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.
Xiaomin WEI
,
Lining JIANG
. JONES TYPE C*-BASIC CONSTRUCTION IN NON-EQUILIBRIUM HOPF SPIN MODELS*[J]. Acta mathematica scientia, Series B, 2023
, 43(6)
: 2573
-2588
.
DOI: 10.1007/s10473-023-0615-4
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