Finite-index extremal II1 subfactor data

Assume AC. A II1 factor is a von Neumann algebra $A$ with center $\mathbb{C}1$, no nonzero minimal projections, and a faithful normal normalized trace ${\tau }_A$, with the operator and trace conventions of Von Neumann algebras and commutants and States, tracial states and faithful normal traces on a von Neumann algebra. A II1 subfactor inclusion is a unital inclusion $N\subseteq M$ of II1 factors, with ${\tau }_N={\tau }_M{\mathrm{\mid }}_N$. Its trace-preserving normal faithful conditional expectation $E_N:M\to N$ is the unique expectation constructed in Tracial standard form, normal functional vectors, and expectation; it is positive, unital, $N$-bimodular, and the restriction of the orthogonal projection $e_N:L^2(M)\to L^2(N)$.

A normal right Hilbert-space $N$-module is a Hilbert space with a unital star representation of $N^{\mathrm{o}\mathrm{p}}$ that is WOT-continuous on the operator-norm unit ball. This is the topological normality convention of the trace definition above. Right multiplication is $R_a(x)=xa$, so $R_a{R}_b=R_{ba}$. The amplification theorem Ordinary Hilbert-module dimension and finite projection models constructs a right-module unitary $H\cong P({\mathrm{\ell }}^2(I)\otimes L^2(N))$ and its model-independent canonical semifinite trace on ${\mathrm{End}}_N(H)$, given in that model by ${\mathrm{Tr}}_H(T)={\sum }_{i\in I}{\tau }_N(T_{ii})$ for positive $T$. Define ${\dim }_{N}H={\mathrm{Tr}}_H(1)$; in particular ${\dim }_N{L}^2(N)=1$. The ordinary Jones index of the inclusion is $[M:N]={\dim }_N{L}^2(M,{\tau }_M)=:\lambda \mathrm{.}$ The inclusion has finite index when $\lambda <\mathrm{\infty }$. No finite bounded frame or complete $N$-valued module model is part of this hypothesis.

For finite index, put $B={\mathrm{End}}_N{L}^2(M)$ and ${\tau }_B={\lambda }^{-1}{\mathrm{Tr}}_{L^2(M)}$. Let $J(x)=x^{\ast }$ on $L^2(M)$, so $R_a=J{L}_{a^{\ast }}J$. The inclusion is extremal when ${\tau }_B(R_a)={\tau }_M(a)(a\in N^{\mathrm{\prime }}\cap M)\mathrm{.}$ Here $N^{\mathrm{\prime }}\cap M=\{a\in M:an=na\text{ for every }n\in N\}$, and $R_a$ belongs to $B$ for these $a$. Equivalently, the normalized trace on the commutant of the left $N$ action and ${\tau }_M$ agree on $N^{\mathrm{\prime }}\cap M$, transported by $J$. This specifies the right-action trace: the normalized trace of $B$ on the left action of $M$ agrees automatically with ${\tau }_M$ and does not define extremality.

For later calculations write $\mathcal{X}=M$ with $N$-valued inner product $⟨x,y{⟩}_N=E_N(x^{\ast }y)$ and norm $\mathrm{\parallel }x{\mathrm{\parallel }}_{\mathcal{X}}=\mathrm{\parallel }{E}_N(x^{\ast }x){\mathrm{\parallel }}^{1\mathrm{/}2}$. Its completeness, finite projection model $\mathcal{X}\cong p{N}^k$, and bounded coordinate frame are consequences proved below from ordinary finite index; they are not additional assumptions. No separability or irreducibility hypothesis is imposed.