Next, choose a a symplectic connection $\nabla$ on $M$, i.e.~a connection on $M$ which satisfies
$\nabla \omega = 0$. The symplectic connection canonically lifts to a connection
\[
  \nabla : \Omega^\bullet \mathbb W (M ) \rightarrow \Omega^{\bullet + 1} \mathbb W ( M ).
\]
By Fedosov's construction, there exists a section $A \in \Omega^1 \mathbb W (M)$
such that the connection
\begin{equation}
  \label{eq:FedConn}
   D : = \nabla + \frac{i}{\hbar} [ - , A ]
\end{equation}
is abelian, i.e.~satisfies $D \circ D = 0$. The $1$-form $A$ is even uniquely
determined by the latter property, if one additionally requires that
$\deg_\textup{F} (A) \geq 2$. The connection $D$ defined by such a  $1$-form $A$
will be called a \emph{Fedosov connection}.

As has been observed by Fedosov \cite{FedDQIT}, the space
\[
 {\mathcal{W}}_D (M) := \{ a \in {\mathcal{W}} (M) \mid Da = 0 \}
\]
of flat sections of the Weyl algebras bundle gives rise to  a deformation quantization of $\mathcal{C}^\infty (M)$ via
 the symbol map
\[
  \sigma :  {\mathcal{W}} (M) \rightarrow \mathcal{C}^\infty (M)[[\hbar]],  \quad a = \sum_{s\in \mathbb{N}, k \in \mathbb{N}} a_{s,k} \hbar^k \mapsto
   \sum_{k \in \mathbb{N}} a_{0,k} \hbar^k .
\]
More precisely, if the $1$-form $A$ has been chosen as above, the restriction
\[
  \sigma_{|{\mathcal{W}}_D (M)} :  {\mathcal{W}}_D (M) \rightarrow \mathcal{C}^\infty (M)[[\hbar]]
\]
is a linear isomorphism. Let
\[
  \mathfrak{q} :   \mathcal{C}^\infty (M)[[\hbar]]  \rightarrow {\mathcal{W}}_D (M)
\]
be its inverse, the so-called \emph{quantization map}.
Then there exist uniquely determined differential operators
$\mathfrak{q}_k :  \mathcal{C}^\infty (M) \rightarrow  \mathcal{C}^\infty (M)$ such that
\begin{equation}
\label{eq:expquantmap}
   \mathfrak{q} (f) = \sum_{k\in \mathbb{N}} \mathfrak{q}_k (f) \hbar^k \quad
   \text{for all $f \in \mathcal{C}^\infty (M)$},
\end{equation}
and
\[
  \star : \mathcal{C}^\infty (M)[[\hbar]]  \times \mathcal{C}^\infty (M)[[\hbar]] , \quad
  (f , g) \mapsto \sigma \big(  \mathfrak{q} (f) \circ \mathfrak{q} (g) \big)
\]
is a star product on $\mathcal{C}^\infty (M)$.

Now observe that the Fedosov connection $D$ leaves the module
$\mathcal{J}^\infty (X;M) \cdot \Omega^\bullet (M;\mathbb W M)$ invariant.
This implies that $D$ factors to the quotient
\[
  \Omega^\bullet_{\mathcal{E}^\infty} (X;\mathbb W M) :=
  \Omega^\bullet (M;\mathbb W M) / \mathcal{J}^\infty (X;M) \cdot \Omega^\bullet (M;\mathbb W M),
\]
and acts on
$\mathcal{E}^\infty (X;\mathbb W M) := {\mathcal{W}} ( M) / \mathcal{J}^\infty (X;M) \cdot {\mathcal{W}}  (M)$.
Moreover, the symbol map $\sigma$ maps
$\mathcal{J}^\infty (X;M)\cdot {\mathcal{W}} (M)$ to $\mathcal{J}^\infty (X;M)[[\hbar]]$, and
$\mathfrak{q} \big( \mathcal{J}^\infty (X;M)[[\hbar]]\big)$ is contained in $\mathcal{J}^\infty (X;M)\cdot {\mathcal{W}} (M)$,
since in the expansion \eqref{eq:expquantmap} the operators $\mathfrak{q}_k$ are all differential operators.
Hence $\sigma$ and $\mathfrak{q}$ factor to $\mathcal{E}^\infty (X;\mathbb W M)$ respectively
$\mathcal{E}^\infty (X)[[\hbar]]$. This entails the following result.

