Preparing infinite factors

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

For infinite factors the proof of strong stability works with prepared pairs: type III$_1$ factors on one Hilbert space with a common cyclic separating vector and a common irreducible hyperfinite II$_1$ subfactor whose left and right actions agree (Definition 5.1). This lesson shows that two close infinite factors on a separable space become a prepared pair after tensoring with one fixed factor $D$ and moving the second algebra by a small unitary, with control of the completely bounded distance (Proposition 5.2). The fixed factor is an infinite tensor product of $2\times2$ matrix algebras with two incommensurable ratios. Its centralizer is irreducible, and tensoring with it makes every factor of type III$_1$ (Proposition 2.3). The subfactor comes from the programme's construction of an injective subalgebra in a type III$_1$ factor (Proposition 3.1), and the common vector from the uniqueness of normal representations of type III factors (Lemma 4.1) [OpenAI-289, Section 6].

We use: Close operator algebras (Lemmas 1.4 and 1.5, Corollary 3.2, Definition 4.1 and Corollary 4.3); Averaging and the completely positive criterion (Theorem 3.1 and Lemma 5.1); the Tomita–Takesaki theorem for a faithful normal state ($S=J\Delta^{1/2}$, $JMJ=M'$, $\Delta^{it}M\Delta^{-it}=M$), Modular time from a faithful normal state; the KMS characterization of the modular group; the fixed-point criterion (the centralizer $P_\varphi=\{a:\varphi(ax)=\varphi(xa)\ \forall x\}$ of a faithful normal state is the fixed-point algebra of $\sigma^\varphi$); the modular expectation theorem; cone vectors of normal positive functionals and the standard form axioms; for continuous cores, the core trace, the factor criterion (F3), the independence (MG10) of the Connes spectrum $\Gamma$ from the weight, the formula (MT25) $S(P)=\{0\}\cup\exp\Gamma$ for type III factors with separable predual, and the definition (HC1) of type III$_1$ as $S(P)=[0,\infty)$; Corollaries 4.2 and 7.4 of Integration for a trace ($P$ and $P'$ have the same type; semifinite normal traces on a semifinite factor are proportional); Theorem 5.2 of An injective subalgebra with the same core commutant; Theorem 1.6 of Uniqueness of the injective II₁ factor and its model of $R$ as the closure of $\bigcup_kM_2(\mathbb C)^{\otimes k}$; injectivity as a norm-one projection; the facts on projections of countably decomposable type III factors, (B8) of Ultraproducts and the asymptotic centralizer (every nonzero projection is equivalent to $1$); Proposition 7.1 and Corollary 11.5(2) of the tensor product lesson; and Kaplansky's density theorem.

1. A fixed tensor factor

Fix $\lambda=e^{-1},\qquad\mu=e^{-\sqrt2}.$ For $0<\nu<1$ let $\rho_\nu$ be the state of $M_2(\mathbb C)$ with density $h_\nu=(1+\nu)^{-1}\operatorname{diag}(1,\nu)$. Choose types $\nu_1,\nu_2,\dots\in\{\lambda,\mu\}$, each occurring infinitely often. Site $k$ is $M_2(\mathbb C)$ acting by left multiplication on the Hilbert–Schmidt space $M_2(\mathbb C)$ with the unit vector $h_{\nu_k}^{1/2}$. Let $K$ be the infinite tensor product of these Hilbert spaces along the unit vectors $h_{\nu_k}^{1/2}$, $\Omega_\rho$ the product vector, $D_n$ the algebra of the first $n$ sites (a copy of $M_{2^n}(\mathbb C)$), and $D=(\bigcup_nD_n)''$. The vector state of $\Omega_\rho$ on $D$ is the product state $\rho$.

