Periodic solutions to the non--isothermal phase separation model with obstacle
Xin Qiaoling
Jiang Lining \thanks{E-mail address: jianglining@bit.edu.cn
{ Jiashan Zheng\footnote{E-mail address: zhengjiashan2008@163.com
%%In this paper, we investigate the existence and uniqueness of time %%periodic solutions to the multi--dimensional non--isothermal phase %%separation model with obstacle by the viscosity %%approach and %%the Leray-Schauder's fixed point theorem. %This paper is concerned with a class of biological models which %consists of nonlinear diffusion equations and a hysteresis operator %describing the relationship between some variables of the equations. %We show the existence of solutions to the problem under consideration by %using the approximation method. %%
1505.06944.tex
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\begin{document}
\openup 0.8\jot
\title{\Large\bf Jones basic construction on field algebras of $G$-spin models \thanks{This work is supported
by National Science Foundation of China (10971011,11371222)} }
\author{ Xin Qiaoling, Jiang Lining \thanks{E-mail address: jianglining@bit.edu.cn}}
\date{}
\maketitle\begin{center}
\begin{minipage}{16cm}
{\small \it School of Mathematics and Statistics, Beijing Institute
of Technology, Beijing 100081, P. R. China}
\end{minipage}
\end{center}
\vspace{0.05cm}
\begin{center}
\begin{minipage}{16cm}
{\small {\bf Abstract}: Let $G$ be a finite group. Starting from the field algebra ${\mathcal{F}}$ of $G$-spin models, one can construct the crossed product $C^*$-algebra ${\mathcal{F}}\rtimes D(G)$
such that it coincides with the $C^*$-basic construction for the field algebra ${\mathcal{F}}$ and the $D(G)$-invariant subalgebra of ${\mathcal{F}}$, where $D(G)$ is the quantum double of $G$. Under the natural $\widehat{D(G)}$-module action on ${\mathcal{F}}\rtimes D(G)$,
the iterated crossed product $C^*$-algebra
can be obtained, which is $C^*$-isomorphic to the $C^*$-basic construction for ${\mathcal{F}}\rtimes D(G)$ and the field algebra ${\mathcal{F}}$. Furthermore, one can show that the iterated crossed product $C^*$-algebra is a new field algebra
and give the concrete structure with the order and disorder operators.
}
\endabstract
\end{minipage}\vspace{0.10cm}
\begin{minipage}{16cm}
{\bf  Keywords}: $G$-spin models, $C^*$-basic construction, field algebras, dual action\\
Mathematics Subject Classification (2010): 46L05, 16S35
\end{minipage}
\end{center}
\begin{center} \vspace{0.01cm}
\end{center}

%%\title{\bf Periodic solutions to the non--isothermal phase separation model  with  obstacle}

%%\author{{ Jiashan Zheng\footnote{E-mail address: zhengjiashan2008@163.com}  }\\[0.2cm]
%%{\it \small School of Mathematics, Beijing Institute of Technology, Beijing 100081, P.R. China}\\
%%$^b${\it\small Department of Mechanical and Aerospace Engineering,}\\
%%{\it \small University of California at San Diego La Jolla, CA 92093-0411, USA}
%}
%%\date{}


%\date{23 March, 2009}

%%\maketitle \vspace{0.3cm} \noindent
%%\begin{abstract}
%%In this paper, we investigate the existence and uniqueness of time
%%periodic solutions to the multi--dimensional non--isothermal phase
%%separation model with obstacle  by the viscosity
%%approach and
 %%the Leray-Schauder's fixed point theorem.



%This paper is concerned with a class of biological models which
%consists of  nonlinear diffusion equations and a hysteresis operator
%describing the relationship between some variables of the equations.
%We show the existence  of solutions to the problem under consideration by
%using the approximation method.
%%\end{abstract}

%%\vspace{0.3cm} \noindent {\bf\em Key words:}~ periodic solution,
%%non--isothermal phase separation model, constraint,  fixed point
%%theorem
%%\vspace{0.3cm}


%%\noindent {\bf\em 2010 Mathematics Subject Classification}:~ 45M15, 47H10, 35K55
\section{Introduction}
In \cite{Jones}, V.F.R.Jones introduced the notion of indices for inclusions of II$_1$ factors,
which opened a completely new aspect of operator algebras involving various other
fields of mathematics and physics, including topology, quantum physics, dynamical systems, noncommutative
geometry, etc.
He established a method to
study the inner structures or the outer structures of given II$_1$-factors, preserving
certain quantity, called the Jones index. And he enlarged the pure algebraic Galois
theory to operator algebra, and this has a significant impact on mathematics and applied
mathematics. The interesting fact is that not only his theory but also the techniques
he used in his theory are important mathematically and applicable to other fields \cite{V.F.R.J}.
Subsequently, there are lots of attempts to extend the original Jones theory
by various mathematicians. For instance, M.Pimsner and S.Popa in \cite{M.PP} introduced the notion of the probabilistic index for
a conditional expectation, which is the best constant of the so-called Pimsner-Popa
inequality. In \cite{H.Ko}, H.Kosaki discussed another way to define index for a normal semifinite faithful conditional expectation of an arbitrary factor
onto a subfactor exploiting spatial theory of Connes and the theory of operatorvalued weights.




Though the probabilistic index works perfectly for analytic purposes even in the case of $C^*$-algebras (see, e.g. \cite{Popa}), it is not always suitable for algebraic operations
such as the basic construction in the $C^*$-case. Inspired by the Pimsner-Popa basis in the sense of \cite{M.PP}, Kosaki's index formula in the sense of \cite{H.Ko}, and the Casimir elements for
semi-simple Lie algebras, Y.Watatani \cite{Y.Wat} proposed to assume existence of a quasi-basis for a conditional expectation, a generalization of
the Pimsner-Popa basis in the von Neumann algebra case, to analyze inclusions of $C^*$-algebras. With a quasi-basis, Watatani successfully introduced a $C^*$-version of
basic construction, which is closely related to $K$-theory of $C^*$-algebras \cite{T.Kaj,Y.Wat}.
Roughly speaking, the (original or extended) Jones index
theory is the study of certain elements measuring the maximal number of disjoint copies of topological $*$-subalgebra in a given topological $*$-algebra.

On the other hand, quantum chains considered as models of $1+1$-dimensional quantum field theory exhibit many interesting features which include the emergence of braid group statistics and quantum symmetry. In particular, one-dimensional $G$-spin models, as a testing ground for applications of quantum group symmetries,
have an order-disorder type of
quantum symmetry given by the double $D(G)$ of a finite group $G$,
defined by Drinfel'd in the context of finding solutions to the quantum
Yang-Baxter equation arising from statistical mechanics \cite{V.G.Dr}, which generalizes the
$Z(2)\times Z(2)$ symmetry of the lattice Ising model.
In \cite{K.Szl}, the implementation of the Doplicher-Haag-Roberts theory of superselection sectors \cite{DHR1,DHR2,DHR3,DHR4} to $G$-spin models has been carried out. In this approach the symmetries can be reflected as the $D(G)$-invariant subalgebra ${\mathcal{A}}$ of the field algebra ${\mathcal{F}}$ in $G$-spin models, called the observable algebra.

The paper studies the Jones basic construction on field algebras of $G$-spin models, and is organized as follows.

In Section 2, we collect the necessary definitions and facts about $G$-spin models, such as the quantum double $D(G)$, the field algebra ${\mathcal{F}}$ and the Hopf action of the symmetry algebra $D(G)$ on ${\mathcal{F}}$, and then we also give a brief description of the $C^*$-basic construction for $C^*$-algebras.

In Section 3, under the conditional expectation $E$ from the field algebra ${\mathcal{F}}$ onto the $D(G)$-invariant subalgebra ${\mathcal{A}}$, we can construct the crossed product $C^*$-algebra
${\mathcal{F}}\rtimes D(G)$, and then prove that this algebra is $C^*$-isomorphic to the $C^*$-algebra $\langle {\mathcal{F}},e_{\mathcal{A}}\rangle_{C^*}$
constructed from the $C^*$-basic construction for
the inclusion ${\mathcal{A}}\subseteq{\mathcal{F}}$.

In Section 4, we show that there is a natural $\widehat{D(G)}$-module algebra structure on ${\mathcal{F}}\rtimes D(G)$, and there exists a conditional expectation $E_2\colon{\mathcal{F}}\rtimes D(G)\rightarrow {\mathcal{F}}$ such that $E_2$ is consistent with the dual conditional expectation of $E\colon{\mathcal{F}}\rightarrow {\mathcal{A}}$ in Proposition 2.1.
Moreover, we give the quasi-basis for the dual conditional expectation of $E$.

The main result of Section 5 is that the $C^*$-algebra constructed from the $C^*$-basic construction
$\langle{\mathcal{F}}\rtimes D(G),e_2\rangle$ for the inclusion ${\mathcal{F}}\subseteq{\mathcal{F}}\rtimes D(G)$ is $C^*$-isomorphic to the iterated crossed product $C^*$-algebra ${\mathcal{F}}\rtimes D(G)\rtimes\widehat{D(G)}$, which is canonically isomorphic to $M_{|G|^2}({\mathcal{F}})$ by Takai duality \cite{H.Tak}. Then we give the concrete description of the new field algebra ${\mathcal{F}}\rtimes D(G)\rtimes\widehat{D(G)}$ by means of the order and disorder operators.



All the algebras in this paper will be unital associative algebras over the complex field ${\Bbb C}$. The unadorned tensor product $\otimes$ will stand for the usual tensor product over ${\Bbb C}$.
 For general results on Hopf algebras please refer to the books of Abe \cite{E.Abe} and Sweedler \cite{M.E.Sw}. We shall follow their notations, such as $S$, $\bigtriangleup$, $\varepsilon$ for the antipode, the comultiplication and the counit, respectively. Also we shall use the so-called
``Sweedler-type notation" for the image of $\bigtriangleup$. That is
\begin{eqnarray*}
    \begin{array}{c}
    \bigtriangleup(a)=\sum\limits_{(a)}a_{(1)}\otimes a_{(2)}.
     \end{array}
\end{eqnarray*}

\section{$G$-spin models and the $C^*$-basic construction}
We first recall the main features of $G$-spin models, considered in
the $C^*$-algebraic framework for quantum lattice systems, and then give the $C^*$-basic construction for $C^*$-algebras in $G$-spin models.

\subsection{Definitions and preliminary results}
Assume that $G$ is a finite group with a unit $u$. The $G$-valued spin configuration on the two-dimensional square lattices is the map $\sigma\colon{\Bbb Z}^2\rightarrow G$ with Euclidean action functional: $$S(\sigma)=\sum \limits_{(x,y)} f(\sigma_x^{-1}\sigma_y),$$ in which the summation runs over the nearest neighbor pairs in ${\Bbb Z}^2$ and $f\colon G\rightarrow {\Bbb R}$ is a function of the positive type. This kind of classical statistical systems or the corresponding quantum field theories are called $G$-spin models \cite{S.Dop,L.N.Jia,Qiao}. And such models provide the simplest examples of lattice field theories exhibiting quantum symmetry.
In general, $G$-spin models with an Abelian group $G$ are known to have a symmetry group $G\times \widetilde{G}$, where $\widetilde{G}$ is the group of characters of $G$. If $G$ is non-Abelian, the models have a symmetry of a quantum double
 $D(G)$ \cite{K.A.Da,G.Mas}, which is defined as follows.

\begin{definition}
Let $C(G)$ be the algebra of complex valued functions on
$G$ and consider the adjoint action of $G$ on $C(G)$ according to $\alpha_g\colon f\mapsto f\circ Ad(g^{-1})$.
The quantum double $D(G)$ is defined as the crossed product $D(G) = C(G)\rtimes_{\alpha} G$ of
$C(G)$ by this action. In terms of generators $D(G)$ is the algebra generated by elements
$U_g$ and $V_h$ $(g,h\in G)$, with the relations
\begin{eqnarray*}
    \begin{array}{rcl}
    U_g U_h&=&\delta_{g,h}U_g\\[5pt]
    V_g V_h&=&V_{gh}\\[5pt]
    V_h U_g&=&U_{hgh^{-1}}V_h,
     \end{array}
\end{eqnarray*}
and the identification $\sum\limits_{g\in G}U_g=V_u=1$, where
$\delta_{g,h}=\left\{\begin{array}{cc}
                          1 & {\mathrm{if}} \   g=h, \\[5pt]
                          0 & {\mathrm{if}} \  g\neq h.
                        \end{array}\right.$
\end{definition}

It is easy to see that $D(G)$ is of finite dimension, where as a convenient basis one
may choose $U_gV_h, g,h\in G$, multiplying according to
$U_{g_1}V_{h_1}U_{g_2}V_{h_2}=
\delta_{g_1h_1,h_1g_2}U_{g_1}V_{h_1h_2}.$

Here and from now on, by $(g,h)$ we always denote the element $U_gV_h$ for notational convenience.

Also, the structure maps are given by
 \begin{eqnarray*}
    \begin{array}{rcll}
\bigtriangleup(g,h)&=&\sum \limits_{t\in H}(t,h)\otimes (t^{-1}g,h),& (\mathrm{coproduct})\\[5pt]
\varepsilon(g,h)&=&\delta_{g,u},& (\mathrm{counit})\\[5pt]
S(g,h)&=&(h^{-1}g^{-1}h,h^{-1}),& (\mathrm{antipode})
     \end{array}
\end{eqnarray*}
on the linear basis $\{(g,h),g,h\in G\}$ and are extended in $D(G)$ by linearity.
One can prove that $D(G)$ is a Hopf algebra, with a unique element $E=\frac{1}{|G|}\sum\limits_{g\in G}(u,g)$, called an integral element, satisfying for any $a\in D(G)$,
$$aE=Ea=\varepsilon(a)E.$$
Moreover, with the definition
 \begin{eqnarray*}
    \begin{array}{c}
    (g,h)^*=(h^{-1}gh,h^{-1}),
         \end{array}
\end{eqnarray*}
and the appropriate extension, $D(G)$ is a semisimple $*$-algebra of finite dimension \cite{M.E.Sw}, which implies that $D(G)$ becomes a Hopf $C^*$-algebra.

As in the traditional case, one can define the local quantum field algebra as follows.

\begin{definition}
 The local field algebra of a $G$-spin model ${\mathcal{F}}_{\mathrm{loc}}$ is an associative algebra with a unit $I$ generated by
 $\{\delta_g(x), \rho_h(l)\colon g, h\in G, x\in {\Bbb Z}, l\in {\Bbb Z}+\frac{1}{2}\}$ subject to
\begin{eqnarray*}
    \begin{array}{rcl}
\sum\limits_{g\in G}\delta_g(x)&=&I=\rho_u(l),\\[5pt]
\delta_{g_1}(x)\delta_{g_2}(x)&=&\delta_{g_1,g_2}\delta_{g_1}(x),\\[5pt]
 \rho_{h_1}(l)\rho_{h_2}(l)&=&\rho_{h_1h_2}(l),\\[5pt]
\delta_{g_1}(x)\delta_{g_2}(x')&=&\delta_{g_2}(x')\delta_{g_1}(x),\\[5pt]
 \rho_h(l)\delta_g(x)&=& \left\{\begin{array}{cc}
                          \delta_{hg}(x)\rho_h(l) &\mbox{if}\ l<x, \\[5pt]
                          \delta_g(x)\rho_h(l) & \mbox{if}\ l>x,
                        \end{array}\right.\\[5pt]
 \rho_{h_1}(l)\rho_{h_2}(l')&=& \left\{\begin{array}{cc}
                          \rho_{h_2}(l')\rho_{{h_2}^{-1}h_1{h_2}}(l) & \mbox{if}\ l>l', \\[5pt]
                          \rho_{{h_1}{h_2}{h_1}^{-1}}(l')\rho_{h_1}(l) & \mbox{if}\ l<l',
                        \end{array}\right.

