Jones type C*-basic construction in non-equilibrium Hopf spin models

Abstract

Let H be a finite dimensional Hopf C*-algebra, and let K be a Hopf *-subalgebra of H. Considering that the field algebra ${{\cal F}_K}$ of a non-equilibrium Hopf spin model carries a D(H, K)-invariant subalgebra ${{\cal A}_K}$, this paper shows that the C*-basic construction for the inclusion ${{\cal A}_K} \subseteq {{\cal F}_K}$ can be expressed as the crossed product C*-algebra ${{\cal F}_K}D(H,K)$. 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 ${{\cal F}_K}D(H,K)D\widehat {(H,K)}$, which coincides with the C*-basic construction for the inclusion ${{\cal F}_K} \subseteq {{\cal F}_K}D(H,K)$. In the end, the Jones type tower of field algebra ${{\cal F}_K}$ is obtained, and the new field algebra emerges exactly as the iterated crossed product.