The vector $\Omega_\rho$ is cyclic for $D$, because $\{xh^{1/2}_\nu:x\in M_2(\mathbb C)\}$ is the whole site space, and likewise cyclic for the right multiplications on the sites, which commute with $D$. So $\Omega_\rho$ is cyclic and separating for $D$, and $\rho$ is a faithful normal state. With $J$ the modular conjugation of $(D,\Omega_\rho)$, $D'=JDJ$ is generated by the increasing finite-dimensional unital $*$-algebras $JD_nJ$. On $D_n$ the modular group of $\rho$ is $\operatorname{Ad}h_{(n)}^{it}$, with $h_{(n)}=h_{\nu_1}\otimes\cdots\otimes h_{\nu_n}$, by the KMS characterization; for instance the off-diagonal matrix unit $e_{12}$ at a site of type $\nu$ satisfies $\begin{aligned} \sigma^\rho_t(e_{12})&=\nu^{-it}e_{12}\\ &=e^{-it\log\nu}e_{12}.\quad(1.1) \end{aligned}$

Lemma 1.1. $D_\rho'\cap D=\mathbb C1$. In particular $D$ is a factor.

Proof. For two sites of the same type, the unitary of $D_n$ exchanging them preserves $\rho$, so it lies in the centralizer $D_\rho$. Let $x\in D_\rho'\cap D$ and $a\in D_m$. By Kaplansky's density theorem choose $x_j\in\bigcup_nD_n$ with $\|x_j\|\le\|x\|$ and $\|(x-x_j)\Omega_\rho\|\to0$; say $x_j\in D_{n_j}$. A product $w_j$ of exchanges of sites of equal type moves the first $m$ sites to sites beyond $n_j$; it lies in $D_\rho$, so it commutes with $x$ and preserves $\rho$. For $a_j=w_jaw_j^*$, $\begin{gathered} \rho(a_jx)=\rho(w_jaxw_j^*)=\rho(ax),\\ \rho(a_jx_j)=\rho(a)\rho(x_j), \end{gathered}$ the second because $a_j$ and $x_j$ sit on disjoint sites of a product state. Also $|\rho(a_jx)-\rho(a_jx_j)|$$\le\rho(aa^*)^{1/2}\|(x-x_j)\Omega_\rho\|\to0$ and $\rho(x_j)\to\rho(x)$. Hence $\rho(ax)=\rho(a)\rho(x)$, that is, $(x-\rho(x)1)\Omega_\rho$ is orthogonal to the dense set $\bigcup_mD_m\Omega_\rho$. Since $\Omega_\rho$ is separating, $x=\rho(x)1$. $\square$

2. Continuous cores and type III₁

Let $P$ be a von Neumann algebra on a Hilbert space $H_P$, with a faithful normal state $\psi$. On $L^2(\mathbb R,H_P)$ put $\begin{gathered} (\pi(x)\xi)(t)=\sigma^\psi_{-t}(x)\xi(t),\\ (\lambda_s\xi)(t)=\xi(t-s), \end{gathered}$ and let $C(P,\psi)=(\pi(P)\cup\{\lambda_s\})''$, the continuous core (the crossed product by the modular group). From the course on flows of weights we use: $C(P,\psi)$ carries a faithful normal semifinite trace $\tau$ with $\tau\circ\theta_s=e^{-s}\tau$, where the dual action $\theta_s$ fixes $\pi(P)$ and multiplies $\lambda_t$ by $e^{-ist}$ [core trace]; if $P$ is a factor, $C(P,\psi)$ is a factor exactly when the Connes spectrum $\Gamma(\sigma^\psi)$ is $\mathbb R$ [(F3)]; $\Gamma(\sigma^\psi)$ does not depend on the faithful normal semifinite weight used [(MG10)]; and the Connes spectrum of the trivial action is $\{0\}$, since it is contained in the spectrum.

Lemma 2.1 (factorial cores). Let $P$ be a factor with separable predual and $\psi$ a faithful normal state. Then $C(P,\psi)$ is a factor exactly when $P$ is of type III$_1$, and then $C(P,\psi)$ is a factor of type II$_\infty$.

Proof. If $P$ is semifinite, with a faithful normal semifinite trace $\tau_P$, then $\sigma^{\tau_P}$ is trivial, so $\Gamma(\sigma^\psi)=\Gamma(\sigma^{\tau_P})=\{0\}$ and $C(P,\psi)$ is not a factor. If $P$ is of type III, (MT25) gives $S(P)=\{0\}\cup\exp\Gamma(\sigma^\psi)$, so $\Gamma(\sigma^\psi)=\mathbb R$ exactly when $S(P)=[0,\infty)$, that is, when $P$ is of type III$_1$.

