The C ^∗ -algebra index for observable algebra in non-equilibrium Hopf spin models

Denote by H a finite dimensional Hopf C ^∗-algebra, K a Hopf ∗ -subalgebra of H. Starting with the observable algebra A_K in non-equilibrium Hopf spin models, in which A_K carries a coaction of relative quantum double D(H, K), the field algebra F_K= A_K⋊ D(H, K) ^ is obtained, where D(H, K) ^ is the dual of D(H, K). This paper shows that the Haar integral of D(H, K) admits a faithful conditional expectation Γ from F_K onto A_K. The index of Γ is calculated by virtue of its quasi-basis provided by the matrix units of D(H, K) ^.