Ambient maximal abelian algebras from modular spectral gaps

A centralizer need not be maximal abelian. The question here is whether it contains an abelian algebra that is maximal in the entire ambient algebra. We construct such algebras by changing a bounded functional within its centralizer, creating a modular gap, and then assembling orthogonal corners. The perturbation estimates are proved on the original GNS space, where their frequency bounds can be checked directly.

The final application concerns faithful strictly semifinite weights on separable factors of type $\mathrm{III}_\lambda$, $0\leq\lambda<1$. Its type-theoretic input is the precise continuous-core spectrum theorem stated in CAM-06. The elementary spectral and perturbation arguments preceding it apply more generally.

A bounded functional with an inner modular period

We first record a consequence of the Fourier filters in RSF-01–02. For a uniformly bounded real action, a vector whose hull spectrum is contained in $\{0\}$ is fixed. Choose $f\in C_c^\infty(\mathbb R)$ equal to one near zero, with inverse Fourier kernel $k$, and put $f_\varepsilon(p)=f(p/\varepsilon)$. The compact-cutoff theorem gives $\alpha_{f_\varepsilon}x=x$. Translating the inverse kernel $\varepsilon k(\varepsilon t)$ therefore gives, for each fixed $s$, $\begin{aligned} \|\alpha_sx-x\| &\leq C\|x\|\, \|k(\,\cdot-\varepsilon s)-k\|_1\\ &\longrightarrow0. \end{aligned} \tag{CAM.1}$ Thus $\alpha_sx=x$. Conversely a fixed nonzero vector has filters $\alpha_fx=f(0)x$, and hence spectrum $\{0\}$.

If $\alpha_T=\operatorname{id}$, with $T>0$, then $\operatorname{Sp}(\alpha)\subseteq \frac{2\pi}{T}\mathbb Z. \tag{CAM.2}$ Indeed translation of a filter shows that $(e^{iTp}-1)g(p)$ annihilates the action for every $g\in A$. At a point outside the displayed lattice choose $g$ nonzero there. The action-annihilator hull excludes that point. Combining (CAM.1) and (CAM.2) proves that a periodic modular action is lacunary: any closed interval about zero of radius smaller than $2\pi/T$ has precisely the fixed algebra as its spectral subspace.

Let $\phi$ now be a faithful bounded normal positive functional on $M$, and suppose $\sigma_T^\phi=\operatorname{Ad}(u), \qquad u\in\mathcal U(M),\quad T>0. \tag{CAM.3}$ Invariance of $\phi$ gives $\phi(uyu^*)=\phi(y)$ for all $y\in M$. Taking $y=xu$ yields $\phi(ux)=\phi(xu)$. The centralizer criterion CZ-05 puts $u$ in $M_\phi$. Equation (CAM.3) fixes $M_\phi$ pointwise, so in fact $u\in Z(M_\phi)$.

Here is an argument construction using only self-adjoint spectral calculus. Put $c=(u+u^*)/2$, $s=(u-u^*)/(2i)$, and $q=1_{[0,\infty)}(s)$. These elements commute and $c^2+s^2=1$. Define $b=\arccos(c)(2q-1)$. The bounded calculus gives $-\pi\leq b\leq\pi$, $\cos b=c$, and $\sin b=(2q-1)(1-c^2)^{1/2}=s$. Thus $e^{ib}=u$, with a definite choice at the negative real axis. Put $h=e^{b/T}$. Then $\begin{gathered} h\in Z(M_\phi),\qquad h^{iT}=u,\\ e^{-\pi/T}1\leq h\leq e^{\pi/T}1. \end{gathered} \tag{CAM.4}$ The spectral projections and commuting bounded calculus are supplied by SK-05 and SK-08; they keep $b,h$ inside $Z(M_\phi)$. The density formula CZ-11 gives a faithful bounded functional $\psi=\phi_{h^{-1}}$ satisfying $\sigma_t^\psi=\operatorname{Ad}(h^{-it})\sigma_t^\phi, \qquad \sigma_T^\psi=\operatorname{id}. \tag{CAM.5}$ The element $h$ is fixed by $\sigma^\psi$. Extend its abelian generated algebra to a maximal abelian von Neumann subalgebra $A$ of $M_\psi$. This extension follows by Zorn: the von Neumann algebra generated by a chain of commuting abelian algebras is abelian, since commutants are weak-star closed. Maximality is equivalent to $A'\cap M_\psi=A$: otherwise the real or imaginary part of an extra commuting element enlarges it.