Let $C=C(P,\psi)$ be a factor, with the trace $\tau$ scaled by $\theta$. It is semifinite. If $C$ were finite, $\tau$ would be a multiple of the finite trace of $C$ (Corollary 7.4 of the integration lesson), so $\tau(1)<\infty$, and $\tau(1)=\tau(\theta_s(1))=e^{-s}\tau(1)$ would force $\tau(1)=0$. If $C$ were of type I, it would have a minimal projection $e$ with $0<\tau(e)<\infty$; $\theta_s(e)$ is again minimal, and minimal projections of a type I factor are equivalent, hence have equal trace, contradicting $\tau(\theta_s(e))=e^{-s}\tau(e)$ for $s\ne0$. So $C$ is of type II$_\infty$. $\square$

Proposition 2.2. For every factor $A$ with separable predual, $A\bar\otimes D$ is a factor of type III$_1$.

Proof. Let $\psi_A$ be a faithful normal state of $A$, $P=A\bar\otimes D$ and $\psi=\psi_A\otimes\rho$. By the KMS characterization, $\sigma^\psi$ acts on $A\otimes1$ as $\sigma^{\psi_A}\otimes\mathrm{id}$ and on $1\otimes D$ as $\mathrm{id}\otimes\sigma^\rho$. Let $Z$ be central in $C=C(P,\psi)$, represented on $L^2(\mathbb R,H_P)$.

A field of the algebra. The Fourier transform in $t$ turns $\lambda_s$ into multiplication by $e^{-is\zeta}$; these functions generate $L^\infty(\mathbb R)$, whose commutant on $L^2(\mathbb R,H_P)$ consists of the decomposable operators, multiplication by bounded measurable fields $\zeta\mapsto a(\zeta)\in B(H_P)$ (Proposition 7.1 of the tensor product lesson). So $Z$ is decomposable. The constant operators $1\otimes P'$ commute with $C$, hence with $Z$, so $a(\zeta)\in P$ almost everywhere. For $y$ in $1\otimes D_\rho$, $\pi(y)$ is the constant operator $y$, because $\sigma^\psi$ fixes $y$; so $a(\zeta)$ commutes with $1\otimes D_\rho$. By Proposition 7.1 and Corollary 11.5(2) of the tensor product lesson and Lemma 1.1, $\begin{aligned} a(\zeta)&\in(A\bar\otimes D)\cap\bigl(B(H_A)\bar\otimes D_\rho'\bigr)\\ &=A\bar\otimes(D\cap D_\rho')=A\otimes1 . \end{aligned}$

Invariance under a dense group. Let $e_{12}$ be the matrix unit at a site of type $\nu\in\{\lambda,\mu\}$, as an element $1\otimes e_{12}$ of $P$. By (1.1), $\pi(1\otimes e_{12})$ is $1\otimes e_{12}$ times multiplication by $e^{it\log\nu}$, which after Fourier transform is $1\otimes e_{12}$ composed with the translation of the frequency by $\log\nu$. Commutation of $Z$ with it, and the fact that $a(\zeta)\in A\otimes1$ commutes with $1\otimes e_{12}$, give $(a(\zeta)-a(\zeta-\log\nu))(1\otimes e_{12})=0$, hence $\begin{gathered} a(\zeta)=a(\zeta-\log\lambda)=a(\zeta-\log\mu)\\ \text{almost everywhere}. \end{gathered}$ The group generated by $\log\lambda=-1$ and $\log\mu=-\sqrt2$ is dense in $\mathbb R$, since $\sqrt2$ is irrational. Test the field on a countable dense set of pairs of vectors: each coefficient is a bounded function invariant under a dense group of translations; translation is continuous in $L^1_{\rm loc}$, so the coefficient is invariant under all translations, and convolution with an approximate identity shows that it is almost everywhere constant. Hence $Z$ is a constant $a_0\otimes1$, $a_0\in A$.