\begin{theorem}
  Let $(M,\omega)$ be a symplectic manifold, $D$ a Fedosov connection on $\Omega^\bullet \mathbb W $,
  and $X\subset M$ a closed subset. Then the space of flat sections
  \[
    {\mathcal{W}}_D (X) := \{ a \in \mathcal{E}^\infty (X;\mathbb W M)  \mid Da = 0\}
  \]
  is a subalgebra of $\mathcal{E}^\infty (X;\mathbb W M)$, and the symbol map induces an isomorphism of linear
  spaces
  $\sigma_X : {\mathcal{W}}_D (X) \rightarrow \mathcal{E}^\infty(X)[[\hbar]]$. Moreover, the unique product $\star$ on
  $\mathcal{E}^\infty(X)[[\hbar]]$ with respect to which $\sigma_X$ becomes an isomorphism of algebras
  is  a formal deformation quantization of $\mathcal{E}^\infty(X)$.
\end{theorem}

\input{section2}

\bibliographystyle{alpha}

\section*{Introduction}
In physics, many interesting systems are described mathematically by
phase spaces with singularities such as for example  the moduli spaces of
flat connections on a Riemann surface. The study of such singular phase spaces
raises a very interesting question in mathematical physics. How does one
quantize a singular Poisson manifold? In his seminal paper \cite{KonDQPM},
Kontsevich completely solved the problem of constructing
deformation quantizations of Poisson manifolds by his famous formality theorem.
However, the problem of proving a general existence theorem for
deformation quantizations over singular spaces is still open 15 years
later (see \cite{BorHerPflHASRDQ,HerPflIyeESPQSLHTA} for progress in this direction).

One of the key difficulties in the quantization theory of
singular phase spaces spaces is that the algebra of smooth functions
over  a space with singularities appears to be complicated to study
since certain crucial results such as a de Rham Theorem or a
Hochschild--Kostant--Rosenberg type theorem do in general not hold true
in the presence of singularities.

In this paper, we propose to replace the algebra of smooth functions by the
so-called Whitney functions, and discuss some examples of quantizations
of Whitney functions.

Let $M$ be a smooth manifold, and $X\subset M$ be a closed subset of $M$.
A Whitney function on $X$, roughly speaking, is the (infinite) jet of a
smooth function $f$ on $M$ at the subset $X$. We denote the algebra of
Whitney functions on $X$ by $\mathcal{E}^\infty(X)$. A Whitney--Poisson structure on
$X$ is a Poisson structure on $\mathcal{E}^\infty(X)$, i.e. an antisymmetric bilinear
bracket $\{ - , - \}$ on $\mathcal{E}^\infty(X)$ which is a derivation in each of its arguments
and satisfies the Jacobi-identity.
Several  interesting questions arise in the study of Whitney--Poisson structures.
\begin{enumerate}
\item
First observe that if a neighborhood of $X$ in $M$ is equipped with a Poisson bivector $\Pi$,
then $\Pi$ naturally defines a Whitney--Poisson structure on  $X$. This construction usually
provides
various different Whitney--Poisson structures on $X$, which we will call global Whitney--Poisson structures.
In general, is every Whitney--Poisson structure on $X$ a global one? This question is closely related to the
existence of a normal form of a Poisson structure near $X$. We expect to see obstructions for a general $X$
in $M$, which is probably connected to the singularities of $X$ and the embedding of $X$ in $M$.
\item
Whitney functions naturally factorize to smooth functions on $X$. In general, a Whitney--Poisson structure
does not factorize to  a Poisson structure on $X$ by which we mean an antisymmetric and bilinear
bracket on $\mathcal{C}^\infty (X)$ which is a derivation in each of its arguments and satisfies the
Jacobi-identity.
It appears to be an interesting question to describe those Whitney--Poisson
structures that do factorize to $X$.
This problem appears to be  closely related to the question under which conditions one
can embed a singular Poisson variety into a smooth Poisson manifold, see
\cite{EgiEBD,DavisEBD,McMillanEBD}.
\end{enumerate}
In this paper, we propose to study the problem of deformation quantization of Whitney--Poisson
structures on $X$. We will construct a natural deformation quantization of a global
Whitney--Poisson structure on $X$. Moreover, we study such a deformation quantization by
computing its  Hochschild homology when the global Whitney--Poisson structure is symplectic
using the methods developed in \cite{PPT2ADV}.