By periodicity, $\psi$ is lacunary, so WC-06 makes $A$ maximal abelian in $M$. Every $a\in A$ commutes with $h$ and is fixed by $\sigma^\psi$; (CAM.5) then makes it fixed by $\sigma^\phi$. We have proved $A\subset M_\phi, \qquad A'\cap M=A. \tag{CAM.6}$ No inner-period centralizer theorem is left implicit in this bounded-functional argument.

One central corner removes an entire annulus

Let $M$ be a factor, let $\alpha$ be a point-ultraweakly continuous real automorphism action, and put $N=M^\alpha$, $D=Z(N)$. Its Connes spectrum is $\Gamma(\alpha)= \bigcap_{0\ne e\in\operatorname{Proj}(N)} \operatorname{Sp}(\alpha^e). \tag{CAM.7}$ Here $\alpha^e$ is the action on $eMe$. Filtering commutes with compression by a fixed projection, so a corner vector has the same spectrum in the corner and in $M$.

We give the central-corner and directedness arguments explicitly. The central carrier in $N$ satisfies $c_N(e)=\bigvee_{v\in\mathcal U(N)}vev^*. \tag{CAM.8}$ The join is invariant under all unitaries of $N$, contains $e$, and is bounded above by every central projection containing $e$, which proves the formula. Projection joins and the unitary commutation test are proved in BK-02 and BK-08.

Write $c=c_N(e)$. If $0\ne x\in cMc$ has spectrum in a closed set $E$, some $v,w\in\mathcal U(N)$ satisfy $vev^*xwew^*\ne0$: if all these products vanished, taking the joins first on the left and then on the right would give $cxc=0$. Hence $ev^*xwe\ne0$ belongs to $eMe$ and still has spectrum in $E$, because the multiplying elements are fixed. The adjoint/product theorem RSF-03 justifies that assertion. Conversely the smaller corner is contained in the larger one. Testing by arbitrarily small compact Fourier neighborhoods therefore gives $\operatorname{Sp}(\alpha^e) =\operatorname{Sp}(\alpha^{c_N(e)}), \qquad \Gamma(\alpha)= \bigcap_{0\ne z\in\operatorname{Proj}(D)} \operatorname{Sp}(\alpha^z). \tag{CAM.9}$ The local detection used here follows directly from the hull definition: if $p$ is in an action spectrum, any compact Fourier cutoff supported in a neighborhood of $p$ and nonzero at $p$ is a nonzero filter. Its nonzero output has spectrum in that neighborhood.

Compare central corners even when they are orthogonal. Given nonzero $e_1,e_2\in\operatorname{Proj}(D)$ and $\eta>0$, there are nonzero $f_i\leq e_i$ in $D$ with $\operatorname{Sp}(\alpha^{f_i}) \subseteq\operatorname{Sp}(\alpha^{f_j})+[-\eta,\eta] \quad(i\ne j). \tag{CAM.10}$ Since $M$ is a factor, its nonzero projections have full ambient central carrier. The carrier test TE-02 gives $e_1Me_2\ne0$. Choose a nonzero element there and a Fourier filter producing $0\ne y=e_1ye_2$ with compact spectrum in an interval $K$ of diameter less than $\eta$. Existence follows from nonempty spectrum and local detection. Form $\mathcal B=\{a\alpha_t(y)b: a,b\in N,\ t\in\mathbb R\}. \tag{CAM.11}$ Let $f_1$ and $f_2$ be the joins of its left and right supports. Translation preserves the family, as do left and right multiplication by fixed unitaries. Thus these joins are fixed and commute with $N$, so they lie in $D$; also $0\ne f_i\leq e_i$. Every member has spectrum in $K$.

