Strongly irreducible operators, $K_0$-groups, and Jones indices in the hyperfinite type ${\mathrm{I}\mathrm{I}}_1$ factor

In this paper, we study the strong irreducibility of a class of operators associated with irrational rotation algebras in the hyperfinite type ${\mathrm{I}\mathrm{I}}_1$ factor $\mathcal{R}$. By developing weighted shift models for their relative commutants, we obtain $K_0$ calculations and splitting criteria for direct sums. In particular, we construct strongly irreducible operators in $\mathcal{R}$ whose relative commutants have $K_0$-groups isomorphic to $\mathbb{Z}$ and ${\mathbb{Z}}^2$. We realize every Jones index of an irreducible subfactor of $\mathcal{R}$ as $[\mathcal{R}:W^{\ast }(T)]$ for some strongly irreducible operator $T$ relative to $\mathcal{R}$. Furthermore, we construct strong generators of $\mathcal{R}$ and strongly irreducible operators in every type ${\mathrm{I}\mathrm{I}}_1$ factor $\mathcal{M}$ with separable predual.