We would like to dedicate this short article to Nicolai Neumaier, who unfortunately passed away
in Spring 2010 after a brave and long battle with cancer.  Nicolai has been a good friend and
excellent collaborator. The idea to study the quantization of Whitney functions goes back to our
collaboration in 2004 on deformation quantization of orbifolds \cite{nppt}.
We are picking up this idea as a memory to Nicolai's important contribution to the subject of
deformation quantization of singular spaces.

\noindent{\bf Acknowledgments:} Pflaum is partially supported by NSF grant DMS 1105670, and
Tang is partially supported by NSF grant DMS 0900985.

\section{Hochschild and cyclic homology}
The Hochschild homology of algebras of Whitney functions $\mathcal{E}^\infty (X)$ has been computed for a large
class of singular subspaces $X\subset M$ in \cite{BraPfl}. In particular, it follows from this work that
for (locally) subanalytic sets $X\subset M$ with $M$ an analytic manifold the Hochschild homology of
$\mathcal{E}^\infty (X)$ is given by
\begin{equation}
  \label{eq:HHWhitneyFunc}
  HH_\bullet \big( \mathcal{E}^\infty (X) \big) =  \Omega^\bullet_{\mathcal{E}^\infty} (X) .
\end{equation}

In case $(M , \omega)$ is symplectic of dimension $2m$, and $\star$
a star product on $\mathcal{C}^\infty (M)$, the
Hochschild homology  of the deformed algebra $\big( \mathcal{C}^\infty (M)((\hbar)),\star \big)$ was first computed in \cite{NesTsyAIT}. (We extend the star product $\star$ on $\mathcal{C}^\infty(M)[[\hbar]]$ to $\mathcal{C}^\infty(M)((\hbar))$). It is given by
\begin{equation}
  \label{eq:HHDefAlgSmooth}
  HH_\bullet \big(  \mathcal{C}^\infty (M)((\hbar))\big) = H^{2m -\bullet}_\textup{dR} (M, \mathbb{C}((\hbar)) ).
\end{equation}
If $X \subset M$ now is closed, the natural question  arises what the Hochschild homology  of the
deformed algebra of Whitney functions  $\big( \mathcal{E}^\infty (X)((\hbar)), \star \big)$ then is.
Observe that via Teleman's localization technique \cite{Tlocal}, the Hochschild and cyclic homology of $\mathcal{E}^\infty(X)$ and $\mathcal{E}^\infty(X)((\star))$ (and also of $\mathcal{C}^\infty(M)$ and $\mathcal{C}^\infty(M)((\hbar))$) can be computed as the sheaf cohomology of the corresponding sheaf complexes for
Hochschild and cyclic complexes on $X$ (and on $M$) as is explained in \cite{BraPfl}.