For $0\ne x\in f_1Mf_1$, there are $b_0,b_1\in\mathcal B$ with $b_0^*xb_1\ne0$. Indeed the ranges of members of $\mathcal B$ span a dense subspace of $f_1H$. If all such sandwiches vanished, all matrix coefficients of $x|_{f_1H}$ on this dense span would vanish. Now localize a nonzero $x\in f_1Mf_1$ to a compact spectral neighborhood of any $p\in\operatorname{Sp}(\alpha^{f_1})$. Its sandwich is a nonzero vector of $f_2Mf_2$ with spectrum in the compact sum $-K+\operatorname{Sp}_\alpha(x)+K$. Shrinking that neighborhood to $p$, and using closedness of a closed subset of $\mathbb R$ plus the compact interval $[-\eta,\eta]$, proves the first inclusion in (CAM.10). Apply the same reasoning to $\mathcal B^*$ for the other inclusion. No intersection of $e_1$ and $e_2$ was required.

For fixed nonzero $r\in\operatorname{Proj}(D)$, consider $\mathcal F_r=\{ \operatorname{Sp}(\alpha^f)+[-\varepsilon,\varepsilon]: 0\ne f\leq r,\ f\in\operatorname{Proj}(D),\ \varepsilon>0\}. \tag{CAM.12}$ Its members are closed. They are downward directed: apply (CAM.10) to two given projections with $\eta>0$ so small that $2\eta$ is less than both of their thickening radii; the first smaller corner thickened by $[-\eta,\eta]$ lies in both original sets. They are also cofinal among the thickenings without the restriction $f\leq r$. Given a nonzero central fixed projection $e$ and radius $\varepsilon$, apply (CAM.10) to $r,e$ with radius $\varepsilon/2$. The corner obtained under $r$, thickened by the same interval, is contained in $\operatorname{Sp}(\alpha^e)+[-\varepsilon,\varepsilon]$. Hence (CAM.9) implies $\bigcap_{F\in\mathcal F_r}F=\Gamma(\alpha). \tag{CAM.13}$

If $\Gamma(\alpha)=\{0\}$, every $\mu>0$ and nonzero $r\in\operatorname{Proj}(D)$ admit a nonzero central fixed $f\leq r$ such that $\operatorname{Sp}(\alpha^f)\cap \bigl([-2\mu,-\mu]\cup[\mu,2\mu]\bigr) =\varnothing. \tag{CAM.14}$ Otherwise the intersections of all members of $\mathcal F_r$ with this compact annulus would be nonempty compact sets with the finite-intersection property, by directedness. Compactness would put a point of the annulus in (CAM.13), a contradiction. This is a simultaneous exclusion of a whole annulus, not a choice of unrelated corners for its individual points.

We also need corner invariance. For $0\ne e\in\operatorname{Proj}(N)$, put $r=c_N(e)$. For every nonzero central $z\leq r$, the projection $ze$ is nonzero and $c_N(ze)=z$. Conversely the central carrier of each nonzero $p\leq e$ in $N$ lies below $r$. Thus (CAM.9) identifies the intersection over all such $p$ with the intersection over nonzero central $z\leq r$. Equation (CAM.13), followed by intersection over the radii, makes that intersection $\Gamma(\alpha)$. Since $(eMe)^{\alpha^e}=eNe$, this proves $\Gamma(\alpha^e)=\Gamma(\alpha). \tag{CAM.15}$ This argument uses factoriality of $M$, not factoriality of its fixed algebra.

A compact action spectrum gives a sharply bounded implementer

Suppose a normal real automorphism group $\alpha$ on a von Neumann algebra $Q$ has $\operatorname{Sp}(\alpha)\subseteq[-\mu,\mu], \qquad \mu>0. \tag{CAM.16}$ Then there is $b\in Z(Q^\alpha)$ such that $0\leq b\leq\mu1, \qquad \alpha_t=\operatorname{Ad}(e^{itb}). \tag{CAM.17}$ Here the sharp bound comes from the action spectrum, not an asserted equality between the generator norm and its spectral radius.

