$C^*$-basic construction on field algebras of $G$-spin models

Abstract

Let $G$ be a finite group. Starting from the field algebra ${\mathcal{F}}$ of $G$-spin models, we show that the $C^*$-basic construction for the field algebra ${\mathcal{F}}$ and the $D(G)$-invariant subalgebra of ${\mathcal{F}}$ can be represented as the crossed product $C^*$-algebra ${\mathcal{F}}\rtimes D(G)$. Moreover, 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 ${\mathcal{F}}$. In addition, it is proved that the iterated crossed product $C^*$-algebra is a new field algebra, and the concrete structures with the order and disorder operators are given.