    \end{array}
\end{eqnarray*}
for $x, x' \in {\Bbb Z},$ $l, l'\in {\Bbb Z}+\frac{1}{2}$ and $h_1, h_2, g_1,g_2\in G$.
\end{definition}

The $*$-operation is defined on the generators as $\delta_g^\ast(x)=\delta_g(x), \ \rho_h^\ast(l)=\rho_{h^{-1}}(l)$ and can be extended to an involution on ${\mathcal{F}}_{\mathrm{loc}}$. In this way, ${\mathcal{F}}_{\mathrm{loc}}$ becomes a unital $*$-algebra.

For any finite subset $\Lambda\subseteq \frac{1}{2}\Bbb{Z}$, let
${\mathcal{F}}(\Lambda)$ be the $*$-subalgebra of ${\mathcal{F}}_{\mathrm{loc}}$ generated by
 \begin{eqnarray*}
\Big\{\delta_g(x), \rho_h(l)\colon
 g, h\in G, x\in\Lambda\cap\Bbb{Z}, l\in\Lambda\cap(\Bbb{Z}+\frac{1}{2})\Big\}.
 \end{eqnarray*}
In particular, we consider an increasing sequence of intervals $\Lambda_n, n\in\Bbb{N}$, where
\begin{eqnarray*}
    \begin{array}{rcl}
    \Lambda_{2n}&=&\{s\in\frac{1}{2}\Bbb{Z}\colon -n+\frac{1}{2}\leq s\leq n\},\\[5pt]
    \Lambda_{2n+1}&=&\{s\in\frac{1}{2}\Bbb{Z}\colon -n-\frac{1}{2}\leq s\leq n\}.
\end{array}
\end{eqnarray*}
In \cite{K.Szl}, the authors have shown that ${\mathcal{F}}(\Lambda_n), n\in\Bbb{N}$ are full matrix algebras, which can be identified with $M_{|G|^n}$. Moreover, under the induced norm, ${\mathcal{F}}(\Lambda_n)$ are finite dimensional $C^*$-algebras.
The natural embeddings $\iota_n\colon {\mathcal{F}}(\Lambda_n)\rightarrow{\mathcal{F}}(\Lambda_{n+1})$, which identify the $\delta$ and $\rho$ generators, are norm preserving.
Using the $C^*$-inductive limit \cite{B.R.Li,K.Szl}, a $C^*$-algebra ${\mathcal{F}}$, called the field algebra of a $G$-spin model,
 can be given by
\begin{eqnarray*}
    \begin{array}{c}
    {\mathcal{F}}=\overline{\bigcup\limits_{n}
{\mathcal{F}}(\Lambda_{n})}.
 \end{array}
\end{eqnarray*}

There is an action $\gamma$ of $D(G)$ on ${\mathcal{F}}$ in the following. For $x\in {\Bbb Z}$, $l\in {\Bbb Z}+\frac{1}{2}$ and $g,h\in G$, set
\begin{eqnarray*}
    \begin{array}{rcll}
(g,h)\delta_f(x)&=&\delta_{g,u}\delta_{hf}(x), & \forall f\in G,\\
(g,h)\rho_t(l)&=&\delta_{g,hth^{-1}}\rho_g(l), & \forall t\in G.
 \end{array}
\end{eqnarray*}
The map $\gamma$ can be extended for products of generators inductively in the number of generators by the rule
\begin{eqnarray*}
    \begin{array}{c}
    (g,h)(fT)=\sum\limits_{(g,h)}(g,h)_{(1)}(f)(g,h)_{(2)}(T),
 \end{array}
\end{eqnarray*}
where $f$ is one of the generators in ${\mathcal{F}}_{\mathrm{loc}}$ and $T$ is a finite product of generators. Finally, it is linearly extended both in $D(G)$ and ${\mathcal{F}}_{\mathrm{loc}}$.

\begin{lemma}$^{\cite{K.Szl}}$
The field algebra ${\mathcal{F}}$ is a $D(G)$-module algebra with respect to the map $\gamma$. Namely, the map $\gamma$ satisfies the following relations:
\begin{eqnarray*}
    \begin{array}{rcl}
(ab)(T)&=&a(b(T)),\\[5pt]
a(T_1T_2)&=&\sum\limits_{(a)}a_{(1)}(T_1)a_{(2)}(T_2),\\[5pt]
a(T^*)&=&(S(a^*)(T))^*
 \end{array}
\end{eqnarray*}
for $ a, b\in D(G), \ T_1, T_2, T\in {\mathcal{F}}$.
\end{lemma}

Set $${\mathcal{A}}\ =\ \{F\in {\mathcal{F}}\colon a(F)=\varepsilon(a)(F), \ \forall a\in D(G)\}.$$ We call it an observable algebra in the field algebra ${\mathcal{F}}$ of $G$-spin models. Furthermore, one can show that ${\mathcal{A}}$ is a nonzero $C^*$-subalgebra of ${\mathcal{F}}$, and $${\mathcal{A}}\ =\ \{F\in {\mathcal{F}}\colon E(F)=F\}\equiv E({\mathcal{F}}).$$
Indeed, from the following proposition, one can see that ${\mathcal{A}}$ is a $C^*$-subalgebra of ${\mathcal{F}}$.

\begin{proposition}$^{\cite{K.Szl}}$
 The map $E\colon{\mathcal{F}}\rightarrow{\mathcal{A}}$ satisfies the following conditions:

 $(1)$\ $E(I)=I$ where $I$ is the unit of ${\mathcal{F}}$;

 $(2)$\ (bimodular property) $\forall \ F_1, F_2\in{\mathcal{A}}, \ F\in {\mathcal{F}}$,
 $$E(F_1FF_2)=F_1E(F)F_2;$$

 $(3)$\ $E$ is positive.
\end{proposition}

In the following a linear map $\Gamma$ from a unital $C^*$-algebra $B$ onto its unital $C^*$-subalgebra $A$ with properties (1)-(3) in Proposition 2.1 is  called a conditional expectation. If $\Gamma$ is a conditional expectation from $B$ onto $A$, then $\Gamma$ is a projection of norm one \cite{O.Bra}. In addition, if $E(Bb)=0$ implies $b=0$, for $b\in B$, then we say $E$ is faithful.

\subsection{The $C^*$-basic construction for the inclusion ${\mathcal{A}}\subseteq{\mathcal{F}}$}
This section will give a concrete description of the $C^*$-basic construction for the inclusion ${\mathcal{A}}\subset{\mathcal{F}}$ and some properties about the Jones projection.

Let $\Gamma\colon B\rightarrow A$ be a faithful conditional expectation. Then $B_A$(viewing $B$ as a right $A$-module) is a pre-Hilbert module over $A$ with an $A$-valued inner product
$\langle x,y\rangle=\Gamma(x^*y)$ for $x,y\in  B_A.$
Let $\overline{B_A}$ be the completion of $B_A$ with respect to the norm on $B_A$ defined by
$$\|x\|_{B_A}=\|\Gamma(x^*x)\|_A^{\frac{1}{2}}, \ x\in B_A.$$
Then $\overline{B_A}$ is a Hilbert $C^*$-module over $A$. Since $\Gamma$ is faithful, the canonical map
$B\rightarrow {\overline{B_A}}$ is injective. Let $L_A(\overline{B_A})$ be the set of all (right) $A$-module homomorphisms
$T\colon \overline{B_A}\rightarrow \overline{B_A}$ with an adjoint $A$-module homomorphism $T^*\colon \overline{B_A}\rightarrow \overline{B_A}$ such that
$$\langle T\xi,\eta\rangle=\langle\xi, T^*\eta\rangle.$$
Then $L_A(\overline{B_A})$ is a $C^*$-algebra with the operator norm
$$\|T\|\ =\ \sup\{\|T\xi\|\colon \|\xi\|=1\}.$$
There is an injective $*$-homomorphism $\lambda\colon B\rightarrow L_A(\overline{B_A})$ defined by
$\lambda(b)x=bx$
for $x\in B_A$ and $b\in B$, so that $B$ can be viewed as a $C^*$-subalgebra of $L_A(\overline{B_A})$.
Note
that the map $\gamma_A \colon B_A \rightarrow B_A$ defined by
$\gamma_A(x)=\Gamma(x)$ for $x\in B_A$
is bounded and thus it can be extended to a bounded linear operator on $\overline{B_A}$, denoted by
$\gamma_A$ again. Then $\gamma_A\in L_A(\overline{B_A})$ and $\gamma_A =\gamma_A^2
=\gamma_A^*$; that is, $\gamma_A$ is a projection in
$L_A(\overline{B_A})$. From now on we call $\gamma_A$ the Jones projection of $\Gamma$.
The (reduced) $C^*$-basic construction is a $C^*$-subalgebra of $L_A(\overline{B_A})$ defined to be
$$ \langle B,\gamma_A\rangle_{C^*}= \overline{\mbox{span}\{\lambda(x)\gamma_A\lambda(y)\in L_A(\overline{B_A}) \colon x, y \in B \}}^{\|\cdot\|}.$$




For the conditional expectation $E\colon {\mathcal{F}}\rightarrow{\mathcal{A}}$, we shall consider the $C^*$-basic construction
$\langle{\mathcal{F}},e_{\mathcal{A}}\rangle_{C^*}$, which is a $C^*$-subalgebra of
$L_{\mathcal{A}}(\overline{{\mathcal{F}}})$
linearly generated by
$\{\lambda(x)e_{\mathcal{A}}\lambda(y)\colon x,y\in {\mathcal{F}}\}$,
where $\overline{{\mathcal{F}}}$ is the completion of ${\mathcal{F}}_{\mathcal{A}}$ with respect to the norm $\|x\|_{{\mathcal{F}}_{\mathcal{A}}}=
\|E(x^*x)\|_{\mathcal{A}}^{\frac{1}{2}}$ and $e_{\mathcal{A}}$ is the Jones projection of $E$.

In order to describe the concrete construction of
$\langle{\mathcal{F}},e_{\mathcal{A}}\rangle_{C^*}$, we first consider the local $C^*$-basic construction $\langle{\mathcal{F}(\Lambda_{\frac{1}{2},2})},
e_{\mathcal{A}}\rangle_{C^*}$ (since $E\colon {\mathcal{F}(\Lambda_{\frac{1}{2},2})}
\rightarrow{\mathcal{A}(\Lambda_{1,\frac{3}{2}})}$ is also a conditional expectation by Proposition 3.1 in \cite{K.Szl}), where
$\Lambda_{\frac{1}{2},2}=\Big\{\frac{1}{2},1,\frac{3}{2},2\Big\}$ and
${\mathcal{F}(\Lambda_{\frac{1}{2},2})}$ is a $C^*$-subalgebra of the field algebra $\mathcal{F}$ generated by
$$\Big\{\delta_{g}(x),\rho_{h}(l)\colon g,h\in G, x=1,2, l=\frac{1}{2},\frac{3}{2}\Big\}.$$
Also the
dimension of ${\mathcal{F}(\Lambda_{\frac{1}{2},2})}$ is finite, then the $C^*$-algebra $\langle{\mathcal{F}(\Lambda_{\frac{1}{2},2})},
e_{\mathcal{A}}\rangle_{C^*}$ is generated by
$$\Big\{\delta_{g}(x),\rho_{h}(l),e_{\mathcal{A}}\in L_{\mathcal{A}}({\mathcal{F}(\Lambda_{\frac{1}{2},2})})\colon g,h\in G,x=1,2,l=\frac{1}{2},\frac{3}{2}\Big\}.$$
By Definition 2.2 and $e_{\mathcal{A}}$ being a projection, we know the linear basis of $\langle{\mathcal{F}(\Lambda_{\frac{1}{2},2})},
e_{\mathcal{A}}\rangle_{C^*}$ is
$$\Big\{\delta_{g_1}(1)\delta_{g_2}(2)
\rho_{h_1}(\frac{1}{2})\rho_{h_2}(\frac{3}{2})e_{\mathcal{A}}
\delta_{s_1}(1)\delta_{s_2}(2)
\rho_{t_1}(\frac{1}{2})\rho_{t_2}(\frac{3}{2})\colon g_i,h_i,s_i,t_i\in G, \ \mbox{for} \ i=1,2\Big\}.$$

We will give some properties about the elements in $L_{\mathcal{A}}({\mathcal{F}(\Lambda_{\frac{1}{2},2})})$ as follows.

\begin{lemma}
(1)\ As operators on ${\mathcal{F}(\Lambda_{\frac{1}{2},2})}$, we have $e_{\mathcal{A}}Te_{\mathcal{A}}=E(T)e_{\mathcal{A}}$.

(2)\ Let $T\in {\mathcal{F}(\Lambda_{\frac{1}{2},2})}$, then
$T\in {\mathcal{A}(\Lambda_{1,\frac{3}{2}})}$ if and only if $e_{\mathcal{A}}T=Te_{\mathcal{A}}$.
\end{lemma}
\begin{proof}
(1)\ Suppose that $T=\delta_{g_1}(1)\delta_{g_2}(2)\rho_{h_1}
(\frac{1}{2})\rho_{h_2}(\frac{3}{2})$. It suffices to show that $$e_{\mathcal{A}}Te_{\mathcal{A}}(F)=E(T)e_{\mathcal{A}}(F)$$ for any $F=\delta_{s_1}(1)\delta_{s_2}(2)\rho_{t_1}(\frac{1}{2})
\rho_{t_2}(\frac{3}{2})\in {\mathcal{F}(\Lambda_{\frac{1}{2},2})}.$

