C$^*$-algebra $A$. We often identify $A$ and $C(\Omega)$. The projection $p \in C(\Omega)$ is a characteristic function of a closed and open subset $\mathrm{supp}_\Omega(p) \subseteq \Omega$. We identify $A p$ and $C(\mathrm{supp}_\Omega(p))$. By Lemma \[Lemma;First\], the topology of $\Omega$ is generated by at most countably many closed and open subsets. For $g \in I^*$, $\mathrm{Ad} (v(g))$ gives an isomorphism from $A v(g)^* v(g)$ to $A v(g) v(g)^*$. This isomorphism defines a homeomorphism $\psi_g \colon \mathrm{supp}_\Omega(v(g)^* v(g)) \rightarrow \mathrm{supp}_\Omega(v(g) v(g)^*)$ by $$\begin{aligned}
[v(g) a v(g)^* ] (\psi_g (\alpha)) = a (\alpha), 
\quad 
a \in A v(g)^* v(g),
\alpha \in \mathrm{supp}_\Omega(v(g)^* v(g)).\end{aligned}$$ We denote by $\mathrm{Dom}_\Omega(\psi_g)$, $\mathrm{Image}_\Omega(\psi_g) \subseteq \Omega$ the domain and the image of $\psi_g$.

We define a set $\Sigma(g)$ by $$\begin{aligned}
\Sigma(g) = \{ (\psi_g (\alpha), g, \alpha) \ 
|\ g \in I^*, \alpha \in \mathrm{Dom}_\Omega(\psi_g)\}\end{aligned}$$ and define its topology by the identification $\Sigma(g) \cong \mathrm{Dom}_\Omega(\psi_g) \subseteq \Omega$. We define $\Sigma$ by the disjoint union $\Sigma = \bigsqcup_{g \in I^*} \Sigma(g)$ and introduce its topology by the product of the compact Hausdorff topology and the discrete topology. We define an relation $\sim$ on $\Sigma$ by the following: two elements $(\psi_g(\alpha), g, \alpha)$ and $(\psi_h(\beta), h, \beta)$ are said to be 