Abstract: Let H be a finite dimensional Hopf ${C}^*$-algebra, and let K be a Hopf *-subalgebra of H. Considering that the field algebra $\mathscr{F}_{K}$ of a non-equilibrium Hopf spin model carries a $D(H,K)$-invariant subalgebra $\mathscr{A}_{K}$, this paper shows that the ${C}^*$-basic construction for the inclusion $\mathscr{A}_{K} \subseteq \mathscr{F}_{K}$ {can be expressed as} the crossed product ${C}^*$-algebra $\mathscr{F}_{K} \rtimes D(H,K)$. Here, $D(H,K)$ is a bicrossed product of the opposite dual $\widehat{H^{op}}$ and $K$. Furthermore, the natural action of $\widehat{D(H,K)}$ on $D(H,K)$ gives rise to the iterated crossed product $\mathscr{F}_{K} \rtimes D(H,K) \rtimes \widehat{D(H,K)}$, which coincides with the ${C}^*$-basic construction for the inclusion $\mathscr{F}_{K} \subseteq \mathscr{F}_{K} \rtimes D(H,K)$. In the end, the Jones type tower of field algebra $\mathscr{F}_{K}$ is obtained, and the new field algebra emerges exactly as the iterated crossed product.

Key words: field algebra, conditional expectation, basic construction, ${C}^*$-tower