We can compute
\begin{eqnarray*}
    \begin{array}{rcl}
&&e_{\mathcal{A}}Te_{\mathcal{A}}(\delta_{s_1}(1)\delta_{s_2}(2)\rho_{t_1}(\frac{1}{2})
\rho_{t_2}(\frac{3}{2}))\\[5pt]
&=&\delta_{t_1t_2,u}e_{\mathcal{A}}T(\frac{1}{|G|}\sum\limits_{f\in G}
\delta_{fs_1}(1)\delta_{fs_2}(2)\rho_{ft_1f^{-1}}(\frac{1}{2})
\rho_{ft_2f^{-1}}(\frac{3}{2}))\\[5pt]
&=&\frac{1}{|G|}\delta_{t_1t_2,u}E(\sum\limits_{f\in G}\delta_{g_1}(1)\delta_{g_2}(2)\rho_{h_1}
(\frac{1}{2})\rho_{h_2}(\frac{3}{2})\delta_{fs_1}(1)\delta_{fs_2}(2)
\rho_{ft_1f^{-1}}(\frac{1}{2})
\rho_{ft_2f^{-1}}(\frac{3}{2}))\\[5pt]
&=&\frac{1}{|G|}\delta_{t_1t_2,u}E(\sum\limits_{f\in G}\delta_{g_1}(1)
\delta_{h_1fs_1}(1)\delta_{g_2}(2)\delta_{fs_2}(2)
\rho_{h_1ft_1f^{-1}}(\frac{1}{2})
\rho_{ft_2f^{-1}h_2}(\frac{3}{2}))\\[5pt]
&=&\frac{1}{|G|}\delta_{t_1t_2,u}E(\delta_{g_1}(1)
\delta_{g_2}(2)\delta_{h_1^{-1}g_1s_1^{-1}s_2}(2)
\rho_{g_1s_1^{-1}t_1s_1g_1^{-1}h_1}(\frac{1}{2})
\rho_{h_1^{-1}g_1s_1^{-1}t_1^{-1}s_1g_1^{-1}h_1h_2}(\frac{3}{2}))\\[5pt]
&=&\frac{1}{|G|^2}\delta_{t_1t_2,u}\delta_{h_1h_2,u}
\delta_{g_2,h_1^{-1}g_1s_1^{-1}s_2}
w_{g_1^{-1}g_2}(\frac{3}{2})
v_{g_1^{-1}h_1^{-1}g_1s_1^{-1}t_1^{-1}s_1}(1),
 \end{array}
\end{eqnarray*}
and
\begin{eqnarray*}
    \begin{array}{rcl}
&&E(\delta_{g_1}(1)\delta_{g_2}(2)\rho_{h_1}
(\frac{1}{2})\rho_{h_2}(\frac{3}{2}))e_{\mathcal{A}}(
\delta_{s_1}(1)\delta_{s_2}(2)\rho_{t_1}(\frac{1}{2})
\rho_{t_2}(\frac{3}{2}))\\[5pt]
&=&E(\delta_{g_1}(1)\delta_{g_2}(2)\rho_{h_1}
(\frac{1}{2})\rho_{h_2}(\frac{3}{2}))E(\delta_{s_1}(1)\delta_{s_2}(2)\rho_{t_1}(\frac{1}{2})
\rho_{t_2}(\frac{3}{2}))\\[5pt]
&=&\frac{1}{|G|^2}\delta_{t_1t_2,u}\delta_{h_1h_2,u}
w_{g_1^{-1}g_2}(\frac{3}{2}))v_{g_1^{-1}h_1^{-1}g_1(1)}
w_{s_1^{-1}s_2}(\frac{3}{2}))v_{s_1^{-1}t_1^{-1}s_1(1)}\\[5pt]
&=&\frac{1}{|G|^2}\delta_{t_1t_2,u}\delta_{h_1h_2,u}
\delta_{g_2,h_1^{-1}g_1s_1^{-1}s_2}
w_{g_1^{-1}g_2}(\frac{3}{2})
v_{g_1^{-1}h_1^{-1}g_1s_1^{-1}t_1^{-1}s_1}(1).
 \end{array}
\end{eqnarray*}

(2)\ For $T\in {\mathcal{A}(\Lambda_{1,\frac{3}{2}})}$, without loss of generality, set $T=w_y(\frac{3}{2})v_x(1)$.
Notice that
\begin{eqnarray*}
    \begin{array}{rcl}
  &&  e_{\mathcal{A}}T(\delta_{g_1}(1)\delta_{g_2}(2)\rho_{h_1}
(\frac{1}{2})\rho_{h_2}(\frac{3}{2}))\\[5pt]
&=&E(w_y(\frac{3}{2})v_x(1)\delta_{g_1}(1)\delta_{g_2}(2)\rho_{h_1}
(\frac{1}{2})\rho_{h_2}(\frac{3}{2}))\\[5pt]
&=&E(\sum\limits_{s}\delta_s(1)\delta_{sy}(2)\rho_{sx^{-1}s^{-1}}(\frac{1}{2})
\rho_{sxs^{-1}}(\frac{3}{2})\delta_{g_1}(1)\delta_{g_2}(2)\rho_{h_1}
(\frac{1}{2})\rho_{h_2}(\frac{3}{2}))\\[5pt]
&=&E(\sum\limits_{s}\delta_s(1)\delta_{sx^{-1}s^{-1}g_1}(1)\delta_{sy}(2)
\delta_{g_2}(2)\rho_{sx^{-1}s^{-1}h_1}(\frac{1}{2})
\rho_{h_1^{-1}sxs^{-1}h_1h_2}(\frac{3}{2}))\\[5pt]
&=&E(\delta_{g_1x^{-1}}(1)\delta_{g_1x^{-1}y}(2)\delta_{g_2}(2)
\rho_{g_1x^{-1}g^{-1}h_1}(\frac{1}{2})
\rho_{h_1^{-1}g_1xg^{-1}h_1h_2}(\frac{3}{2}))\\[5pt]
&=&\frac{1}{|G|}\delta_{g_1x^{-1}y,g_2}\delta_{h_1h_2,u}
w_{xg_1^{-1}g_2}(\frac{3}{2})
v_{xg_1^{-1}h_1^{-1}g_1}(1),
     \end{array}
\end{eqnarray*}
and
\begin{eqnarray*}
    \begin{array}{rcl}
&&Te_{\mathcal{A}}(\delta_{g_1}(1)\delta_{g_2}(2)\rho_{h_1}
(\frac{1}{2})\rho_{h_2}(\frac{3}{2}))\\[5pt]
&=&\frac{1}{|G|}\delta_{h_1h_2,u}w_y(\frac{3}{2})v_x(1)
w_{g_1g_2^{-1}}(\frac{3}{2})
v_{g_1^{-1}h_1^{-1}g_1}(1)\\[5pt]
&=&\frac{1}{|G|}\delta_{h_1h_2,u}w_y(\frac{3}{2})
w_{xg_1g_2^{-1}}(\frac{3}{2})v_{xg_1^{-1}h_1^{-1}g_1}(1)\\[5pt]
&=&\frac{1}{|G|}\delta_{g_1x^{-1}y,g_2}\delta_{h_1h_2,u}
w_{xg_1^{-1}g_2}(\frac{3}{2})
v_{xg_1^{-1}h_1^{-1}g_1}(1),
     \end{array}
\end{eqnarray*}
we can conclude that $e_{\mathcal{A}}$ commutes with $T$ if $T\in {\mathcal{A}(\Lambda_{1,\frac{3}{2}})}$.

For the converse, since $E(I)=I$ where $I$ is the unit of ${\mathcal{F}}$, then
$$T=TE(I)=(Te_{\mathcal{A}})(I)=(e_{\mathcal{A}}T)(I)=E(TI)=E(T),$$
which yields that $T\in {\mathcal{A}(\Lambda_{1,\frac{3}{2}})}$.
\end{proof}

\section{$C^*$-isomorphism between ${\mathcal{F}}\rtimes D(G)$ and $\langle{\mathcal{F}},e_{\mathcal{A}}\rangle_{C^*}$}
In this section, we will construct the crossed product $C^*$-algebra
${\mathcal{F}}\rtimes D(G)$ extending ${\mathcal{F}}\equiv{\mathcal{F}}\rtimes I_{D(G)}$ by means of a
Hopf module left action of $D(G)$ on ${\mathcal{F}}$, such that ${\mathcal{F}}\rtimes D(G)$ coincides with the $C^*$-algebra $\langle{\mathcal{F}},e_{\mathcal{A}}\rangle_{C^*}$ constructed from the $C^*$-basic construction for the inclusion ${\mathcal{A}}\subseteq{\mathcal{F}}$.



As we have known, the field algebra ${\mathcal{F}}$ of $G$-spin models is a $D(G)$-module algebra, and one can construct the crossed product $*$-algebra ${\mathcal{F}}_{\mathrm{loc}}\rtimes D(G)$, as a vector space ${\mathcal{F}}_{\mathrm{loc}}\otimes D(G)$ with the $*$-algebra structure
\begin{eqnarray*}
    \begin{array}{rcl}
  \Big(T\otimes (g,h)\Big)\Big(F\otimes (s,t)\Big)
  &=& \sum\limits_{(g,h)}T(g,h)_{(1)}(F)\otimes(g,h)_{(2)}(s,t),\\[5pt]
  \Big(T\otimes (g,h)\Big)^*&=&\Big(I_{\mathcal{F}}\otimes (g,h)^*\Big)\Big(T^*\otimes I_{D(G)}\Big).
       \end{array}
\end{eqnarray*}

The crossed product ${\mathcal{F}}_{\mathrm{loc}}\rtimes D(G)$ can be extended naturally to a $C^*$-algebra ${\mathcal{F}}\rtimes D(G)$ in the following way.
Firstly, for any finite subset
$\Lambda\subseteq \frac{1}{2}\Bbb{Z}$, let
$\mathcal{F}(\Lambda)\rtimes D(G)$ be the subalgebra of ${\mathcal{F}}_{\mathrm{loc}}\rtimes D(G)$ generated by
 \begin{eqnarray*}
\Big\{T\otimes (g,h)\colon T\in \mathcal{F}(\Lambda), \
 (g,h)\in D(G)\Big\}.
 \end{eqnarray*}
In particular, we consider an increasing sequence of intervals $\Lambda_n$ for any $n\in\Bbb{N}$, where
\begin{eqnarray*}
    \begin{array}{rcl}
    \Lambda_{2n}&=&\{s\in\frac{1}{2}\Bbb{Z}\colon -n+\frac{1}{2}\leq s\leq n\},\\[5pt]
    \Lambda_{2n+1}&=&\{s\in\frac{1}{2}\Bbb{Z}\colon -n-\frac{1}{2}\leq s\leq n\}.
\end{array}
\end{eqnarray*}
In \cite{CF} the authors have shown that smash
product $A\#H$ for a finite dimensional Hopf algebra $H$ acting on an algebra $A$ is semisimple if $H$ and $A$ are semisimple.
Hence, $\mathcal{F}(\Lambda_n)\rtimes D(G)$ are semisimple.
Moreover, $\mathcal{F}(\Lambda_n)\rtimes D(G)$ are finite dimensional $C^*$-algebras.
The natural embeddings $\iota_n\colon {\mathcal{F}}(\Lambda_n)\rtimes D(G)\rightarrow{\mathcal{F}}(\Lambda_{n+1})\rtimes D(G)$, which identify the $T$ and $(g,h)$, are norm preserving.
Using the $C^*$-inductive limit \cite{B.R.Li}, a crossed product $C^*$-algebra ${\mathcal{F}}\rtimes D(G)$ can be given by
\begin{eqnarray*}
    \begin{array}{c}
    {\mathcal{F}}\rtimes D(G)\ =\ \overline{\bigcup\limits_{n}
(\mathcal{F}(\Lambda_{n})\rtimes D(G))}.
 \end{array}
\end{eqnarray*}


To make further investigation about a $C^*$-algebra  ${\mathcal{F}}\rtimes D(G)$, we first study a local net structure. For $\Lambda_{\frac{1}{2},2}\subset \frac{1}{2}\Bbb{Z}$, we can construct the crossed product $C^*$-algebra ${\mathcal{F}}(\Lambda_{\frac{1}{2},2})\rtimes D(G)$.

Using the linear basis $\delta_{g_1}(1)\delta_{g_2}(2)\rho_{h_1}
(\frac{1}{2})\rho_{h_2}(\frac{3}{2})\rtimes (g,h)\in{\mathcal{F}}(\Lambda_{\frac{1}{2},2})\rtimes D(G)$, the
 multiplication and $*$-operation
are given as follows
\begin{eqnarray*}
    \begin{array}{rcl}
&&\Big(\delta_{g_1}(1)\delta_{g_2}(2)\rho_{h_1}
(\frac{1}{2})\rho_{h_2}(\frac{3}{2})\rtimes (g,h)\Big)
\Big(\delta_{s_1}(1)\delta_{s_2}(2)\rho_{t_1}(\frac{1}{2})
\rho_{t_2}(\frac{3}{2})\rtimes (s,t)\Big)\\[5pt]
&=&\sum\limits_{(g,h)}\delta_{g_1}(1)\delta_{g_2}(2)\rho_{h_1}
(\frac{1}{2})\rho_{h_2}(\frac{3}{2}) (g,h)_{(1)}(\delta_{s_1}(1)\delta_{s_2}(2)\rho_{t_1}(\frac{1}{2})
\rho_{t_2}(\frac{3}{2}))\rtimes (g,h)_{(2)}(s,t)\\[5pt]
&=&\sum\limits_{f\in G}\delta_{t_1t_2,h^{-1}fh}
\delta_{g_1}(1)\delta_{g_2}(2)\rho_{h_1}
(\frac{1}{2})\rho_{h_2}(\frac{3}{2})
\delta_{hs_1}(1)\delta_{hs_2}(2)\rho_{ht_1h^{-1}}(\frac{1}{2})
\rho_{ht_2h^{-1}}(\frac{3}{2})\\[5pt]
&=&\sum\limits_{f\in G}\delta_{t_1t_2,h^{-1}fh}
\delta_{g_1}(1)\delta_{h_1hs_1}(1)\delta_{g_2}(2)\delta_{h_1h_2hs_2}(2)
\rho_{h_1ht_1h^{-1}}(\frac{1}{2})
\rho_{ht_1^{-1}h^{-1}h_2ht_1t_2h^{-1}}(\frac{3}{2})\rtimes
\delta_{f^{-1}gh,hs}(f^{-1}g,ht)\\[5pt]
&=&\delta_{h^{-1}gh,t_1t_2s}\delta_{g_1,h_1hs_1}\delta_{g_2,h_1h_2hs_2}
\delta_{g_1}(1)\delta_{g_2}(2)
\rho_{h_1ht_1h^{-1}}(\frac{1}{2})
\rho_{ht_1^{-1}h^{-1}h_2ht_1t_2h^{-1}}(\frac{3}{2})\rtimes
(hsh^{-1},ht),
     \end{array}
\end{eqnarray*}
where the first equation uses the relation
\begin{eqnarray*}
    \begin{array}{rcl}
&& (f,h)\Big(\delta_{s_1}(1)\delta_{s_2}(2)\rho_{t_1}(\frac{1}{2})
\rho_{t_2}(\frac{3}{2})\Big)\\[5pt]
&=&\sum\limits_{(f)}f_{(1)}\delta_{s_1}(1)
f_{(2)}\delta_{s_2}(2)
 f_{(3)}\rho_{t_1}(\frac{1}{2})
 f_{(4)}\rho_{t_2}(\frac{3}{2})\\[5pt]
&=&\sum\limits_{x_1,x_2,x_3\in G}(x_3,h)\delta_{s_1}(1)
(x_3^{-1}x_2,h)\delta_{s_2}(2)
 (x_2^{-1}x_1,h)\rho_{t_1}(\frac{1}{2}) (x_1^{-1}f,h)\rho_{t_2}(\frac{3}{2})\\[5pt]
&=&\delta_{x_3,u}\delta_{hs_1}(1)\delta_{x_3^{-1}x_2,u}\delta_{hs_2}(2)
\delta_{x_2^{-1}x_1,ht_1h^{-1}}\rho_{ht_1h^{-1}}(\frac{1}{2})
\delta_{x_1^{-1}f,ht_2h^{-1}}\rho_{ht_2h^{-1}}(\frac{3}{2})\\[5pt]
&=&\delta_{h^{-1}fh,t_1t_2}\delta_{hs_1}(1)\delta_{hs_2}(2)
\rho_{ht_1h^{-1}}(\frac{1}{2})\rho_{ht_2h^{-1}}(\frac{3}{2}),
    \end{array}
\end{eqnarray*}
and
\begin{eqnarray*}
    \begin{array}{rcl}
    &&\Big(\delta_{g_1}(1)\delta_{g_2}(2)\rho_{h_1}
(\frac{1}{2})\rho_{h_2}(\frac{3}{2})\rtimes (g,h)\Big)^*\\[5pt]
&=&\Big(\sum\limits_{s_1,s_2\in G}\delta_{s_1}(1)\delta_{s_2}(2)\rtimes (h^{-1}gh,h^{-1})\Big)
\Big(\delta_{h_1g_1^{-1}}(1)\delta_{h_2^{-1}h_1^{-1}g_2}(2)\rho_{h_1^{-1}}
(\frac{1}{2})\rho_{h_1h_2^{-1}h_1^{-1}}(\frac{3}{2})\rtimes \sum\limits_{s\in G}(s,u) \Big)\\[5pt]
&=&\sum\limits_{s_1,s_2,s\in G}\delta_{g,h_2^{-1}h_1^{-1}s}\delta_{s_1,h^{-1}h_1^{-1}g_1}
\delta_{s_2,h^{-1}h_2^{-1}h_1^{-1}g_2}
\delta_{s_1}(1)\delta_{s_2}(2)\rho_{h^{-1}h_1^{-1}h}(\frac{1}{2})
\rho_{h^{-1}h_1h_2^{-1}h_1^{-1}h}(\frac{3}{2})\rtimes (h^{-1}sh,h^{-1})\\[5pt]
&=&\delta_{h^{-1}h_1^{-1}g_1}(1)\delta_{h^{-1}h_2^{-1}h_1^{-1}g_2}(2)
\rho_{h^{-1}h_1^{-1}h}(\frac{1}{2})
\rho_{h^{-1}h_1h_2^{-1}h_1^{-1}h}(\frac{3}{2})
\rtimes (h^{-1}h_1h_2gh,h^{-1}).
         \end{array}
\end{eqnarray*}