Conclusion. $Z$ commutes with $\pi(x\otimes1)$, which is the field $t\mapsto\sigma^{\psi_A}_{-t}(x)\otimes1$; by strong continuity of the modular group, $a_0$ commutes with $x$. So $a_0\in A\cap A'=\mathbb C1$, and $C$ is a factor. Lemma 2.1 shows that $P$ is of type III$_1$. $\square$

3. Irreducible hyperfinite subfactors in type III₁ factors

Proposition 3.1. Let $P$ be a type III$_1$ factor with separable predual, acting on a separable Hilbert space. There are a faithful normal state $\varphi$ on $P$ and a hyperfinite II$_1$ factor $R=(\bigcup_jQ_j)''$, with increasing full matrix subalgebras $Q_j\cong M_{2^j}(\mathbb C)$, such that $R\subseteq P_\varphi,\qquad R'\cap P=\mathbb C1 .$

Proof. Theorem 5.2 of the lesson on core commutants, applied to the inclusion $P\subseteq P$ with the identity expectation, gives a faithful normal state $\varphi$ and an approximately finite-dimensional, hence injective, subalgebra $P_0\subseteq P$ with $\sigma^\varphi_t(P_0)=P_0$ and $(P_0)_\varphi'\cap P=\mathbb C1$, where $(P_0)_\varphi$ is the centralizer of $\varphi|_{P_0}$. Put $R=(P_0)_\varphi$. By the modular expectation theorem, the modular group of $\varphi|_{P_0}$ is the restriction of $\sigma^\varphi$, so by the fixed-point criterion $R=P_0\cap P_\varphi\subseteq P_\varphi$.

The center of $R$ lies in $R'\cap P=\mathbb C1$, so $R$ is a factor, and $\varphi|_R$ is a faithful normal tracial state, so $R$ is finite. It is not finite-dimensional: if it were, with matrix units $e_{ij}$, the elements $\sum_ie_{i1}ye_{1i}$, $y\in e_{11}Pe_{11}$, would form a copy of the corner $e_{11}Pe_{11}$ inside $R'\cap P$, and a corner of a type III factor is not one-dimensional. So $R$ is a II$_1$ factor. It is injective: the modular expectation theorem, applied to $\varphi|_{P_0}$ and its fixed-point algebra $R$, gives a faithful normal conditional expectation $P_0\to R$, and composing it with a norm-one projection of $B(H)$ onto $P_0$ gives a norm-one projection onto $R$. By Theorem 1.6 of the uniqueness lesson, $R$ is isomorphic to the closure of $\bigcup_kM_2(\mathbb C)^{\otimes k}$; the isomorphism is normal, and the images $Q_j$ of the algebras $M_2(\mathbb C)^{\otimes j}$ generate $R$. $\square$

4. Normal representations of type III factors

Lemma 4.1. Let $P$ be a type III factor with separable predual and $\pi$ a nonzero normal representation of $P$ on a separable Hilbert space $H$. Then $\pi$ is unitarily equivalent to the standard representation. Consequently, for every faithful normal state $\varphi$ of $P$ there is a unit vector $\Omega\in H$, cyclic and separating for $\pi(P)$, with $\varphi=\langle\pi(\cdot)\Omega,\Omega\rangle$.

Proof. Let $P$ act standardly on $L^2(P)$ with conjugation $J$ and positive cone $\mathcal P$. By Zorn's lemma $H$ is an orthogonal sum of cyclic subspaces $[\pi(P)\xi_k]$, countably many since $H$ is separable. For each $k$, the normal positive functional $\langle\pi(\cdot)\xi_k,\xi_k\rangle$ is $\omega_{\eta_k}$ for a cone vector $\eta_k\in\mathcal P$, and $\pi(x)\xi_k\mapsto x\eta_k$ extends to a unitary from $[\pi(P)\xi_k]$ onto $[P\eta_k]\subseteq L^2(P)$ intertwining the actions. So $\pi$ is equivalent to a subrepresentation of $L^2(P)\otimes\ell^2$. The commutant $P'=JPJ$ is a countably decomposable type III factor, hence properly infinite; as in Section 4 of Close operator algebras it contains isometries $s_k$ with orthogonal ranges summing to $1$, and $V(\xi\otimes\delta_k)=s_k\xi$ is a unitary from $L^2(P)\otimes\ell^2$ onto $L^2(P)$ with $V(x\otimes1)=xV$. Hence $\pi$ is equivalent to the subrepresentation on $qL^2(P)$ for a nonzero projection $q\in P'$. In $P'$, $q$ is equivalent to $1$: an isometry $w\in P'$ with $ww^*=q$ is a unitary from $L^2(P)$ onto $qL^2(P)$ intertwining the actions.

