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

Abstract

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 subfactors of the hyperfinite $\mathrm{II}_1$-factor $\mathscr{R}$, we establish \[ \mathrm{d}_{\mathrm{MT}}(\mathscr{R}_{\mathsf{H}_α},\mathscr{R}_{\mathsf{H}_β}) =|\sin(α-β)|. \] We prove that two such subfactors form a commuting square over their intersection if and only if they are maximally distant ($\mathrm{d}_{\mathrm{MT}}=1$). More generally, we show 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 prove that diffuse subalgebras orthogonal in the sense of Popa are 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 orthogonality via interior angle differs from Popa orthogonality. Finally, in the free group factor $L(\mathbb{F}_2)=L(\langle a,b\rangle)$, for $u\in L(\langle b\rangle)$, we establish \[ \mathrm{d}_{\mathrm{MT}}(L(\langle a\rangle),uL(\langle a\rangle)u^*) =\sqrt{1-|τ(u)|^4}. \] Finally, we construct maximally distant masas in $L(\mathbb{F}_2)$ that do not arise from subgroups of $\mathbb{F}_2$.