arXiv2026
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, together with the interior angle. For each $n\in\mathbb{N}$, we show that the hyperfinite $\mathrm{II}_1$-factor $\mathscr{R}$ contains an uncountable family of pairwise distinct regular $n\times n$ spin model subfactors. In particular, we obtain a continuous family $(\mathscr{R}_{\mathsf{H}_α})_{α\in[0,π)}$ of $2\times2$ spin model subfactors satisfying \[ \mathrm{d}_{\mathrm{MT}}(\mathscr{R}_{\mathsf{H}_α},\mathscr{R}_{\mathsf{H}_β}) =|\sin(α-β)|. \] We prove that two such subfactors form a commuting square over their intersection if and only if they are maximally distant, i.e., $\mathrm{d}_{\mathrm{MT}}=1$. More generally, we establish structural results showing that commuting squares of $\mathrm{II}_1$-factors, under natural index conditions, force maximal distance, yielding $\mathrm{d}_{\mathrm{KK}}=1=\mathrm{d}_{\mathrm{MT}}$. We also show that two diffuse subalgebras orthogonal in the sense of Popa must be maximally distant. In contrast, no two members of $(\mathscr{R}_{\mathsf{H}_α})_{α\in[0,π)}$ are Popa-orthogonal. Nevertheless, whenever two are maximally distant, their interior angle over their intersection is $π/2$, demonstrating that interior-angle orthogonality differs from Popa orthogonality. Finally, in the free group factor $L(\mathbb{F}_2)=L(\langle a,b\rangle)$, we prove \[ \mathrm{d}_{\mathrm{MT}}(L(\langle a\rangle),uL(\langle a\rangle)u^*) =\sqrt{1-|τ(u)|^4} \] for $u\in L(\langle b\rangle)$, and construct maximally distant masas in $L(\mathbb{F}_2)$ that do not arise from subgroups of $\mathbb{F}_2$.