For a faithful normal state $\varphi$, let $\xi_\varphi\in\mathcal P$ be its cone vector. It is separating because $\varphi$ is faithful. Since $J\xi_\varphi=\xi_\varphi$ and $JPJ=P'$, the closures of $P\xi_\varphi$ and $P'\xi_\varphi$ are exchanged by $J$; a vector separating for $P$ is cyclic for $P'$, so $\xi_\varphi$ is also cyclic for $P$. Transport $\xi_\varphi$ to $H$. $\square$

5. Prepared pairs

Definition 5.1 (prepared pair). A prepared pair consists of type III$_1$ factors $M,N\subseteq B(H)$ on a separable Hilbert space, a unit vector $\Omega\in H$ cyclic and separating for both, and a hyperfinite II$_1$ factor $R=(\bigcup_jR_j)''\subseteq M\cap N$, generated by increasing full matrix subalgebras $R_j$, such that:

$R'\cap M=R'\cap N=\mathbb C1$;
the vector state $\varphi=\langle\cdot\,\Omega,\Omega\rangle$ centralizes $R$ on $M$ and on $N$: $R\subseteq M_\varphi\cap N_\varphi$;
$J_MrJ_M=J_NrJ_N$ for every $r\in R$, where $J_M,J_N$ are the modular conjugations of $(M,\Omega)$ and $(N,\Omega)$.

If $\varphi$ centralizes $R$ on $M$ and $r\in R$, then $r^*$ is fixed by $\sigma^\varphi$, so $\Delta_M^{it}r^*\Omega=r^*\Omega$ for all $t$; hence $r^*\Omega$ lies in the kernel of $\log\Delta_M$ and $\Delta_M^{1/2}r^*\Omega=r^*\Omega$. Since $S_M(r^*\Omega)=r\Omega$, $S_M=J_M\Delta_M^{1/2}$ and $J_M\Omega=\Omega$, $\begin{gathered} r\Omega=J_Mr^*\Omega=J_Mr^*J_M\Omega\\ (r\in R).\quad(5.1) \end{gathered}$

Proposition 5.2 (preparing an infinite pair). Let $M_0,N_0\subseteq B(H)$ be factors on a separable Hilbert space, $M_0$ infinite, with $d(M_0,N_0)<t<4\cdot10^{-9}$. There are unitaries $u_0\in B(H)$ and $v\in B(H\otimes K)$ with $\|u_0-1\|+\|v-1\|\le3\cdot10^7t$ such that $\begin{gathered} M=M_0\bar\otimes D,\\ N=v\bigl((u_0N_0u_0^*)\bar\otimes D\bigr)v^* \end{gathered}$ form a prepared pair and $d_{\rm cb}(M,N)\le6\cdot10^7t$.

Proof. A common infinite family of isometries. $M_0$ has separable predual, so it is countably decomposable, and an infinite factor is properly infinite; it contains an infinite family of isometries, which generates a copy $S_0$ of $B(\ell^2)$, an injective algebra. Since $S_0\subset_tN_0$, Christensen's theorem gives a unitary $u_0$ with $\|u_0-1\|\le150t$ and $S_0\subseteq u_0N_0u_0^*$. In particular $N_0$ is infinite. By Lemma 1.5 and Corollary 4.3 of Close operator algebras, $d_{\rm cb}(M_0,u_0N_0u_0^*)$$=d(M_0,u_0N_0u_0^*)<301t$. By Lemma 5.1(2) of the averaging lesson, applicable since $D'$ is generated by the increasing algebras $JD_nJ$, $\begin{gathered} d_{\rm cb}(M,N_1)<301t,\\ N_1=(u_0N_0u_0^*)\bar\otimes D . \end{gathered}$ Both $M$ and $N_1$ are type III$_1$ factors by Proposition 2.2.

