Jones type basic construction on field algebras of $G$-spin models

Let $G$ be a finite group. Starting from the field algebra $\mathcal{F}$ of $G$-spin models, one can construct the crossed product $C^{\ast }$-algebra $\mathcal{F}⋊D(G)$ such that it coincides with the $C^{\ast }$-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 $\mathcal{F}⋊D(G)$,the iterated crossed product $C^{\ast }$-algebra can be obtained, which is $C^{\ast }$-isomorphic to the $C^{\ast }$-basic construction for ${$\mathcal{F}$}\rtimes D(G)$andthefieldalgebra${$\mathcal{F}$}$. Furthermore, one can show that the iterated crossed product $C^{\ast }$-algebra is a new field algebra and give the concrete structure with the order and disorder operators.