Now, we consider a special element $I_{\mathcal{F}}\rtimes \frac{1}{|G|}\sum\limits_{g\in G}(u,g)$ in ${\mathcal{F}}\rtimes D(G)$.

\begin{lemma}
The element $I_{\mathcal{F}}\rtimes \frac{1}{|G|}\sum\limits_{g\in G}(u,g)$ is a self-adjoint idempotent element. That is
\begin{eqnarray*}
 \begin{array}{c}
\Big(I_{\mathcal{F}}\rtimes \frac{1}{|G|}\sum\limits_{g\in G}(u,g)\Big)^2\ =\
I_{\mathcal{F}}\rtimes \frac{1}{|G|}\sum\limits_{g\in G}(u,g) \ =\
\Big(I_{\mathcal{F}}\rtimes \frac{1}{|G|}\sum\limits_{g\in G}(u,g)\Big)^*.
       \end{array}
\end{eqnarray*}
\end{lemma}
\begin{proof}
We can compute that
\begin{eqnarray*}
 \begin{array}{rcl}
&&\Big(I_{\mathcal{F}}\rtimes \frac{1}{|G|}\sum\limits_{g\in G}(u,g)\Big)^2\\[5pt]
&=&\Big(\sum\limits_{g_1,g_2\in G}\delta_{g_1}(1)\delta_{g_2}(2)\rtimes
 \frac{1}{|G|}\sum\limits_{h\in G}(u,h) \Big)
 \Big(\sum\limits_{s_1,s_2\in G}\delta_{s_1}(1)\delta_{s_2}(2)\rtimes
 \frac{1}{|G|}\sum\limits_{t\in G}(u,t) \Big)\\[5pt]
&=&\frac{1}{|G|^2}\delta_{g_1,hs_1}\delta_{g_2,hs_2}
\sum\limits_{g_1,g_2,s_1,s_2,h,t\in G}
\delta_{g_1}(1)\delta_{g_2}(2)\rtimes (u,ht)\\[5pt]
&=&\frac{1}{|G|^2}\sum\limits_{g_1,g_2,h,t\in G}
\delta_{g_1}(1)\delta_{g_2}(2)\rtimes (u,ht)\\[5pt]
&=&\frac{1}{|G|}\Big(I_{\mathcal{F}}\rtimes \sum\limits_{h\in G}(u,h)\Big),
        \end{array}
\end{eqnarray*}
and
\begin{eqnarray*}
 \begin{array}{rcl}
\Big(I_{\mathcal{F}}\rtimes \frac{1}{|G|}\sum\limits_{g\in G}(u,g)\Big)^*
&=&\frac{1}{|G|}\Big(I_{\mathcal{F}}\rtimes\sum\limits_{g\in G}(u,g)^*\Big)
\Big(I_{\mathcal{F}}\rtimes I_{D(G)}\Big)\\[5pt]
&=&\frac{1}{|G|}\Big(I_{\mathcal{F}}\rtimes\sum\limits_{g\in G}(u,g^{-1})\Big)
\Big(I_{\mathcal{F}}\rtimes I_{D(G)}\Big)\\[5pt]
&=&\frac{1}{|G|}\Big(I_{\mathcal{F}}\rtimes\sum\limits_{g\in G}(u,g)\Big).
        \end{array}
\end{eqnarray*}
Hence, $I_{\mathcal{F}}\rtimes \frac{1}{|G|}\sum\limits_{g\in G}(u,g)$ is a self-adjoint idempotent element.
\end{proof}


\begin{lemma}
The element
$T\rtimes I_{D(G)}$ in ${\mathcal{F}}\rtimes D(G)$ satisfies the following covariant relation

$\Big(I_{\mathcal{F}}\rtimes \frac{1}{|G|}\sum\limits_{g\in G}(u,g)\Big)\Big(T\rtimes I_{D(G)}\Big)\Big(I_{\mathcal{F}}\rtimes \frac{1}{|G|}\sum\limits_{g\in G}(u,g)\Big)
=\Big(E(T)\rtimes I_{D(G)}\Big)\Big(I_{\mathcal{F}}\rtimes \frac{1}{|G|}\sum\limits_{g\in G}(u,g)\Big)$.
\end{lemma}

\begin{proof}
Suppose that $T=\delta_{s_1}(1)\delta_{s_2}(2)
\rho_{t_1}(\frac{1}{2})\rho_{t_2}(\frac{3}{2})$, we can obtain
\begin{eqnarray*}
 \begin{array}{rcl}
&&\Big(I_{\mathcal{F}}\rtimes \frac{1}{|G|}\sum\limits_{g\in G}(u,g)\Big)\Big(T\rtimes I_{D(G)}\Big)\\[5pt]
&=&\frac{1}{|G|}\Big(\sum\limits_{g_1,g_2\in G} \delta_{g_1}(1)\delta_{g_2}(2)\rtimes\sum\limits_{g\in G}(u,g) \Big)\Big(\delta_{s_1}(1)\delta_{s_2}(2)
\rho_{t_1}(\frac{1}{2})\rho_{t_2}(\frac{3}{2})\rtimes\sum\limits_{s\in G}(s,u)\Big)\\[5pt]
&=&\frac{1}{|G|}\sum\limits_{g_1,g_2,g,s\in G}
\delta_{u,t_1t_2s}\delta_{g_1,gs_1}\delta_{g_2,gs_2}
\delta_{g_1}(1)\delta_{g_2}(2)\rho_{gt_1g^{-1}}(\frac{1}{2})
\rho_{gt_2g^{-1}}(\frac{3}{2})\rtimes(gsg^{-1},g)\\[5pt]
&=&\frac{1}{|G|}\sum\limits_{g\in G}
\delta_{gs_1}(1)\delta_{gs_2}(2)\rho_{gt_1g^{-1}}(\frac{1}{2})
\rho_{gt_2g^{-1}}(\frac{3}{2})\rtimes(gt_2^{-1}t_1^{-1}g^{-1},g),
       \end{array}
\end{eqnarray*}
and then
\begin{eqnarray*}
 \begin{array}{rcl}
&&\Big(I_{\mathcal{F}}\rtimes \frac{1}{|G|}\sum\limits_{g\in G}(u,g)\Big)\Big(T\rtimes I_{D(G)}\Big)\Big(I_{\mathcal{F}}\rtimes \frac{1}{|G|}\sum\limits_{g\in G}(u,g)\Big)\\[5pt]
&=&\frac{1}{|G|}\sum\limits_{g\in G}
\delta_{gs_1}(1)\delta_{gs_2}(2)\rho_{gt_1g^{-1}}(\frac{1}{2})
\rho_{gt_2g^{-1}}(\frac{3}{2})\rtimes(gt_2^{-1}t_1^{-1}g^{-1},g)
(\sum\limits_{g_1,g_2\in G}\delta_{g_1}(1)\delta_{g_2}(2)\rtimes\frac{1}{|G|}\sum\limits_{g\in G}(u,g) )\\[5pt]
&=&\frac{1}{|G|^2}\sum\limits_{g,g_1,g_2,f\in G}
\delta_{t_1t_2,u}\delta_{s_1,t_1g_1}\delta_{s_2,g_2}
\delta_{gs_1}(1)\delta_{gs_2}(2)\rho_{gt_1g^{-1}}(\frac{1}{2})
\rho_{gt_2g^{-1}}(\frac{3}{2})\rtimes(u,gf)\\[5pt]
&=&\frac{1}{|G|^2}\delta_{t_1t_2,u}\sum\limits_{g,f\in G}
\delta_{gs_1}(1)\delta_{gs_2}(2)\rho_{gt_1g^{-1}}(\frac{1}{2})
\rho_{gt_2g^{-1}}(\frac{3}{2})\rtimes(u,gf)\\[5pt]
&=&\frac{1}{|G|^2}\delta_{t_1t_2,u}\sum\limits_{g,f\in G}
\delta_{gs_1}(1)\delta_{gs_2}(2)\rho_{gt_1g^{-1}}(\frac{1}{2})
\rho_{gt_2g^{-1}}(\frac{3}{2})\rtimes(u,f).
       \end{array}
\end{eqnarray*}
On the other hand,
\begin{eqnarray*}
 \begin{array}{rcl}
&& E(T)=E\Big(\delta_{s_1}(1)\delta_{s_2}(2)
\rho_{t_1}(\frac{1}{2})\rho_{t_2}(\frac{3}{2})\Big)\\[5pt]
&=&\frac{1}{|G|}\delta_{t_1t_2,u}\sum\limits_{f\in G}\delta_{fs_1}(1)\delta_{fs_2}(2)
\rho_{ft_1f^{-1}}(\frac{1}{2})\rho_{ft_2f^{-1}}(\frac{3}{2}),
        \end{array}
\end{eqnarray*}
and then
\begin{eqnarray*}
 \begin{array}{rcl}
&&\Big(E(T)\rtimes I_{D(G)}\Big)\Big(I_{\mathcal{F}}\rtimes \frac{1}{|G|}\sum\limits_{g\in G}(u,g)\Big)\\[5pt]
&=&\Big(\frac{1}{|G|}\delta_{t_1t_2,u}\sum\limits_{f\in G}\delta_{fs_1}(1)\delta_{fs_2}(2)
\rho_{ft_1f^{-1}}(\frac{1}{2})\rho_{ft_2f^{-1}}(\frac{3}{2})\rtimes
 \sum\limits_{s\in G}(s,u)\Big)\Big(\sum\limits_{g_1,g_2\in G} \delta_{g_1}(1)\delta_{g_2}(2)\rtimes \frac{1}{|G|}\sum\limits_{t\in G}(u,t)\Big)\\[5pt]
&=&\frac{1}{|G|^2}\delta_{t_1t_2,u}
\sum\limits_{f,t,g_1,g_2,s\in G}
\delta_{s,u}\delta_{s_1,t_1f^{-1}g_1}\delta_{fs_2,g_2}
\delta_{fs_1}(1)\delta_{fs_2}(2)
\rho_{ft_1f^{-1}}(\frac{1}{2})\rho_{ft_2f^{-1}}(\frac{3}{2})
\rtimes(u,t)\\[5pt]
&=&\frac{1}{|G|^2}\delta_{t_1t_2,u}
\sum\limits_{f,t\in G}
\delta_{fs_1}(1)\delta_{fs_2}(2)
\rho_{ft_1f^{-1}}(\frac{1}{2})\rho_{ft_2f^{-1}}(\frac{3}{2})
\rtimes(u,t).
        \end{array}
\end{eqnarray*}
From the above, we can obtain the desired result.
\end{proof}

The following theorem is one of main results of this paper, which gives a characterization of the $C^*$-algebra $\langle{\mathcal{F}},e_{\mathcal{A}}\rangle_{C^*}$ constructed from the $C^*$-basic construction for the inclusion ${\mathcal{A}}\subseteq {\mathcal{F}}$.

\begin{theorem}
There exists a $C^*$-isomorphism
between the crossed product $C^*$-algebra ${\mathcal{F}}\rtimes D(G)$ and
the $C^*$-algebra $\langle{\mathcal{F}},e_{\mathcal{A}}\rangle_{C^*}$. That is,
$${\mathcal{F}}\rtimes D(G)\cong \langle{\mathcal{F}},e_{\mathcal{A}}\rangle_{C^*}.$$
\end{theorem}
\begin{proof}
Let $$\Phi_{\frac{1}{2},2}\colon \langle {\mathcal{F}}(\Lambda_{\frac{1}{2},2}),e_{\mathcal{A}}
\rangle_{C^*}\rightarrow {\mathcal{F}}(\Lambda_{\frac{1}{2},2})\rtimes D(G)$$ be a map with
$$\begin{array}{ccc}
T&\mapsto&T\rtimes I_{D(G)},\\[5pt]
e_{\mathcal{A}}&\mapsto&I_{\mathcal{F}}\rtimes\frac{1}{|G|}\sum\limits_{g\in G}(u,g).
  \end{array}$$

Firstly, $\Phi_{\frac{1}{2},2}$ is well-defined and can be linearly extended in $\langle {\mathcal{F}}(\Lambda_{\frac{1}{2},2}),e_{\mathcal{A}}\rangle_{C^*}$ preserving algebraic structure. Actually, by Lemma 2.2 and Lemma 3.2, we obtain that $$\Phi_{\frac{1}{2},2}(e_{\mathcal{A}}Te_{\mathcal{A}})=
\Phi_{\frac{1}{2},2}(e_{\mathcal{A}})
\Phi_{\frac{1}{2},2}(T)\Phi_{\frac{1}{2},2}(e_{\mathcal{A}})
=\Phi_{\frac{1}{2},2}(E(T)e_{\mathcal{A}}).$$


 Also we have for $T_1=\delta_{g_1}(1)\delta_{g_2}(2)
\rho_{h_1}(\frac{1}{2})\rho_{h_2}(\frac{3}{2}),T_2=\delta_{s_1}(1)\delta_{s_2}(2)
\rho_{t_1}(\frac{1}{2})\rho_{t_2}(\frac{3}{2})$,
\begin{eqnarray*}
    \begin{array}{rcl}
T_1T_2
&=&\delta_{g_1}(1)\delta_{g_2}(2)
\rho_{h_1}(\frac{1}{2})\rho_{h_2}(\frac{3}{2})\delta_{s_1}(1)\delta_{s_2}(2)
\rho_{t_1}(\frac{1}{2})\rho_{t_2}(\frac{3}{2})\\[5pt]
&=&\delta_{g_1,h_1s_1}\delta_{g_2,h_1h_2s_2}
\delta_{g_1}(1)\delta_{g_2}(2)
\rho_{h_1t_1}(\frac{1}{2})\rho_{t_1^{-1}h_2t_1t_2}(\frac{3}{2})
  \end{array}
\end{eqnarray*}
and
\begin{eqnarray*}
    \begin{array}{rcl}
&&\Phi_{\frac{1}{2},2}(T_1)\Phi_{\frac{1}{2},2}(T_2)\\[5pt]
&=&\Big(\delta_{g_1}(1)\delta_{g_2}(2)
\rho_{h_1}(\frac{1}{2})\rho_{h_2}(\frac{3}{2})\rtimes I_{D(G)}\Big)
\Big(\delta_{s_1}(1)\delta_{s_2}(2)
\rho_{t_1}(\frac{1}{2})\rho_{t_2}(\frac{3}{2})\rtimes I_{D(G)}\Big)\\[5pt]
&=&\Big(\delta_{g_1}(1)\delta_{g_2}(2)
\rho_{h_1}(\frac{1}{2})\rho_{h_2}(\frac{3}{2})\rtimes \sum\limits_{g\in G}(g,u)\Big)
\Big(\delta_{s_1}(1)\delta_{s_2}(2)
\rho_{t_1}(\frac{1}{2})\rho_{t_2}(\frac{3}{2})\rtimes \sum\limits_{s\in G}(s,u)\Big)\\[5pt]
&=&\sum\limits_{g,s\in G}
\delta_{g,t_1t_2s}\delta_{g_1,h_1s_1}\delta_{g_2,h_1h_2s_2}
\delta_{g_1}(1)\delta_{g_2}(2)
\rho_{h_1t_1}(\frac{1}{2})\rho_{t_1^{-1}h_2t_1t_2}(\frac{3}{2})
\rtimes (s,u)\\[5pt]
&=&\delta_{g_1,h_1s_1}\delta_{g_2,h_1h_2s_2}
\delta_{g_1}(1)\delta_{g_2}(2)
\rho_{h_1t_1}(\frac{1}{2})\rho_{t_1^{-1}h_2t_1t_2}(\frac{3}{2})
\rtimes I_{D(G)}\\[5pt]
&=&\Phi_{\frac{1}{2},2}(T_1T_2).
  \end{array}
\end{eqnarray*}
Thus, $\Phi_{\frac{1}{2},2}$ is an algebra homomorphism.

