Distances Between von Neumann Subalgebras: Spin Models, Commuting Squares, and Free Group Factors

We investigate the relative position of von Neumann subalgebras through their Mashood-Taylor ($\mathrm{d}_{\mathrm{MT}}$) and Kadison-Kastler ($\mathrm{d}_{\mathrm{KK}}$) distances and their interior angle. For the continuous family $(\mathscr{R}_{\mathsf{H}_α})_{α\in[0,π)}$ of $2\times2$ spin model