We start the computation of the homology groups by first noting that $\mathcal{E}^\infty (X)((\hbar))$ carries a filtration
$\big( \mathcal{F}^k_\hbar \mathcal{E}^\infty (X)((\hbar))\big)_{k\in \mathbb{Z}}$ by the $\hbar$-degree. More precisely,
\[
   \mathcal{F}^k_\hbar \mathcal{E}^\infty (X)((\hbar)) =
   \{ F \in \mathcal{E}^\infty (X)((\hbar)) \mid \deg_\hbar F \geq k  \},
\]
where the $\hbar$-degree of $F = \sum_{k\in\mathbb{Z}} F_k \hbar^k \in  \mathcal{E}^\infty (X)((\hbar))$ with
$F_k  \in  \mathcal{E}^\infty (X)$ is given by
\[
  \deg_\hbar (F) = \min \{ k\in \mathbb{Z} \mid F_k \neq 0 \}.
\]
The $\hbar$-filtration of $\mathcal{E}^\infty (X)((\hbar))$ induces  a filtration
$\big( \mathcal{F}^k_\hbar C^\bullet (\mathcal{E}^\infty (X)((\hbar)) \big)_{k\in \mathbb{Z}}$ of the
Hochschild chain complex $C^\bullet (\mathcal{E}^\infty (X)((\hbar)))$ which then gives rise to a
spectral sequence $E^\bullet_{pq}$. Since
\[
  \mathcal{F}^{k+1}_\hbar \mathcal{E}^\infty (X)((\hbar)) / \mathcal{F}^k_\hbar \mathcal{E}^\infty (X)((\hbar)))
  \cong \mathcal{E}^\infty (X) ,
\]
the $E^1$-term has to coincide with the Hochschild homology of $\mathcal{E}^\infty (X)$, hence
\begin{equation}
  \label{eq:E1term}
  E^1_{pq} = \Omega^q_{\mathcal{E}^\infty} (X) .
\end{equation}
Since $\mathcal{E}^\infty (X)$ is the quotient of $\mathcal{C}^\infty (M)$ by the ideal
$\mathcal{J}^\infty (X;M)$, it follows from \cite[Sec.~3]{BryDCPM}  that the differential
$d^1_{pq} : \Omega^q_{\mathcal{E}^\infty} (X) \rightarrow \Omega^{q-1}_{\mathcal{E}^\infty} (X)$
coincides with the canonical derivative
\[
\begin{split}
   \delta : \: & \Omega^q_{\mathcal{E}^\infty} (X) \rightarrow \Omega^{q-1}_{\mathcal{E}^\infty} (X), \quad
     f_0 \, df_1 \wedge \ldots \wedge df_q \mapsto \\
     & \sum_{i=1}^q (-1)^{i+1} \{ f_o,f_i \}
     df_1 \wedge \ldots \wedge \widehat{d f_i} \wedge \ldots \wedge df_q +  \\
     & \sum_{1 \leq i < j \leq q} (-1)^{i+j}  f_o d \{ f_i , f_j \} \wedge
     df_1 \wedge \ldots \wedge \widehat{d f_i} \wedge \ldots \wedge
     \widehat{d f_j} \wedge \ldots \wedge df_q .
\end{split}
\]
Next let us recall Brylinski's definition of the symplectic Hodge $*$-operator
(see \cite{BryDCPM}).
Let $\nu$ be the volume form $\frac{1}{m!} \omega^m$ over $M$, and
$\Lambda^k\Pi $ the operator
\[
\begin{split}
  & \Omega^k M \times \Omega^k M \rightarrow \mathcal{C}^\infty (M),\quad \\
  & (f_0df_1 \wedge \ldots \wedge df_k , g_0dg_1  \wedge \ldots \wedge dg_k) \mapsto
   f_0 g_0 \, \big( \Pi  \lrcorner  \, df_1 \wedge dg_1 \big) \cdot \ldots \cdot
   \big( \Pi \lrcorner \, df_k \wedge dg_k \big)  .
\end{split}
\]
The symplectic $*$-operator
$* : \Omega^k (M) \rightarrow \Omega^{2m-k} (M)$  now is uniquely defined by
requiring that $\alpha \wedge (* \beta) = \Lambda^k\Pi (\alpha , \beta) \, \nu$ for all
$\alpha,\beta \in \Omega^k (M)$. Obviously, $*$ leaves the
$\mathcal{J}^\infty (X;M) \cdot \Omega^\bullet (M)$ invariant, hence induces an operator
$* : \Omega^k_{\mathcal{E}^\infty} (M) \rightarrow \Omega^{2m-k}_{\mathcal{E}^\infty} (M)$ which by the
properties of the corresponding operator on $\Omega^\bullet (M)$ satisfies the
equality $* \circ * = \operatorname{id} $. By \cite{BryDCPM} it also follows that on $\Omega^k_{\mathcal{E}^\infty} (X)$
the canonical differential $\delta$ is equal to $( - 1 )^{k+1} * d * $. But this implies
by \cite{BraPfl} that
\begin{equation}
  \label{eq:E2term}
  E^2_{pq} = H^{2m-q} (X) .
\end{equation}
Under the assumption that $X$ is compact subanalytic, there exists a finite triangulation of $X$,
hence the singular cohomology with values in $\mathbb{R}$, and by \cite{BraPfl} the
periodic cyclic homology of $\mathcal{E}^\infty (X)$ then have to be finite dimensional.
Arguing like in \cite{NesTsyAIT}, one concludes that under this assumption on $X$,
the spectral sequence degenerates at $E^2$, and the Hochschild homology of the deformed
algebra $\mathcal{E}^\infty (X)[[\hbar]]$ is given by \eqref{eq:E2term}. Let us show that this holds
even in more generality.