Secondly, $\Phi_{\frac{1}{2},2}$ preserves the $*$-operation.
In fact, we can show
for $T=\delta_{g_1}(1)\delta_{g_2}(2)
\rho_{h_1}(\frac{1}{2})\rho_{h_2}(\frac{3}{2})$,
$T^*=\delta_{h_1^{-1}g_1}(1)\delta_{h_2^{-1}h_1^{-1}g_2}(2)
\rho_{h_1^{-1}}(\frac{1}{2})\rho_{h_1h_2^{-1}h_1^{-1}}(\frac{3}{2})$ and
\begin{eqnarray*}
    \begin{array}{rcl}
   \Big(\Phi_{\frac{1}{2},2}(T)\Big)^*
    &=&\sum\limits_{g\in G}
    \delta_{h_1^{-1}g_1}(1)\delta_{h_2^{-1}h_1^{-1}g_2}(2)
\rho_{h_1^{-1}}(\frac{1}{2})\rho_{h_1h_2^{-1}h_1^{-1}}(\frac{3}{2})
\rtimes (h_1h_2g,u)\\[5pt]
 &=&T^*\rtimes I_{D(G)}\\[5pt]
 &=&\Phi_{\frac{1}{2},2}(T^*).
     \end{array}
\end{eqnarray*}
It follows from
Lemma 3.1 that $\Phi_{\frac{1}{2},2}(e_{\mathcal{A}})$ is self-adjoint, which together with $e_{\mathcal{A}}$ is a projection yields that $(\Phi_{\frac{1}{2},2}(e_{\mathcal{A}}))^*
=\Phi_{\frac{1}{2},2}(e_{\mathcal{A}})
=\Phi_{\frac{1}{2},2}(e_{\mathcal{A}}^*)$.

In order to complete the proof we need to show that $\Phi_{\frac{1}{2},2}$ is bijective.
Indeed,
For any $\delta_{g_1}(1)\delta_{g_2}(2)
\rho_{h_1}(\frac{1}{2})\rho_{h_2}(\frac{3}{2})\rtimes(g,h)\in {\mathcal{F}}(\Lambda_{\frac{1}{2},2})\rtimes D(G)$, choose

$\Big(|G|\delta_{g_1}(1)\delta_{g_2}(2)
\rho_{h_1}(\frac{1}{2})\rho_{h_2g}(\frac{3}{2})\Big)
e_{\mathcal{A}}\Big(\delta_{h^{-1}h_1^{-1}g_1}(1)
\delta_{h^{-1}g^{-1}h_2^{-1}h_1^{-1}g_2}(2)
\rho_{u}(\frac{1}{2})\rho_{h^{-1}g^{-1}h}(\frac{3}{2})\Big)\in \langle {\mathcal{F}}(\Lambda_{\frac{1}{2},2}),e_{\mathcal{A}}\rangle_{C^*}$ such that \begin{eqnarray*}
    \begin{array}{rcl}
&&\Phi_{\frac{1}{2},2}
\Big((|G|\delta_{g_1}(1)\delta_{g_2}(2)
\rho_{h_1}(\frac{1}{2})\rho_{h_2g}(\frac{3}{2}))
e_{\mathcal{A}}(\delta_{h^{-1}h_1^{-1}g_1}(1)
\delta_{h^{-1}g^{-1}h_2^{-1}h_1^{-1}g_2}(2)
\rho_{u}(\frac{1}{2})\rho_{h^{-1}g^{-1}h}(\frac{3}{2}))
\Big)\\[5pt]
&=&\delta_{g_1}(1)\delta_{g_2}(2)
\rho_{h_1}(\frac{1}{2})\rho_{h_2}(\frac{3}{2})\rtimes(g,h),
      \end{array}
\end{eqnarray*}
which implies that $\Phi_{\frac{1}{2},2}$ is surjective,
and then
$\dim{\langle {\mathcal{F}}(\Lambda_{\frac{1}{2},2}),e_{\mathcal{A}}\rangle_{C^*}}\geq |G|^6$. Also we can show that every element
$$\delta_{g_1}(1)\delta_{g_2}(2)
\rho_{h_1}(\frac{1}{2})\rho_{h_2}(\frac{3}{2}))e_{\mathcal{A}}
(\delta_{s_1}(1)\delta_{s_2}(2)
\rho_{t_1}(\frac{1}{2})\rho_{t_2}(\frac{3}{2})\in\langle {\mathcal{F}}(\Lambda_{\frac{1}{2},2}),e_{\mathcal{A}}\rangle_{C^*},$$ as a operator on ${\mathcal{F}}(\Lambda_{\frac{1}{2},2})$, is equal to the element
$$\delta_{f_1}(1)\delta_{f_2}(2)
\rho_{\varphi_1}(\frac{1}{2})\rho_{\varphi_2g}(\frac{3}{2}))e_{\mathcal{A}}
(\delta_{h^{-1}\varphi_1^{-1}f_1}(1)
\delta_{h^{-1}g^{-1}\varphi_2^{-1}\varphi_1^{-1}f_2}(2)
\rho_{u}(\frac{1}{2})\rho_{h^{-1}g^{-1}h}(\frac{3}{2})\in\langle {\mathcal{F}}(\Lambda_{\frac{1}{2},2}),e_{\mathcal{A}}\rangle_{C^*},$$ where
\begin{eqnarray*}
    \begin{array}{rcl}
    f_1&=&g_1, \ \ \ \  f_2=g_2, \\[5pt]
    \varphi_1&=&g_1s_1^{-1}t_1s_1g_1^{-1}h_1,\\[5pt]
    \varphi_2&=&h_1^{-1}g_1s_1^{-1}t_1^{-1}s_1g_1^{-1}h_1h_2h_1^{-1}g_1 s_1^{-1}t_1t_2s_1g_1^{-1}h_1,\\[5pt]
    g&=&h_1^{-1}g_1s_1^{-1}t_2^{-1}t_1^{-1}s_1g_1^{-1}h_1,\\[5pt]
    h&=&h_1^{-1}g_1s_1^{-1}.
      \end{array}
\end{eqnarray*}

Hence, $\Phi_{\frac{1}{2},2}$ is a $C^*$-homomorphism and is bijective. By Theorem 2.1.7 in \cite{GJM}, $\Phi_{\frac{1}{2},2}$ is a $C^*$-isomorphism from $\langle {\mathcal{F}}(\Lambda_{\frac{1}{2},2}),e_{\mathcal{A}}
\rangle_{C^*}$ onto ${\mathcal{F}}(\Lambda_{\frac{1}{2},2})\rtimes D(G)$.

By induction, we can build a $C^*$-isomorphism
$\Phi_{\frac{n}{2},m}$ between $\langle {\mathcal{F}}(\Lambda_{\frac{n}{2},m}),e_{\mathcal{A}}\rangle_{C^*}$ and ${\mathcal{F}}(\Lambda_{\frac{n}{2},m})\rtimes D(G)$ for any $n,m\in\Bbb{Z}$ with $n<m$.

Now, because of the last relation we can define a $C^*$-isomorphism
\begin{eqnarray*}
    \begin{array}{c}
\Phi\colon \bigcup\limits_{n<m}
\langle {\mathcal{F}}(\Lambda_{\frac{n}{2},m}),e_{\mathcal{A}}\rangle_{C^*}
\rightarrow
\bigcup\limits_{n<m}({\mathcal{F}}(\Lambda_{\frac{n}{2},m})\rtimes D(G))
       \end{array}
\end{eqnarray*}
by $\Phi|_{\langle {\mathcal{F}}(\Lambda_{\frac{n}{2},m}),e_{\mathcal{A}}\rangle_{C^*}}
=\Phi_{\frac{n}{2},m}$.
Since each $\Phi_{\frac{n}{2},m}$ is an isometry, $\Phi$ is an isometry between $\bigcup\limits_{n<m}
\langle {\mathcal{F}}(\Lambda_{\frac{n}{2},m}),e_{\mathcal{A}}\rangle_{C^*}$ and $\bigcup\limits_{n<m}({\mathcal{F}}(\Lambda_{\frac{n}{2},m})\rtimes D(G))$.
Then by Theorem 2.7 in \cite{O.B}, The map $\Phi$
can be extended to an isomorphism
from $\overline{\bigcup\limits_{n<m}\langle {\mathcal{F}}(\Lambda_{\frac{n}{2},m}),e_{\mathcal{A}}\rangle_{C^*}}$ onto $\overline{\bigcup\limits_{n<m}({\mathcal{F}}(\Lambda_{\frac{n}{2},m})\rtimes D(G))}$.

Finally, the uniqueness of the $C^*$-inductive limit \cite{B.R.Li} implies that $\langle {\mathcal{F}},e_{\mathcal{A}}\rangle_{C^*}
=\overline{\bigcup\limits_{n<m}\langle {\mathcal{F}}(\Lambda_{\frac{n}{2},m}),e_{\mathcal{A}}\rangle_{C^*}}$
and ${\mathcal{F}}\rtimes D(G)=\overline{\bigcup\limits_{n<m}
({\mathcal{F}}(\Lambda_{\frac{n}{2},m})
\rtimes D(G))}$.
Consequently, the $C^*$-algebra $\langle {\mathcal{F}},e_{\mathcal{A}}\rangle_{C^*}$ constructed from the $C^*$-basic construction for the inclusion ${\mathcal{A}}\subseteq {\mathcal{F}}$
 is $C^*$-isomorphic to the crossed product $C^*$-algebra ${\mathcal{F}}\rtimes D(G)$.
\end{proof}

\begin{remark}
From Theorem 3.1, we know that the $C^*$-basic constructions do not depend on the choice of conditional expectations, which can also be seen in Proposition 2.10.11 \cite{Y.Wat}.
\end{remark}

\section{The conditional expectation from ${\mathcal{F}}\rtimes D(G)$ onto ${\mathcal{F}}$}

Let $H(m,\iota,S,\bigtriangleup,\varepsilon)$ be a Hopf $C^*$-algebra with finite dimension,
where $m,\iota,S,\bigtriangleup,\varepsilon$ denote  multiplication, unit, antipode, comultiplication and counit, respectively.
 The dual $\widehat{H}$ of $H$ is also a finite Hopf $C^*$-algebra with multiplication $\widetilde{m}$, unit $\widetilde{\iota}$, antipode $\widetilde{S}$, comultiplication $\widetilde{\bigtriangleup}$,
and counit $\widetilde{\varepsilon}$ defined by
\begin{eqnarray*}
 \begin{array}{rcl}
    \langle\widetilde{\bigtriangleup}(\varphi),x\otimes y\rangle
&:=&\langle\varphi,m(x\otimes y)\rangle,\\[5pt]
    \langle\widetilde{m}(\varphi\otimes\phi),x\rangle
&:=&\langle\varphi\otimes\phi,\bigtriangleup(x)\rangle, \\[5pt]
    \langle\varphi^*,x\rangle
&:=&\overline{\langle\varphi,S(x)^*\rangle},\\[5pt]
    \widetilde{\varepsilon}(\varphi)
&:=&\langle\varphi,1_H\rangle,\\[5pt]
    \langle 1_{\widehat{H}},x\rangle
&:=&\varepsilon(x),\\[7pt]
    \langle\widetilde{S}(\varphi),x\rangle
&:=&\langle\varphi,S(x)\rangle.
 \end{array}
\end{eqnarray*}

Since $D(G)$ is of finite dimension and $\widehat{{C(G)\otimes \Bbb{C}G}}\cong \Bbb{C}G\otimes C(G)$ as algebras, $\{(y,\delta_x)\colon y,x\in G\}$ can be viewed as a linear basis of $\widehat{D(G)}$. As the above states, the structure maps on $\widehat{D(G)}$ are the following
\begin{eqnarray*}
 \begin{array}{rcll}
   \widetilde{\bigtriangleup}(y,\delta_x)
&=&\sum \limits_{t\in G}(y,\delta_{t^{-1}})\otimes(tyt^{-1},\delta_{tx}),
&(\mbox{coproduct})\\
   (y,\delta_x)(w,\delta_z)
&=&\delta_{x,z}(yw,\delta_x),&(\mbox{multiplication})\\[5pt]
   (y,\delta_x)^*
&=&(y^{-1},\delta_x),&(\mbox{*-operation})\\[5pt]
    \widetilde{\varepsilon}(y,\delta_x)
&=&\delta_{x,u},&(\mbox{counit})\\[5pt]
   \widetilde{S}(y,\delta_x)
&=&(x^{-1}y^{-1}x,\delta_{x^{-1}}).&(\mbox{antipode})
 \end{array}
\end{eqnarray*}
It is easy to see that $I_{\widehat{D(G)}}=
\sum\limits_{x\in G}(u,\delta_x)$, and there is a unique element $E_2=\frac{1}{|G|}\sum\limits_{y\in G}(y,\delta_u)$, called an integral element, satisfying for any $b\in \widehat{D(G)}$,
$$bE_2=E_2b=\varepsilon(b)E_2.$$

The map $\sigma\colon \widehat{D(G)}\times ({\mathcal{F}}\rtimes D(G))\rightarrow {\mathcal{F}}\rtimes D(G)$ given on the generating elements of ${\mathcal{F}}\rtimes D(G)$ as
$$\sigma\Big((y,\delta_x)\times(F\otimes (g,h))\Big)=\delta_{x,h}\Big(F\otimes(gy^{-1},h)\Big)$$
for $(g,h)\in D(G)$, can be linearly extended both in $\widehat{D(G)}$ and ${\mathcal{F}}\rtimes D(G)$.
Here and from now on, by $(y,\delta_x)(F\otimes (g,h))$ we always denote $\sigma((y,\delta_x)\times(F\otimes (g,h)))$ for notational
convenience.