The subfactor and the vector. Proposition 3.1 gives a faithful normal state $\varphi$ of $M$ and $R=(\bigcup_jR_j)''\subseteq M_\varphi$ with $R'\cap M=\mathbb C1$. By Lemma 4.1 there is a unit vector $\Omega\in H\otimes K$, cyclic and separating for $M$, with $\varphi=\langle\cdot\,\Omega,\Omega\rangle$ on $M$; no algebra is moved. By (5.1), $r\Omega=J_Mr^*J_M\Omega$ for $r\in R$. Put $S=J_MRJ_M\subseteq M'$.

Moving $N_1$. Since $R\subseteq M$, $R\subset_{301t}N_1$, and Christensen's theorem gives $v_1$ with $v_1Rv_1^*\subseteq N_1$ and $\|v_1-1\|\le45150t$. Put $N_2=v_1^*N_1v_1\supseteq R$; then $d_{\rm cb}(M,N_2)\le90601t$, and Corollary 3.2 of Close operator algebras gives $d(M',N_2')\le181202t$. The algebra $S$ is generated by the increasing finite-dimensional algebras $J_MR_jJ_M$, so it is injective. So $S\subset_{181203t}N_2'$, and Christensen's theorem gives a unitary $v_2\in(S\cup N_2')''$ with $v_2Sv_2^*\subseteq N_2'$ and $\|v_2-1\|\le150\cdot181203t$. Both $S$ and $N_2'$ commute with $R$, so $v_2\in R'$. Put $N=v_2^*N_2v_2$ and $v=v_2^*v_1^*$. Then $R\subseteq N$, $S\subseteq N'$, $\|v-1\|\le27225600t$, and $\begin{aligned} d_{\rm cb}(M,N)&\le90601t+2\cdot150\cdot181203t\\ &\le6\cdot10^7t . \end{aligned}$ Also $d(M',N')\le2d_{\rm cb}(M,N)<1/2$ and $d(M,N)<1/2$, by the bound on $t$.

Irreducibility. For $x\in M$ let $E_j(x)=\int_{\mathcal U(R_j)}uxu^*\,du$. Every weak operator cluster point of $E_j(x)$ commutes with all $R_j$, hence lies in $R'\cap M=\mathbb C1$, and $\varphi(E_j(x))=\varphi(x)$ because $R\subseteq M_\varphi$; so $E_j(x)\to\varphi(x)1$ weakly. Let $p\in R'\cap N$ be a projection and choose $x\in M_1$ with $\|p-x\|<1/2$. Since $E_j(p)=p$, $\|E_j(x)-p\|\le\|x-p\|$, and in the limit $\|\varphi(x)1-p\|<1/2$. A projection other than $0,1$ has distance $1/2$ from the scalars, so $p\in\{0,1\}$ and $R'\cap N=\mathbb C1$. The same argument in the commutants gives $S'\cap N'=\mathbb C1$. It uses $S'\cap M'=J_M(R'\cap M)J_M=\mathbb C1$ and the invariance of the vector state on $M'$ under the unitaries $s=J_MuJ_M$ of $S$: by (5.1), $s^*\Omega=u\Omega$, so $\langle sx's^*\Omega,\Omega\rangle=\langle u^*x'u\Omega,\Omega\rangle=\langle x'\Omega,\Omega\rangle$ for $x'\in M'$.