For this more refined computation of the Hochschild and cyclic homology of $\mathcal{E}^\infty(X)$, we use a specific quasi-isomorphism implementing the isomorphism \eqref{eq:HHDefAlgSmooth} above. In \cite{PPT2ADV}, we have constructed morphisms
\[
\Psi^i_{2k}:\mathcal{C}_{2k-i}\left(\mathcal{C}^\infty(M)((\hbar)),\star\right)\to\Omega^i(M)((\hbar)),
\]
satisfying the property
\begin{equation}\label{eq:psi}
(-1)^id\circ \Psi^i_{2k}=\Psi^{i+1}_{2k}\circ b+\Psi^{i+1}_{2k+2}\circ B,
\end{equation}
where $b$ and $B$ are the Hochschild and Connes' $B$-operator computing cyclic homology.
>From \cite[Thm 2.4]{PPTLMP}, it follows by Eq.~(\ref{eq:psi}) that the combination $\Psi_i:=\sum_{l\geq 0}\Psi^{2m-2l-i}_{2m-2l}$ defines an S-morphism $\Psi_\bullet $ of complexes of sheaves
\[
\Psi_\bullet:{\rm Tot}_\bullet\left(\mathcal{BC}\left(\mathcal{C}^\infty(M)((\hbar)),\star\right),b+B\right)\to \left(\bigoplus_{l\geq 0}\Omega^{2m-2l-\bullet}(M)((\hbar)), (-1)^{2m-2l-\bullet}d\right),
\]
where on the left we have the total sheaf complex of Connes' $(b,B)$-complex
(cf. \cite[Prop. 2.5.15]{Loday} for more on S-morphisms).

\begin{proposition} \label{prop:psi}
$\Psi^i_{2k}$ maps $C_{2k-i}\left(\mathcal{J}^\infty(X,M)((\hbar))\right)$ to
$\Omega^i_{\mathcal{J}^\infty}(M)((\hbar))$.
\end{proposition}
\begin{proof}
The proof is given by two observations: first, since the Fedosov--Taylor series defining the
quantization map $\mathfrak{q}:\mathcal{C}^\infty(M)\to {\mathcal{W}}(M)$ only involves partial derivatives, it will
map $\mathcal{J}^\infty(X,M)$ to $\mathcal{J}^\infty(X,M)\cdot{\mathcal{W}}(M)$. Second, we see from \cite{PPT2ADV}
that
$\Psi^i_{2k}$ is given by contraction of an explicitly given cyclic cocycle on the formal Weyl
algebra acting fiberwise on $\mathbb W(M)$, with the Fedosov connection $D$. From this, the result
is obvious.
\end{proof}

Proposition \ref{prop:psi} proves that the S-morphism $\Psi_\bullet$ descends to define an
S-morphism of complexes of sheaves on $X$
\[
\Psi_\bullet:{\rm Tot}_\bullet\left(\mathcal{BC}\left(\mathcal{E}^\infty(X)((\hbar)),\star\right),b+B\right)
\to \left(\bigoplus_{l\geq 0}\Omega^{2m-2l-\bullet}_{\mathcal{E}^\infty}(X)((\hbar)), (-1)^{2m-2l-\bullet}d\right).
\]
\begin{theorem}\label{thm:homology}
Let $(M,\omega)$ be a real analytic symplectic manifold, and $X \subset M$ a subanalytic subset.
Then the S-morphism $\Psi_\bullet$ defined above is a quasi-isomorphism, and therefore
\begin{eqnarray*}
&HH_\bullet(\mathcal{E}^\infty(X)((\hbar)), \star)&=H^{2m-\bullet}(X)((\hbar)),\\
&HC_\bullet(\mathcal{E}^\infty(X)((\hbar)),\star)&=\bigoplus_{k\geq 0}H^{2m-\bullet-2k}(X)((\hbar)).
\end{eqnarray*}
\end{theorem}
\begin{proof}
The proof is essentially a repetition of the arguments \cite[Theorem 3.9]{PPT2ADV}. Since $\Psi$
is an S-morphism, it suffices to check that
$\Psi^i_{2m}: C_{i}(\mathcal{E}^\infty(X)((\hbar)))\to \Omega^{2m-i}_{\mathcal{E}^\infty}((\hbar))$ is a
quasi-isomorphism. Since $\Psi^i_{2m}$ is a morphism of complexes of sheaves, we only need to
check that $\Psi^i_{2m}$ is a quasi-isomorphism on a sufficiently nice local chart of $X$,
which we choose to be the  intersection of a Darboux chart $U$ of $M$ with $X$.