First choose $f\in C_c^\infty$ equal to one near $[-\mu,\mu]$, with inverse kernel $k$. Every vector has compact spectrum in that interval; RSF-02 gives $\alpha_f=\operatorname{id}$. Hence $\alpha_s$ is the filter with kernel $k(\,\cdot-s)$. Translation is differentiable in $L^1$ for a Schwartz kernel: the difference quotient is the average of translates of $-k'$, converging in $L^1$ by translation continuity. The filter estimate proves differentiability in operator norm and gives a bounded generator $d$. Differentiating the automorphism identities makes $d$ a star derivation. The group law gives $\alpha'_t=d\alpha_t$, so differentiating $e^{-td}\alpha_t$ proves $\alpha_t=e^{td}$.

The smooth-test spectrum used in BI-04–08 equals the hull spectrum here. If a hull point is absent, choose $g$ in the annihilator nonzero there. The local inverse in UW-03, multiplied by a smooth cutoff, gives smooth annihilators on a whole neighborhood. Conversely such a neighborhood contains a smooth annihilator nonzero at the point, excluding it from the hull. Thus (CAM.16) is exactly the spectral bound required by the tail construction.

Apply the construction of BI-06 with $D=\mu$. Its hypotheses at that stage are that every spectrum is contained in $[-D,D]$, that filters obey the product and adjoint rules, and that $\alpha=e^{td}$ has bounded generator. They have just been proved; the preceding estimate $D=\|d\|$ in BI-05 is not needed. More explicitly, its tails are the left-support joins of $Q^\alpha([s,\infty))$, made left continuous by taking the meet over parameters below $s$. They are central in $Q^\alpha$, equal one for $s\leq0$, and zero for $s>\mu$. The dyadic lower and upper sums in (GI.13a)–(GI.13d), now on $[0,\mu]$, differ by at most $\mu2^{-n}$. Their common norm limit is the $b$ in (CAM.17), with precisely those spectral tails.

BI-07 proves the transfer of each frequency band to $\operatorname{Ad}(e^{itb})$, using these same tails. BI-08 identifies that action with $\alpha$: the two actions commute, each vector is a finite sum of arbitrarily narrow bands, and the difference action has singleton spectrum and bounded generator. Its singleton lemma, or (CAM.1), makes the difference action trivial. This proves (CAM.17) with the stated bound. Subtracting $\mu1/2$ gives the equally valid implementer $a=b-\mu1/2\in Z(Q^\alpha), \qquad \|a\|\leq\mu/2. \tag{CAM.18}$

The modular perturbation shifts frequencies by at most twice its norm

Let $\phi$ be a faithful bounded normal positive functional on $M$, put $\alpha=\sigma^\phi$, and take self-adjoint $a\in M_\phi$ with $\|a\|\leq c$. The faithful bounded perturbation $\psi=\phi_{e^{-a}}$ has $\beta_t=\sigma_t^\psi =\operatorname{Ad}(e^{-ita})\alpha_t. \tag{CAM.19}$ For every closed $E\subseteq\mathbb R$, we prove $M^\alpha(E)\subseteq M^\beta(E+[-2c,2c]). \tag{CAM.20}$ The sum on the right is closed, because the second set is compact.

Use the original bounded GNS map $\Lambda=\Lambda_\phi$, which is bounded and injective, and the modular group $U_t=\Delta_\phi^{it}$. On its dense GNS vectors define $L_a\Lambda(x)=\Lambda(ax), \qquad R_a\Lambda(x)=\Lambda(xa). \tag{CAM.21}$ Both extend to bounded self-adjoint operators of norm at most $\|a\|$. The assertion for $L_a$ is the GNS representation. For $R_a$, centralizer cyclicity gives $\phi(ax^*xa)=\phi(x^*xa^2)$; the difference $\|a\|^2\phi(x^*x)-\phi(ax^*xa)$ equals the nonnegative number $\phi(d x^*x d)$ with $d=(\|a\|^2-a^2)^{1/2}\in M_\phi$. Also $\phi(y^*xa)=\phi(ay^*x) =\langle\Lambda(x),\Lambda(ya)\rangle,$ which proves self-adjointness. Left and right multiplication commute. Since $\alpha_t(a)=a$, both operators commute with $U_t$. Put $D=L_a-R_a=D^*,\qquad \|D\|\leq2c. \tag{CAM.22}$ Power series and GNS covariance give $\Lambda(\beta_t x)=e^{-itD}U_t\Lambda(x). \tag{CAM.23}$ No change to a different GNS space is made.