The common vector. Let $p$ project onto the closure of $N'\Omega$; then $p\in N$. For $r\in R$ and $x'\in N'$, $rx'\Omega=x'r\Omega=x'(J_Mr^*J_M)\Omega\in N'\Omega$, because $J_Mr^*J_M\in S\subseteq N'$. So $p$ commutes with $R$, $p\in R'\cap N$, and $p=1$: $\Omega$ is cyclic for $N'$, hence separating for $N$. Similarly the projection onto the closure of $N\Omega$ lies in $N'$ and commutes with $S$, since $(J_MrJ_M)x\Omega=x\,r^*\Omega$ for $x\in N$; it is $1$, and $\Omega$ is cyclic for $N$.

The centralizer and the right actions. For a unitary $u\in R$ and $x\in N$, $u^*\Omega=J_MuJ_M\Omega$ with $J_MuJ_M$ a unitary of $N'$, so $\langle uxu^*\Omega,\Omega\rangle$$=\langle x\,J_MuJ_M\Omega,J_MuJ_M\Omega\rangle$$=\langle x\Omega,\Omega\rangle$. Thus $\varphi\circ\operatorname{Ad}u=\varphi$ on $N$, and $R\subseteq N_\varphi$. By (5.1) for $N$, $J_NrJ_N\Omega=r^*\Omega=J_MrJ_M\Omega$; both operators lie in $N'$, and $\Omega$ is separating for $N'$, so $J_NrJ_N=J_MrJ_M$. Finally $N=vN_1v^*$ is a type III$_1$ factor. All conditions of Definition 5.1 hold. $\square$

6. Exercises

Exercise 6.1. Show that the subgroup of $\mathbb R$ generated by $1$ and $\sqrt2$ is dense.

Exercise 6.2. Compute the modular operator of $(M_2(\mathbb C),\rho_\nu)$ on the Hilbert–Schmidt space with vector $h_\nu^{1/2}$, and check (1.1).

Exercise 6.3. Show that a unitary $u$ with $\varphi(uxu^*)=\varphi(x)$ for all $x$ lies in the centralizer $P_\varphi$.

Exercise 6.4. In the proof of Lemma 4.1, check that $V(\xi\otimes\delta_k)=s_k\xi$ is unitary and intertwines $x\otimes1$ with $x$.

Exercise 6.5. Let $P$ be a factor and $e\ne0$ a projection. Show that the corner $ePe$ is one-dimensional exactly when $e$ is a minimal projection, and conclude that corners of type III factors are not one-dimensional.

7. Solutions

Solution 6.1. A subgroup of $\mathbb R$ is dense or of the form $c\mathbb Z$. If $1=mc$ and $\sqrt2=m'c$ with integers $m,m'$, then $\sqrt2=m'/m$ is rational.

Solution 6.2. $S(xh^{1/2})=x^*h^{1/2}=h^{1/2}\,(h^{-1/2}x^*h^{1/2})$; one checks $J\xi=\xi^*$ and $\Delta\xi=h\xi h^{-1}$. Then $\sigma_t(x)=h^{it}xh^{-it}$, and $h^{it}e_{12}h^{-it}=1^{it}\nu^{-it}e_{12}$.

Solution 6.3. Replace $x$ by $xu$: $\varphi(ux)=\varphi(uxuu^*)=\varphi(xu)$.

Solution 6.4. $\langle V(\xi\otimes\delta_k),V(\eta\otimes\delta_l)\rangle$$=\langle s_l^*s_k\xi,\eta\rangle=\delta_{kl}\langle\xi,\eta\rangle$, so $V$ is isometric, and its range contains $s_ks_k^*L^2(P)$ for every $k$, whose sum is $L^2(P)$. Since $s_k\in P'$, $V(x\xi\otimes\delta_k)=s_kx\xi=xs_k\xi$.

Solution 6.5. If $ePe=\mathbb Ce$, then every subprojection of $e$ in $P$ is $0$ or $e$, so $e$ is minimal; conversely a minimal projection of a factor has $ePe=\mathbb Ce$. A type III factor has no minimal projections, since a minimal projection is abelian and finite.

References

[OpenAI-289] OpenAI, Universal strong Kadison–Kastler stability, preprint, 23 September 2026, Section 6. https://github.com/openai/math/tree/main/preprints/Universal-strong-Kadison-Kastler-stability-September-23-2026