In particular, considering the action of $E_2$ on ${\mathcal{F}}\rtimes D(G)$, we can obtain that
\begin{eqnarray*}
 \begin{array}{c}
E_2(F\otimes (g,h))
 =\frac{1}{|G|}\sum\limits_{y\in G}(y,\delta_u)(F\otimes (g,h))
 =\frac{1}{|G|}\sum\limits_{y\in G}\delta_{h,u}(F\otimes(gy^{-1},h))
 =\frac{1}{|G|}\delta_{h,u}(F\otimes I_{D(G)}),
\end{array}
\end{eqnarray*}
which means that the range of $E_2$ on ${\mathcal{F}}\rtimes D(G)$ is contained in ${\mathcal{F}}$. Moreover, we can show that
$E_2$ is a positive map preserving the unit and possessing the bimodular property. Namely,

\begin{proposition}
The map $E_2\colon {\mathcal{F}}\rtimes D(G)\rightarrow {\mathcal{F}}$ is a conditional expectation.
\end{proposition}
\begin{proof}
(1)\ $E_2(I_{{\mathcal{F}}\rtimes D(G)})=E_2(I_{\mathcal{F}}\otimes \sum\limits_{g\in G}(g,u))=\frac{1}{|G|}\sum\limits_{g\in G}(I_{\mathcal{F}}\otimes I_{D(G)})=I_{\mathcal{F}}.$

 (2)\ $\forall T_1, T_2\in{\mathcal{F}}, \widetilde{T}\in {\mathcal{F}}\rtimes D(G)$, we have
 \begin{eqnarray*}
    \begin{array}{rcl}
 E_2(T_1\widetilde{T}T_2)
 &=&\frac{1}{|G|}\sum\limits_{y\in G}(y,\delta_u)(T_1\widetilde{T}T_2)\\[5pt]
 &=&\frac{1}{|G|}\sum\limits_{y,t_1,t_2\in G}(y,\delta_{t_1^{-1}})(T_1)(t_1yt_1^{-1},\delta_{t_1t_2^{-1}})
 (\widetilde{T})
 (t_2yt_2^{-1},\delta_{t_2})(T_2)\\[5pt]
 &=&\frac{1}{|G|}\sum\limits_{y,t_1,t_2\in G}
 \widetilde{\varepsilon}(y,\delta_{t_1^{-1}})(T_1)
 (t_1yt_1^{-1},\delta_{t_1t_2^{-1}})(\widetilde{T})
 \widetilde{\varepsilon}(t_2yt_2^{-1},\delta_{t_2})(T_2)\\[5pt]
 &=&\frac{1}{|G|}\sum\limits_{y,t_1,t_2\in G}
 \delta_{t_1^{-1},u}(T_1)
 (t_1yt_1^{-1},\delta_{t_1t_2^{-1}})(\widetilde{T})
 \delta_{t_2,u}(T_2)\\[5pt]
 &=&\frac{1}{|G|}\sum\limits_{y\in G}T_1(y,\delta_u)(\widetilde{T})T_2\\[5pt]
 &=&T_1E_2(\widetilde{T})T_2.
 \end{array}
\end{eqnarray*}

(3)\ We note the relation for any
 $\widetilde{T}\in {\mathcal{F}}\rtimes D(G)$,
 \begin{eqnarray*}
    \begin{array}{rcl}
    E_2(\widetilde{T}^*\widetilde{T})
 &=&\frac{1}{|G|}\sum\limits_{y\in G}(y,\delta_u)(\widetilde{T}^*\widetilde{T})\\[5pt]
 &=&\frac{1}{|G|}\sum\limits_{y,t\in G}(y,\delta_{t^{-1}})(\widetilde{T}^*)
 (tyt^{-1},\delta_t)(\widetilde{T})\\[5pt]
 &=&\frac{1}{|G|}\sum\limits_{y,t\in G}(\widetilde{S}(y,\delta_{t^{-1}})^*
 (\widetilde{T}))^*(tyt^{-1},\delta_t)(\widetilde{T})\\[5pt]
 &=&\frac{1}{|G|}\sum\limits_{y,t\in G}
 ((tyt^{-1},\delta_t)(F))^*(tyt^{-1},\delta_t)(\widetilde{T}),
  \end{array}
\end{eqnarray*}
which means $E_2$ is a positive map on ${\mathcal{F}}\rtimes D(G)$.
\end{proof}

We recall some basic facts about the index for $C^*$-algebras in \cite{Y.Wat}.

\begin{definition}
Let $\Gamma$ be a conditional expectation from a unital $C^*$-algebra $B$ onto its unital $C^*$-subalgebra $A$. A finite family $\{(u_1,v_1), (u_2,v_2),\cdots, (u_n,v_n)\}\subseteq B\times B$ is called a quasi-basis for $\Gamma$ if for all $b\in B$,
 \begin{eqnarray*}
    \begin{array}{c}
\sum\limits_{i=1}^{n}u_i\Gamma(v_ib)=b=\sum\limits_{i=1}^{n}\Gamma(b u_i)v_i.
    \end{array}
\end{eqnarray*}
Furthermore, if there exists a quasi-basis for $\Gamma$, we call $\Gamma$ of index-finite type. In this case we define the index of $\Gamma$ by
 \begin{eqnarray*}
    \begin{array}{c}
{\mathrm{Index}}\ \Gamma=\sum\limits_{i=1}^nu_iv_i.
      \end{array}
\end{eqnarray*}
\end{definition}

\begin{remark}
(1) If $\Gamma$ is a conditional expectation of index-finite type, then the $C^*$-index ${\mathrm{Index}}\ \Gamma$ is a central element of $B$ and does not depend on the choice of quasi-basis.
In particular, if $A\subset B$ are simple unital $C^*$-algebras, then ${\mathrm{Index}}\ \Gamma$ is a positive scalar.
Moreover, we can choose one of the form $\{(w_i,w_i^*)\colon i=1,2,\cdots, n\}$, which shows that ${\mathrm{Index}}\ \Gamma$ is a positive element \cite{Y.Wat}.

(2) Let $N \subseteq M$ be factors of type II$_1$ and $\Gamma\colon M\rightarrow N$ the canonical conditional expectation determined by the unique normalized trace on $M$, then ${\mathrm{Index}}\ \Gamma$ is exactly Jones index $[M,N]$ based on the coupling constant \cite{M.PP}. More generally, let $M$ be a ($\sigma$-finite) factor with a
subfactor $N$ and $\Gamma$ a normal conditional expectation from $M$
onto $N$, then $\Gamma$ is of index-finite if and only if ${\mathrm{Index}}\ \Gamma$ is finite in the sense of Ref. \cite{H.Ko}, and the values of ${\mathrm{Index}}\ \Gamma$ are equal.
\end{remark}

Next, we consider the conditional expectation from
$\langle {\mathcal{F}},e_{\mathcal{A}}\rangle_{C^*}$ onto the $C^*$-subalgebra ${\mathcal{F}}$ with the common unit, where  $\langle{\mathcal{F}},e_{\mathcal{A}}\rangle_{C^*}$ is the $C^*$-algebra constructed from the $C^*$-basic construction for the inclusion ${\mathcal{A}}\subseteq {\mathcal{F}} $.

\begin{theorem}
The map $\widetilde{E}\colon\langle {\mathcal{F}},e_{\mathcal{A}}\rangle_{C^*}\rightarrow {\mathcal{F}}$ defined on the basis elements of $\langle {\mathcal{F}},e_{\mathcal{A}}\rangle_{C^*}$ by
\begin{eqnarray*}
    \begin{array}{c}
\widetilde{E}(Te_{\mathcal{A}}F)=\frac{1}{|G|^2}TF ,
 \end{array}
   \end{eqnarray*}
linearly extended in $\langle {\mathcal{F}},e_{\mathcal{A}}\rangle_{C^*}$, is a conditional expectation, and we call it the dual conditional expectation of $E\colon{\mathcal{F}}\rightarrow {\mathcal{A}}$.
More precisely, set $$u_{x,y}=|G|^{\frac{3}{2}}\delta_x(k)\rho_y(k+\frac{1}{2})e_{\mathcal{A}}$$ for any $k\in \Bbb{Z}$, then $\{(u_{x,y},u_{x,y}^*)\colon x,y\in G\}$ is a quasi-basis for $\widetilde{E}$.
\end{theorem}
\begin{proof}
Without loss of generality, one can consider the case of $k=1$.

Firstly, one can show that
$\{(u_{x,y},u_{x,y}^*)\colon x,y\in G\}$ is a quasi-basis of $\widetilde{E}\colon\langle {\mathcal{F}}(\Lambda_{{\frac{1}{2}},2}),e_{\mathcal{A}}\rangle_{C^*}\rightarrow {\mathcal{F}}(\Lambda_{{\frac{1}{2}},2})$.
By hypothesis, we have
\begin{eqnarray*}
    \begin{array}{rcl}
&&\widetilde{E}\Big((\delta_{g_1}(1)\delta_{g_2}(2)
   \rho_{h_1}(\frac{1}{2})\rho_{h_2}(\frac{3}{2}))e_{\mathcal{A}}
   (\delta_{s_1}(1)\delta_{s_2}(2)
   \rho_{t_1}(\frac{1}{2})\rho_{t_2}(\frac{3}{2}))\Big)\\[5pt]
&=&\frac{1}{|G|^2}\delta_{g_1,h_1s_1}\delta_{g_2,h_1h_2s_2}
   \delta_{g_1}(1)\delta_{g_2}(2)
   \rho_{h_1t_1}(\frac{1}{2})\rho_{t_1^{-1}h_2t_1t_2}(\frac{3}{2}).
 \end{array}
   \end{eqnarray*}

Note that
 \begin{eqnarray*}
    \begin{array}{rcl}
&&\sum\limits_{x,y\in G}
u_{x,y}\widetilde{E}\Big[u_{x,y}^*\delta_{g_1}(1)\delta_{g_2}(2)
\rho_{h_1}(\frac{1}{2})\rho_{h_2}(\frac{3}{2})e_{\mathcal{A}}
\delta_{s_1}(1)\delta_{s_2}(2)
\rho_{t_1}(\frac{1}{2})\rho_{t_2}(\frac{3}{2})\Big]\\[5pt]
&=&|G|^{\frac{3}{2}}\sum\limits_{x,y\in G}
u_{x,y}\widetilde{E}\Big[e_{\mathcal{A}}\delta_x(1)\rho_{y^{-1}}(\frac{3}{2})
\delta_{g_1}(1)\delta_{g_2}(2)
\rho_{h_1}(\frac{1}{2})\rho_{h_2}(\frac{3}{2})e_{\mathcal{A}}
\delta_{s_1}(1)\delta_{s_2}(2)
\rho_{t_1}(\frac{1}{2})\rho_{t_2}(\frac{3}{2}) \Big]\\[5pt]
&=&|G|^{\frac{3}{2}}\sum\limits_{x,y\in G}
u_{x,y}\widetilde{E}\Big[e_{\mathcal{A}}\delta_x(1)\delta_{g_1}(1)
\delta_{y^{-1}g_2}(2)
\rho_{h_1}(\frac{1}{2})\rho_{h_1^{-1}y^{-1}h_1h_2}(\frac{3}{2})e_{\mathcal{A}}
\delta_{s_1}(1)\delta_{s_2}(2)
\rho_{t_1}(\frac{1}{2})\rho_{t_2}(\frac{3}{2}) \Big]\\[5pt]
&=&|G|\sum\limits_{y\in G}\delta_{g_1}(1)\rho_y(\frac{3}{2})e_{\mathcal{A}}
E(\delta_{g_1}(1)
\delta_{y^{-1}g_2}(2)
\rho_{h_1}(\frac{1}{2})\rho_{h_1^{-1}y^{-1}h_1h_2}(\frac{3}{2}))
\delta_{s_1}(1)\delta_{s_2}(2)
\rho_{t_1}(\frac{1}{2})\rho_{t_2}(\frac{3}{2})\\[5pt]
&=&\sum\limits_{y,f\in G}\delta_{y,h_1h_2}\delta_{g_1}(1)\rho_y(\frac{3}{2})
\delta_{fg_1}(1)\delta_{fy^{-1}g_2}(2)
\rho_{fh_1f^{-1}}(\frac{1}{2})\rho_{fh_1^{-1}f^{-1}}(\frac{3}{2})
e_{\mathcal{A}}
\delta_{s_1}(1)\delta_{s_2}(2)
\rho_{t_1}(\frac{1}{2})\rho_{t_2}(\frac{3}{2})\\[5pt]
&=&\delta_{g_1}(1)\rho_{h_1h_2}(\frac{3}{2})
\delta_{h_2^{-1}h_1^{-1}g_2}(2)\rho_{h_1}(\frac{1}{2})
\rho_{h_1^{-1}}(\frac{3}{2})e_{\mathcal{A}}
\delta_{s_1}(1)\delta_{s_2}(2)
\rho_{t_1}(\frac{1}{2})\rho_{t_2}(\frac{3}{2})\\[5pt]
&=&\delta_{g_1}(1)\delta_{g_2}(2)
\rho_{h_1}(\frac{1}{2})\rho_{h_2}(\frac{3}{2})
e_{\mathcal{A}}
\delta_{s_1}(1)\delta_{s_2}(2)
\rho_{t_1}(\frac{1}{2})\rho_{t_2}(\frac{3}{2}),
 \end{array}
\end{eqnarray*}
which yields that for any $a\in \langle {\mathcal{F}}(\Lambda_{{\frac{1}{2}},2}),e_{\mathcal{A}}\rangle_{C^*}$,
 \begin{eqnarray*}
    \begin{array}{c}
    \sum\limits_{x,y\in G}
    u_{x,y}\widetilde{E}(u_{x,y}^*a)=a.
      \end{array}
\end{eqnarray*}

Similarly, one can verify
 \begin{eqnarray*}
    \begin{array}{c}
    \sum\limits_{x,y\in G}\widetilde{E}(a u_{x,y})u_{x,y}^*=a, \ \ \forall a\in \langle{\mathcal{F}}(\Lambda_{{\frac{1}{2}},2}),e_{\mathcal{A}}\rangle_{C^*}.
      \end{array}
\end{eqnarray*}

By induction, we can show that $\{(u_{x,y},u_{x,y}^*)\colon x,y\in G\}$ is a quasi-basis of $\widetilde{E}\colon
\langle{\mathcal{F}}(\Lambda_{{n-\frac{1}{2}},m}),e_{\mathcal{A}}
\rangle_{C^*}
\rightarrow{\mathcal{F}}(\Lambda_{{n-\frac{1}{2}},m})$ for any $n,m\in\Bbb{Z}$ and $n<m$.