We note that $\Psi^i_{2m}$ is compatible with the $\hbar$-filtrations on the Hochschild complexes
$C_{i}(\mathcal{E}^\infty(X)((\hbar)))$ and $\Omega^{2m-i}_{\mathcal{E}^\infty}((\hbar))$, and therefore induces a natural morphism between the spectral sequences associated to the $\hbar$-filtrations. To prove that $\Psi^\bullet_{2m}$ is a quasi-isomorphism, it suffices to check that $\Psi^\bullet_{2m}$ is a quasi-isomorphism at the $E^2$-level of the spectral sequences associated to the $\hbar$-filtrations. Over $U$, the algebra $(\mathcal{C}^\infty(U)((\hbar)), \star )$ can be identified with the standard Weyl algebra.
In addition, the $E^2$-level of the spectral sequence associated to the Hochschild complex of
$(\mathcal{C}^\infty(U)((\hbar)), \star) $ is the Poisson homology complex $(\Omega^\bullet(U)((\hbar)),\delta)$. Similarly, the $E^2$-level associated to $(\mathcal{E}^\infty(X)((\hbar)),\star )$ is again the Poisson homology complex $(\Omega^\bullet_{\mathcal{E}^\infty}(X)((\hbar)), \delta)$. Under this identification,  $\Psi^i_{2m}$ becomes the symplectic Hodge star operator, which is an isomorphism between the Poisson homology and the de Rham cohomology in (\ref{eq:E2term}).
\end{proof}
\begin{remark}
Theorem \ref{thm:homology} has a natural generalization to deformation quantizations of
global Whitney--Poisson structures on $X$ using the method in \cite{DolgADV}, i.e.
\begin{eqnarray*}
&HH_\bullet(\mathcal{E}^\infty(X)((\hbar)), \star)&=H_\bullet^\pi(X)((\hbar)),\\
&HP_\bullet(\mathcal{E}^\infty(X)((\hbar)),\star)&=H_\bullet (X)((\hbar)),
\end{eqnarray*}
where $H_\bullet^\pi(X)((\hbar))$ is the Poisson homology of $(X, \pi)$. We leave the details to diligent readers.
\end{remark}

\begin{remark}
It is easy to see that the so-called ``algebraic index theorem'' \cite{NesTsyAIT} descends to the level of Whitney functions: consider the morphism
\[
  \mu: C_\bullet\left(\mathcal{E}^\infty(X)[[\hbar]],\star \right)\to \Omega^\bullet_{\mathcal{E}^\infty}(X)
\]
given by
\[
\mu(f_0\otimes\ldots\otimes f_k):=\left. f_0df_1\wedge\ldots\wedge df_k\right|_{\hbar=0},
\]
where $\mathcal{E}^\infty(X)[[\hbar]]$ is viewed as an algebra over $\mathbb{C}$.
This map sends the Hochschild differential $b$ to zero and intertwines $B$ with the Whitney--de Rham operator $d$. The previously defined quasi-isomorphism $\Psi$ naturally extends to define a chain morphism
\[
 \Psi: \operatorname{Tot}_\bullet\left(\mathcal{BC}(\mathcal{E}^\infty(X)[[\hbar]],\star)\right)\longrightarrow \bigoplus_{l\geq 0}\Omega^{2m-2l-\bullet}_{\mathcal{E}^\infty}(X)((\hbar)).
\]
The algebraic index theorem gives the defect of the map $\mu$ to agree with the
morphism $\Psi$:
\begin{theorem}
Under the assumptions of Thm.~\ref{thm:homology}
the following diagram commutes after taking homology:
\[
  \xymatrix{\operatorname{Tot}_\bullet\left(\mathcal{BC}(\mathcal{E}^\infty(X)[[\hbar]],\star)\right)\ar[rr]^{\mu}\ar[drr]_{\Psi}&&
  \bigoplus_{l\geq 0}\Omega^{2m-2l-\bullet}_{\mathcal{E}^\infty}(X)\ar[d]^{\wedge\hat{A}(M)e^{-\Omega\slash 2\pi\sqrt{-1}\hbar}}
\\
& &
\bigoplus_{l\geq 0}\Omega^{2m-2l-\bullet}_{\mathcal{E}^\infty}(X)((\hbar))} .
\]
Hereby, $\hat{A}(M)$ is the standard $\hat{A}$-class of $M$ associated to the symplectic structure, and $\Omega$ is the characteristic class of the star product $\star$ on $M$.
\end{theorem}
As a consequence,  the following equality holds true in $H^\bullet(X)((\hbar))$:
\[
 \Psi(a)= \left([\hat{A}(M)]\cup [e^{\sqrt{-1}\Omega\slash 2\pi\hbar}]\right)\cup \mu(a),
\]
for all $a=a_0\otimes\ldots\otimes a_k\in C_k(\mathcal{E}^\infty(X),\star)$.
\end{remark}