The structure of the observable algebra determined by a Hopf $*$-subalgebra in Hopf spin models

Abstract

Let $H$ be a finite dimensional Hopf $C^*$- algebra, $H_{1}$ a Hopf $*$-subalgebra of $H$. This paper considers the observable algebra $\mathcal{A}_{H_{1}}$ of Hopf spin models determined by $H_{1}$. Further, using the iterated tensor product of finite algebras, one can prove that the observable algebra $\mathcal{A}_{H_{1}}$ is $C^*$-isomorphic to the $C^*$-inductive limit $\cdots \rtimes H_{1} \rtimes \widehat{H} \rtimes H_{1} \rtimes \widehat{H} \rtimes H_{1} \rtimes \cdots$, where $\widehat{H}$ denotes the dual of $H$.