Finally, by the continuity of the conditional expectation $\widetilde{E}$ and the uniqueness of the $C^*$-inductive limit \cite{B.R.Li}, we conclude that $\{(u_{x,y},u_{x,y}^*)\colon x,y\in G\}$ is a quasi-basis for $\widetilde{E}\colon\langle {\mathcal{F}},e_{\mathcal{A}}\rangle_{C^*}\rightarrow {\mathcal{F}}$.
\end{proof}

\begin{remark}
$E_2$ is consistent with the dual conditional expectation $\widetilde{E}$ of $E\colon{\mathcal{F}}\rightarrow {\mathcal{A}}$ in Theorem 3.2. Indeed, it suffices to show that the restriction of $E_2$ on ${\mathcal{F}}(\Lambda_{\frac{1}{2},2})\rtimes D(G)$ is consistent with that of $\widetilde{E}$ on $\langle{\mathcal{F}}(\Lambda_{\frac{1}{2},2}),e_{\mathcal{A}}\rangle_{C^*}$ because of the continuity of conditional expectations and the uniqueness of the $C^*$-inductive limit. Again, it follows from Theorem 3.1 that
${\mathcal{F}}(\Lambda_{\frac{1}{2},2})\rtimes D(G)$ is $C^*$-isomorphic to $\langle{\mathcal{F}}(\Lambda_{\frac{1}{2},2}),e_{\mathcal{A}}\rangle_{C^*}$ by the map
 \begin{eqnarray*}
    \begin{array}{rcl}
    &&\delta_{g_1}(1)\delta_{g_2}(2)
\rho_{h_1}(\frac{1}{2})\rho_{h_2}(\frac{3}{2})\rtimes(g,h)\\[5pt]
&\mapsto&
|G|\delta_{g_1}(1)\delta_{g_2}(2)
\rho_{h_1}(\frac{1}{2})\rho_{h_2g}(\frac{3}{2}))
e_{\mathcal{A}}(\delta_{h^{-1}h_1^{-1}g_1}(1)
\delta_{h^{-1}g^{-1}h_2^{-1}h_1^{-1}g_2}(2)
\rho_{u}(\frac{1}{2})\rho_{h^{-1}g^{-1}h}(\frac{3}{2}).
  \end{array}
\end{eqnarray*}
Thus, we have
 \begin{eqnarray*}
    \begin{array}{rcl}
   && E_2(\delta_{g_1}(1)\delta_{g_2}(2)
\rho_{h_1}(\frac{1}{2})\rho_{h_2}(\frac{3}{2})\rtimes(g,h))\\[5pt]
&=&\frac{1}{|G|}\delta_{h,u}(\delta_{g_1}(1)\delta_{g_2}(2)
\rho_{h_1}(\frac{1}{2})\rho_{h_2}(\frac{3}{2})\otimes I_{D(G)})\\[5pt]
&=&\widetilde{E}(|G|\delta_{g_1}(1)\delta_{g_2}(2)
\rho_{h_1}(\frac{1}{2})\rho_{h_2g}(\frac{3}{2}))
e_{\mathcal{A}}(\delta_{h^{-1}h_1^{-1}g_1}(1)
\delta_{h^{-1}g^{-1}h_2^{-1}h_1^{-1}g_2}(2)
\rho_{u}(\frac{1}{2})\rho_{h^{-1}g^{-1}h}(\frac{3}{2})).
      \end{array}
\end{eqnarray*}
\end{remark}


\section{The $C^*$-basic construction for the inclusion ${\mathcal{F}}\subseteq {\mathcal{F}}\rtimes D(G)$}
In this section, we continue to investigate the crossed
 product $C^*$-algebra ${\mathcal{F}}\rtimes D(G)$, and the natural $\widehat{D(G)}$-module algebra action on ${\mathcal{F}}\rtimes D(G)$, which gives rise to the iterated crossed product $C^*$-algebra ${\mathcal{F}}\rtimes D(G)\rtimes\widehat{D(G)}$.
 The fixed point algebra under this action is given by ${\mathcal{F}}\equiv{\mathcal{F}}\rtimes I{_D(G)}$, which is consistent with
the range of the conditional expectation $E_2$.
 We then prove that the $C^*$-algebra $\langle{\mathcal{F}}\rtimes D(G),e_2\rangle_{C^*}$ constructed from the
 $C^*$-basic construction
  for the inclusion
 ${\mathcal{F}}\subseteq{\mathcal{F}}\rtimes D(G)$ is precisely $C^*$-isomorphic to the iterated crossed product $C^*$-algebra ${\mathcal{F}}\rtimes D(G)\rtimes\widehat{D(G)}$.

\begin{proposition}
The map $\sigma$ defines a Hopf module left action of $\widehat{D(G)}$ on ${\mathcal{F}}\rtimes D(G)$. That is ${\mathcal{F}}\rtimes D(G)$ is a left $\widehat{D(G)}$-module algebra.
\end{proposition}

\begin{proof}
It suffices to check that the map $\sigma\colon \widehat{D(G)}\times ({\mathcal{F}}\rtimes D(G))\rightarrow {\mathcal{F}}\rtimes D(G)$ satisfies the following relations:
\begin{eqnarray*}
    \begin{array}{rcl}
\Big((y,\delta_x)(w,\delta_z)\Big)\Big(F\otimes(g,h)\Big)&=&
(y,\delta_x)\Big((w,\delta_z)(F\otimes(g,h))\Big),\\[5pt]
    (y,\delta_x)\Big((F\otimes(g_1,h_1))(T\otimes(g_2,h_2))\Big)&=&
\sum\limits_{(y,\delta_x)}\Big((y,\delta_x)_{(1)}(F\otimes(g_1,h_1))\Big)
\Big((y,\delta_x)_{(2)}(T\otimes(g_2,h_2))\Big),\\[5pt]
    (y,\delta_x)\Big(F\otimes(g,h)\Big)^*&=&
\Big(\widetilde{S}(y,\delta_x)^*(F\otimes(g,h))\Big)^*,
    \end{array}
\end{eqnarray*}
 for $(y,\delta_x),(w,\delta_z)\in \widehat{D(G)}$, $T,F\in{\mathcal{F}}$ and $(g_i,h_i),(g,h)\in D(G)$ for $i=1,2$.

As to the first equality, we compute
\begin{eqnarray*}
    \begin{array}{clcl}
    \Big((y,\delta_x)(w,\delta_z)\Big)(F\otimes(g,h))
&=&\delta_{x,z}(yw,\delta_x)(F\otimes(g,h))\\[5pt]
&=&\delta_{x,z}\delta_{x,h}(F\otimes(gw^{-1}y^{-1},h))\\[5pt]
&=&\delta_{z,h}\delta_{x,h}(F\otimes(gw^{-1}y^{-1},h))\\[5pt]
&=&\delta_{z,h}(y,\delta_x)(F\otimes(gw^{-1},h))\\[5pt]
&=&(y,\delta_x)\Big((w,\delta_z)(F\otimes(g,h))\Big).
    \end{array}
\end{eqnarray*}

Next,
\begin{eqnarray*}
    \begin{array}{rcl}
  && (y,\delta_x)\Big((F\otimes(g_1,h_1))(T\otimes(g_2,h_2))\Big)\\[5pt]
&=&\sum\limits_{(g_1,h_1)}(y,\delta_x)
 \Big(F(g_1,h_1)_{(1)}T\otimes(g_1,h_1)_{(2)}(g_2,h_2)\Big)\\[5pt]
&=&\sum\limits_{f\in G}(y,\delta_x)
 \Big(F(f,h_1)T\otimes(f^{-1}g_1,h_1)(g_2,h_2)\Big)\\[5pt]
&=&\sum\limits_{f\in G}(y,\delta_x)
 \Big(F(f,h_1)T\otimes\delta_{f^{-1}g_1h_1,h_1g_2}(f^{-1}g_1,h_1h_2)\Big)\\[5pt]
&=&(y,\delta_x)
 \Big(F(g_1h_1g_2^{-1}h_1^{-1},h_1)T\otimes(h_1g_2h_1^{-1},h_1h_2)\Big)\\[5pt]
&=&\delta_{x,h_1h_2}F(g_1h_1g_2^{-1}h_1^{-1},h_1)T\otimes
(h_1g_2h_1^{-1}y^{-1},h_1h_2),
    \end{array}
\end{eqnarray*}
and
\begin{eqnarray*}
    \begin{array}{rcl}
   &&\sum\limits_{(y,\delta_x)}\Big((y,\delta_x)_{(1)}
   (F\otimes(g_1,h_1))\Big)
\Big((y,\delta_x)_{(2)}(T\otimes(g_2,h_2))\Big)\\[5pt]
&=&\sum\limits_{t\in G}\Big((y,\delta_{t^{-1}})(F\otimes(g_1,h_1))\Big)
\Big((tyt^{-1},\delta_{tx})(T\otimes(g_2,h_2))\Big)\\[5pt]
&=&\sum\limits_{t\in G}\delta_{t^{-1},h_1}\delta_{tx,h_2}
\Big(F\otimes(g_1y^{-1},h_1)\Big)\Big(T\otimes(g_2ty^{-1}t^{-1},h_2)\Big)\\[5pt]
&=&\delta_{x,h_1h_2}
\Big(F\otimes(g_1y^{-1},h_1)\Big)
\Big(T\otimes(g_2h_1^{-1}y^{-1}h_1,h_2)\Big)\\[5pt]
&=&\sum\limits_{f\in G}\delta_{x,h_1h_2}
F(f,h_1)T\otimes(f^{-1}g_1y^{-1},h_1)(g_2h_1^{-1}y^{-1}h_1,h_2)\\[5pt]
&=&\delta_{x,h_1h_2}\delta_{f^{-1}g_1,h_1g_2h_1^{-1}}
F(f,h_1)T\otimes(f^{-1}g_1y^{-1},h_1h_2)\\[5pt]
&=&\delta_{x,h_1h_2}F(g_1h_1g_2^{-1}h_1^{-1},h_1)T\otimes
(h_1g_2h_1^{-1}y^{-1},h_1h_2).
    \end{array}
\end{eqnarray*}
Thus, we obtain that
\begin{eqnarray*}
    \begin{array}{c}
       (y,\delta_x)\Big((F\otimes(g_1,h_1))(T\otimes(g_2,h_2))\Big)=
\sum\limits_{(y,\delta_x)}\Big((y,\delta_x)_{(1)}(F\otimes(g_1,h_1))\Big)
\Big((y,\delta_x)_{(2)}(T\otimes(g_2,h_2))\Big).
    \end{array}
\end{eqnarray*}

To prove the third equation, we can calculate
\begin{eqnarray*}
    \begin{array}{rcl}
(y,\delta_x)\Big(F\otimes(g,h)\Big)^*&=&(y,\delta_x)\Big((I_{\mathcal{F}}
\otimes(g,h)^*)(F^*\otimes I_{D(G)})\Big)\\[5pt]
&=&(y,\delta_x)\Big((I_{\mathcal{F}}
\otimes(h^{-1}gh,h^{-1}))(F^*\otimes I_{D(G)})\Big)\\[5pt]
&=&\sum\limits_{f\in G}
(y,\delta_x)\Big((f,h^{-1})F^*\otimes(f^{-1}h^{-1}gh,h^{-1})\Big)\\[5pt]
&=&\sum\limits_{f\in G}\delta_{x,h^{-1}}
(f,h^{-1})F^*\otimes(f^{-1}h^{-1}ghy^{-1},h^{-1}),
    \end{array}
\end{eqnarray*}
and
\begin{eqnarray*}
    \begin{array}{rcl}
\Big(\widetilde{S}(y,\delta_x)(F\otimes(g,h))\Big)^*
&=&\Big((x^{-1}yx,\delta_{x^{-1}})(F\otimes(g,h))\Big)^*\\[5pt]
&=&\delta_{x^{-1},h}\Big(F\otimes(gx^{-1}y^{-1}x,h)\Big)^*\\[5pt]
&=&\sum\limits_{f\in G}\delta_{x,h^{-1}}
(f,h^{-1})F^*\otimes(f^{-1}h^{-1}ghy^{-1},h^{-1}).
    \end{array}
\end{eqnarray*}
\end{proof}

From Proposition 5.1, we can construct the crossed product $({\mathcal{F}}\rtimes D(G))\rtimes\widehat{D(G)}$,
which is called the iterated crossed product $C^*$-algebra.


In the following, we will consider the $\widehat{D(G)}$-invariant subalgebra of ${\mathcal{F}}\rtimes D(G)$. To do this, set
$$({\mathcal{F}}\rtimes D(G))^{\widehat{D(G)}}\ =\
\{\widetilde{T}\in {\mathcal{F}}\rtimes D(G)\colon b(\widetilde{T})=\widetilde{\varepsilon}(b)(\widetilde{T}), \ \ \forall b\in \widehat{D(G)}\}.$$
 One can show that $({\mathcal{F}}\rtimes D(G))^{\widehat{D(G)}}$ is a $C^*$-subalgebra of ${\mathcal{F}}\rtimes D(G)$. Furthermore, $$({\mathcal{F}}\rtimes D(G))^{\widehat{D(G)}}\ =\ \{ \widetilde{T}\in {\mathcal{F}}\rtimes D(G)\colon E_2(\widetilde{T})=\widetilde{T}\}.$$
In fact, for $\widetilde{T}\in ({\mathcal{F}}\rtimes D(G))^{\widehat{D(G)}}\subseteq {\mathcal{F}}\rtimes D(G)$, we can compute that
\begin{eqnarray*}
    \begin{array}{c}
    E_2(\widetilde{T})=\frac{1}{|G|}\sum\limits_{y\in G}(y,\delta_u)
    (\widetilde{T})
    =\frac{1}{|G|}\sum\limits_{y\in G}\widetilde{\varepsilon}(y,\delta_u)(\widetilde{T})
    =\frac{1}{|G|}\sum\limits_{y\in G}\widetilde{T}
    =\widetilde{T}.
   \end{array}
\end{eqnarray*}
For the converse, suppose that $ \widetilde{T}\in {\mathcal{F}}\rtimes D(G)$ with $E_2(\widetilde{T})=\widetilde{T}$. Then for any $(w,\delta_z)\in \widehat{D(G)}$, we have
\begin{eqnarray*}
    \begin{array}{rcl}
    (w,\delta_z)(\widetilde{T})
    &=&(w,\delta_z)E_2(\widetilde{T})\\[5pt]
    &=&(w,\delta_z)\frac{1}{|G|}\sum\limits_{y\in G}(y,\delta_u)
    (\widetilde{T})\\[5pt]
    &=&\frac{1}{|G|}\sum\limits_{y\in G}\delta_{z,u}(wy,\delta_u)(\widetilde{T})\\[5pt]
    &=&\widetilde{\varepsilon}(w,\delta_z)E_2(\widetilde{T})\\[5pt]
    &=&\widetilde{\varepsilon}(w,\delta_z)(\widetilde{T}).
   \end{array}
\end{eqnarray*}
This can be linearly extended in $\widehat{D(G)}$. Hence, $\widetilde{T}\in ({\mathcal{F}}\rtimes D(G))^{\widehat{D(G)}}.$





\begin{remark}

$({\mathcal{F}}\rtimes D(G))^{\widehat{D(G)}}$ is the subalgebra of ${\mathcal{F}}\rtimes D(G)$ corresponding to the trivial representation $\widetilde{\varepsilon}$ of $\widehat{D(G)}$. In fact, $\widehat{D(G)}$ is semisimple and ${\mathcal{F}}\rtimes D(G)$ is a $\widehat{D(G)}$-module algebra, then ${\mathcal{F}}\rtimes D(G)$ is completely reducible, which means ${\mathcal{F}}\rtimes D(G)$ can be decomposed into a direct sum
$${\mathcal{F}}\rtimes D(G)=\bigoplus\limits_{r\in [\widehat{D(G)}]}(\mathcal{F}\rtimes D(G))^r, \ \ \ ({\mathcal{F}\rtimes D(G)})^r=M^r({\mathcal{F}}\rtimes D(G)),$$
where $[\widehat{D(G)}]$ denotes the set of equivalence classes of irreducible representations of $\widehat{D(G)}$, and $\{M^r, r\in [\widehat{D(G)}]\}$ the set of minimal central idempotents in $\widehat{D(G)}$.

 Clearly, $(\widetilde{\varepsilon}, {\mathcal{F}}\rtimes D(G))$ is an irreducible representation of $\widehat{D(G)}$. As we have known that the representations of $\widehat{D(G)}$ are in one-to-one correspondence with the $\widehat{D(G)}$-modules, then $${(\mathcal{F}\rtimes D(G))}^{\widetilde{\varepsilon}}
  \ \triangleq\ \{\widetilde{F}\in {\mathcal{F}}\rtimes D(G) \colon b(\widetilde{F})=\widetilde{\varepsilon}(b)\widetilde{F}, \forall \ b\in \widehat{D(G)}\}\ =\ E_2({\mathcal{F}}\rtimes D(G))$$ is a $\widehat{D(G)}$-module. As a consequence, $({\mathcal{F}}\rtimes D(G))^{\widehat{D(G)}}=({\mathcal{F}\rtimes D(G)})^{\widetilde{\varepsilon}}$ corresponds to the trivial representation $\widetilde{\varepsilon}$, where $M^{\widetilde{\varepsilon}}$ is just $E_2$.
\end{remark}

Naturally, we consider the $C^*$-algebra $\langle{\mathcal{F}}\rtimes D(G),e_2\rangle_{C^*}$ constructed from the
$C^*$-basic construction for the inclusion ${\mathcal{F}}\subseteq{\mathcal{F}}\rtimes D(G)$
 in the following, where $e_2$ is the Jones projection of $E_2$.

\begin{theorem}
There exists a $C^*$-isomorphism
of $C^*$-algebras between ${\mathcal{F}}\rtimes D(G)\rtimes \widehat{D(G)}$
and $\langle{\mathcal{F}}\rtimes D(G),e_2\rangle_{C^*}$. That is,
$${\mathcal{F}}\rtimes D(G)\rtimes \widehat{D(G)}\cong \langle{\mathcal{F}}\rtimes D(G),e_2\rangle_{C^*}.$$
\end{theorem}
\begin{proof}
The proof is similar to that of Theorem 3.1. Indeed, considering the map $\Psi$ given by
\begin{eqnarray*}
\begin{array}{cccc}
\Psi\colon &
\langle{\mathcal{F}}\rtimes D(G),e_2\rangle_{C^*}
&\rightarrow & {\mathcal{F}}\rtimes D(G)\rtimes \widehat{D(G)}\\[5pt]
&T\rtimes(g,h)&\mapsto&T\rtimes(g,h)\rtimes I_{\widehat{D(G)}},\\[5pt]
&e_2&\mapsto&I_{\mathcal{F}\rtimes D(G)}\rtimes\frac{1}{|G|}\sum\limits_{y\in G}(y,\delta_u),
  \end{array}
  \end{eqnarray*}
one can show that $\Psi$ is a $C^*$-isomorphism
from ${\mathcal{F}}\rtimes D(G)\rtimes \widehat{D(G)}$
onto $\langle{\mathcal{F}}\rtimes D(G),e_2\rangle_{C^*}$.
\end{proof}

Moreover, by Takai duality \cite{H.Tak}, the iterated crossed product $C^*$-algebra ${\mathcal{F}}\rtimes D(G)\rtimes \widehat{D(G)}$
is canonically isomorphic to $M_{|G|^2}({\mathcal{F}})$.

\begin{remark}
In the following we will give the concrete construction for $M_{|G|^2}({\mathcal{F}})$.

The local field $M_{|G|^2}({\mathcal{F}}_{\mathrm{loc}})$ of a $M_{|G|^2}(G)$-spin model is a $*$-algebra with a unit $I_{M_{|G|^2}({\mathcal{F}})}$ generated by
 $\Big\{\delta_g(x)\otimes M, \rho_h(l)\otimes N\colon g, h\in G, \ x\in {\Bbb Z},\ l\in {\Bbb Z}+\frac{1}{2},\ M, N\in \mbox{LB} (M_{|G|^2})\Big\}$ satisfying the following relations
\begin{eqnarray*}
    \begin{array}{rcl}
O_M^g(x)O_N^h(x)&=&\delta_{g,h}O_{MN}^g(x),\\[5pt]
 D_M^g(l)D_N^h(l)&=&D_{MN}^{gh}(l),\\[5pt]
 \sum\limits_{g\in G}
O_I^g(x)&=&I_{M_{|G|^2}({\mathcal{F}})}
=D_I^u(l),\\[5pt]
O_M^g(x)O_N^h(x')&=&O_M^h(x')O_N^g(x),\\[5pt]
 D_M^g(l)O_N^h(x)&=& \left\{\begin{array}{cc}
                          O_M^{gh}(x)D_N^g(l), & \mbox{if}\ l<x, \\[5pt]
                          O_M^h(x)D_N^g(l), & \mbox{if}\ l>x,
                        \end{array}\right.\\[5pt]
 D_M^g(l)D_N^h(l')&=& \left\{\begin{array}{cc}
                          D_M^h(l')D_N^{h^{-1}gh}(l), & \mbox{if}\ l>l',\\[5pt]
                          D_M^{ghg^{-1}}(l')D_N^{g}(l), & \mbox{if}\ l<l',
                        \end{array}\right.\\[5pt]
 (O_M^g(x))^*&=&O_{M^*}^g(x),\\[5pt]
  (D_N^h(l))^*&=&D_{N^*}^{h^{-1}}(l),
    \end{array}
\end{eqnarray*}
for $x, x' \in {\Bbb Z},\ l, l'\in {\Bbb Z}+\frac{1}{2} \ {\mbox{and}} \ g, h\in G$, where by $O_M^g(x)$, $D_N^h(l)$ and $\mbox{LB}(M_{|G|^2})$ we denote $\delta_g(x)\otimes M$, $\rho_h(l)\otimes N$ and the linear basis of $M_{|G|^2}(\Bbb{C})$ for convenience, respectively.
\end{remark}

For any finite subset $\Lambda\subseteq \frac{1}{2}\Bbb{Z}$, let
$M_{|G|^2}({\mathcal{F}}(\Lambda))$ be the $*$-subalgebra of
$M_{|G|^2}({\mathcal{F}}_{\mathrm{loc}})$
generated by
 \begin{eqnarray*}
\Big\{O_M^g(x),\ D_N^h(l)\colon
 g, h\in G,\ x, l\in\Lambda,\ M, N\in \mbox{LB}(M_{|G|^2})\Big\}.
 \end{eqnarray*}
Similar to the case of the field algebra ${\mathcal{F}}$ of a $G$-spin model, one can show that $M_{|G|^2}({\mathcal{F}})$ is the $C^*$-algebra given by the $C^*$-inductive limit
\begin{eqnarray*}
    \begin{array}{c}
    M_{|G|^2}({\mathcal{F}})
=\overline{\bigcup\limits_{n}M_{|G|^2}({\mathcal{F}}(\Lambda_n))}.
 \end{array}
\end{eqnarray*}
From now on, we call $M_{|G|^2}({\mathcal{F}})$ the field algebra of a $M_{|G|^2}(G)$-spin model, and we call $O_M^g(x)$ and $D_N^h(l)$ order and disorder operators, respectively.

Now one can show that the field algebra ${\mathcal{F}}\rtimes D(G)\rtimes \widehat{D(G)}$ is $D(G)$-module algebra. Indeed,
the map $$\tau\colon D(G)\times ({\mathcal{F}}\rtimes D(G)\rtimes\widehat{D(G)}) \rightarrow {\mathcal{F}}\rtimes D(G)\rtimes\widehat{D(G)}$$
given on the generating elements of ${\mathcal{F}}\rtimes D(G)\rtimes\widehat{D(G)}$ as
$$\tau((g,h)\times(\widetilde{F}\otimes(y,\delta_x) ))=\delta_{h^{-1}gh,x^{-1}yx}(\widetilde{F}\otimes(y,\delta_{xh^{-1}}))$$
for any $\widetilde{F}\in {\mathcal{F}}\rtimes D(G)$, can be linearly extended both in $D(G)$ and ${\mathcal{F}}\rtimes D(G)\rtimes \widehat{D(G)}$.


The observable algebra of $M_{|G|^2}(G)$-spin models is defined as  $({\mathcal{F}}\rtimes D(G)\rtimes \widehat{D(G)})^{D(G)}$.
So it is clear that $({\mathcal{F}}\rtimes D(G)\rtimes \widehat{D(G)})^{D(G)}=
E_2({\mathcal{F}}\rtimes D(G)\rtimes \widehat{D(G)})={\mathcal{F}}\rtimes D(G)$.

\begin{remark}
Let ${\mathcal{A}}\subseteq{\mathcal{F}}$ be an inclusion of unital $C^*$-algebras with a conditional expectation $E\colon{\mathcal{F}}\rightarrow{\mathcal{A}}$ of index-finite type \cite{XINQL}. Set
${\mathcal{F}}_{-1}={\mathcal{A}}, \ {\mathcal{F}}_0={\mathcal{F}}$, and
$E_1=E$, and recall the $C^*$-basic construction (the $C^*$-algebra version of the basic construction). We inductively define $e_{k+1}=e_{_{\mathcal{F}_{k-1}}}$
and ${\mathcal{F}}_{k+1}=\langle{\mathcal{F}}_k,e_{k+1}\rangle_{C^*}$, the Jones projection and $C^*$-basic
construction applied to $E_{k+1}\colon {\mathcal{F}}_{k}\rightarrow{\mathcal{F}}_{k-1}$, and take
$E_{k+2}\colon {\mathcal{F}}_{k+1}\rightarrow{\mathcal{F}}_{k}$ to be the
dual conditional expectation $E_{{\mathcal{F}}_{k}}$ of Definition 2.3.3 in \cite{Y.Wat}. Then this gives the inclusion tower
of iterated basic constructions
$${\mathcal{A}}\subseteq{\mathcal{F}}\subseteq{\mathcal{F}}\rtimes D(G)\subseteq {\mathcal{F}}\rtimes D(G)\rtimes \widehat{D(G)}\subseteq
{\mathcal{F}}\rtimes D(G)\rtimes\widehat{D(G)}\rtimes D(G)\subseteq\cdots.$$
 It follows from Proposition 2.10.11 of \cite{Y.Wat} that this tower
does not depend on the choice of $E$.

Notice that ${\mathcal{F}}_2={\mathcal{F}}\rtimes D(G)\rtimes \widehat{D(G)}$ is $C^*$-isomorphic to $M_{|G|^2}({\mathcal{F}})$,
 the field algebra of a $M_{|G|^2}(G)$-spin model, and
${\mathcal{F}}_4={\mathcal{F}}\rtimes D(G)\rtimes \widehat{D(G)}\rtimes D(G)\rtimes \widehat{D(G)}$ is $C^*$-isomorphic to $M_{|G|^4}({\mathcal{F}})$, called the field algebra of a $M_{|G|^4}(G)$-spin model, where the order and disorder operators can be defined similar to those in Remark 5.2.
\end{remark}


\begin{thebibliography}{}

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\bibitem{DHR1} S.Doplicher, R.Haag, J.E.Roberts, ``Fields, observables and gauge transformations. I," Comm. Math. Phys. \textbf{13} 1-23 (1969).
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\bibitem{G.Mas} G.Mason, {\it The quantum double of a finite group and its role in conformal field theory}, London Mathematical Society Lecture Notes, 212, 405-417, (Cambridge Univ. Press, Cambridge, 1995).
\bibitem{GJM} G.J.Murphy, $C^*$-algebras and operator theory (Academic Press, New York, 1990).
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\bibitem{Qiao} Q.L.Xin and L.N.Jiang, ``Symmetric structure of field algebra of $G$-spin models determined by a normal subgroup," J. Math. Phys. \textbf{55}, 091703 (2014).
\bibitem{XINQL} Q.L.Xin and L.N.Jiang, ``$C^*$-index of observable algebra in the field algebra  determined by a normal group," math.OA/\textbf{1505.04784} (2015).
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\end{thebibliography}
 \end{document}

1505.06944.extracted_bib
{E.Abe E.Abe, {Hopf Algebras, Cambridge Tracts in Mathematics, No. 74, Cambridge (Cambridge Univ. Press, New York, 1980).
1505.06944.extracted_bib
{O.B O.Bratteli, {Inductive limits of finite dimensional $C^*$-algebras, Trans. Amer. Math. Soc. 171 195-234 (1972).
1505.06944.extracted_bib
{O.Bra O.Bratteli, D.W.Robinson, {Operator algebras and quantum statistical mechanics, Vol. 1. (Providence: Springer, New York, 1987).
1505.06944.extracted_bib
{CF M.Cohen, D.Fischman, ``Hopf algebra action," J. Algebra 100 363-379 (1986).
1505.06944.extracted_bib
{K.A.Da K.A.Dancer, P.S.Isaac, J.Links, ``Representations of the quantum doubles of finite group algebras and spectral parameter dependent solutions of the Yang-Baxter equations," J. Math. Phys. 47, 103511 (2006).
1505.06944.extracted_bib
{DHR1 S.Doplicher, R.Haag, J.E.Roberts, ``Fields, observables and gauge transformations. I," Comm. Math. Phys. 13 1-23 (1969).
1505.06944.extracted_bib
{DHR2 S.Doplicher, R.Haag, J.E.Roberts, ``Fields, observables and gauge transformations. II," Comm. Math. Phys. 15 173-200 (1969).
1505.06944.extracted_bib
{DHR3 S.Doplicher, R.Haag, J.E.Roberts, ``Local observables and particle statistics. I," Comm. Math. Phys. 23 199-230 (1971).
1505.06944.extracted_bib
{DHR4 S.Doplicher, R.Haag, J.E.Roberts, ``Local observables and particle statistics. II," Comm. Math. Phys. 35 49-85 (1974).
1505.06944.extracted_bib
{S.Dop S.Doplicher and J.E.Roberts, ``Fields, statistics and non-abelian gauge group," Comm. Math. Phys. 28, 331-348 (1972).
1505.06944.extracted_bib
{L.N.Jia L.N.Jiang, ``Towards a quantum Galois theory for quantum double algebras of finite groups," Proc. Amer. Math. Soc. 138, 2793-2801 (2010).
1505.06944.extracted_bib
{Jones V.F.R.Jones, ``Index for subfactors," Invent. Math. 72, 1-25 (1983).
1505.06944.extracted_bib
{V.F.R.J V.F.R.Jones, {Subfactors and Knots, CBMS, No. 80, (American Mathematical Society Providence, Rhode Island, 1991).
1505.06944.extracted_bib
{T.Kaj T.Kajiwara, Y.Watatani, ``Jones index theory by Hilbert $C^*$-bimodules and $K$-theory," Trans. Amer. Math. Soc. 352 3429-3472 (2000).
1505.06944.extracted_bib
{H.Ko H.Kosaki, ``Extension of Jones' theory on index to arbitrary factors," J. Func. Anal. 66, 123-140 (1986).
1505.06944.extracted_bib
{B.R.Li B.R.Li, {Operator Algebras (Scientific Press, Beijing, 1998) (in Chinese).
1505.06944.extracted_bib
{G.Mas G.Mason, {The quantum double of a finite group and its role in conformal field theory, London Mathematical Society Lecture Notes, 212, 405-417, (Cambridge Univ. Press, Cambridge, 1995).
1505.06944.extracted_bib
{GJM G.J.Murphy, $C^*$-algebras and operator theory (Academic Press, New York, 1990).
1505.06944.extracted_bib
{M.PP M.Pimsner and S.Popa, ``Entropy and index for subfactors," Ann. Sci. Ecole. Norm. Sup. 19, 57-106 (1986).
1505.06944.extracted_bib
{Popa S.Popa, ``On the relative Dixmier property for inclusions of $C^*$-algebras," J. Funct. Anal. 171, 139-154 (2000).
1505.06944.extracted_bib
{M.E.Sw M.E.Sweedler, {Hopf algebras (W.A. Benjamin, New York, 1969).
1505.06944.extracted_bib
{K.Szl K.Szlach\'{anyi and P.Vecserny\'{es, ``Quantum symmetry and braided group statistics in $G$-spin models," Comm. Math. Phys. 156, 127-168 (1993).
1505.06944.extracted_bib
{H.Tak H.Takai, ``On a duality for crossed products of $C^*$-algebras," J. Funct. Anal. 19, 25-39 (1975).
1505.06944.extracted_bib
{Y.Wat Y.Watatani, {Index for $C^*$-subalgebras, Memoirs of the AMS, No. 424, (Providence: AMS, 1990).
1505.06944.extracted_bib
{Qiao Q.L.Xin and L.N.Jiang, ``Symmetric structure of field algebra of $G$-spin models determined by a normal subgroup," J. Math. Phys. 55, 091703 (2014).
1505.06944.extracted_bib
{XINQL Q.L.Xin and L.N.Jiang, ``$C^*$-index of observable algebra in the field algebra determined by a normal group," math.OA/1505.04784 (2015).
1505.06944.extracted_bib
{V.G.Dr V.G.Drinfel'd, Quantum groups, in ``Proceedings International Congress of Mathematicians, Berkeley," pp. 798-820, American Mathematical Society, Providence, Rhode Island, 1986.