Let $B=\log\Delta_\phi$. For every Fourier filter $h$, the normal coefficient argument of WC-06, (WC.43) gives $\Lambda(\alpha_hx)=h(B)\Lambda(x)$. If $x\in M^\alpha(E)$, smooth cutoffs outside $E$ therefore show that the spectral measure of $\xi=\Lambda(x)$ for $B$ is supported in $E$. To pass from compact cutoffs to the complement, exhaust it by the compact sets $\{s:|s|\leq n,\ d(s,E)\geq1/n\}$; for an empty $E$, RSF-02 already gives $x=0$.

Approximate $D$, by its bounded spectral calculus, with finite step operators $D_n=\sum_jd_jP_j, \qquad d_j\in[-2c,2c], \qquad \|D_n-D\|\longrightarrow0. \tag{CAM.24}$ Each $P_j$ commutes with every $U_t$, because it is a bounded spectral projection of $D$ and $D$ commutes with every $U_t$. Consequently $P_j$ commutes with every Fourier filter $h(B)$. To obtain the spectral projections of $B$, approximate $1_{(-\infty,r)}$ pointwise by uniformly bounded smooth compactly supported functions: take cutoffs on $[-n,n]$, multiply by a smooth transition that is one below $r-1/n$ and zero at and above $r$, and let $n\to\infty$. Spectral dominated convergence gives the strong limit $1_{(-\infty,r)}(B)$. Thus $P_j$ commutes with these half-line projections. Differences, monotone limits and complements give commutation with every Borel spectral projection. In particular $P_j\xi$ still has $B$-spectral measure supported in $E$. Take $h=\mathcal Fk\in A_c$ with support disjoint from $E+[-2c,2c]$. For every $j$, bounded functional calculus gives $h(B-d_j)P_j\xi=0$. Consequently $\int k(t)e^{-itD_n}U_t\xi\,dt =\sum_jh(B-d_j)P_j\xi=0. \tag{CAM.25}$ The integrals are norm integrals of bounded continuous vector orbits. The estimate $\|e^{-itD_n}-e^{-itD}\| \leq\min(2,|t|\|D_n-D\|)$ follows here from their common spectral calculus, and dominated convergence with $|k(t)|$ passes (CAM.25) to the integral for $D$. Equation (CAM.23) identifies that integral with $\Lambda(\beta_hx)$; alternatively pair with GNS vectors to justify passage through the bounded GNS map. Injectivity of $\Lambda$ gives $\beta_hx=0$. The exact closed-space criterion RSF-02 proves (CAM.20). In particular the argument covers unbounded closed half-lines and does not assume simultaneous weak-star continuity of multiplication.

Turning an annular gap into a centralizer MASA

Suppose $\phi$ is a faithful bounded normal positive functional and $\alpha=\sigma^\phi$ satisfies $\operatorname{Sp}(\alpha)\cap \bigl([-2\mu,-\mu]\cup[\mu,2\mu]\bigr) =\varnothing, \qquad \mu>0. \tag{CAM.26}$ The middle spectral space $Q=M^\alpha([-\mu,\mu]) \tag{CAM.27}$ is a von Neumann algebra. It is weak-star closed by RSF-02, contains the identity, and is closed under adjoints. The product theorem initially places a product in $[-2\mu,2\mu]$; intersecting with the action spectrum and using (CAM.26) puts it back in $[-\mu,\mu]$. Its restricted action has spectrum in that interval and fixed algebra exactly $M_\phi$.

CAM-03 gives an implementer $a\in Z(M_\phi)$ for the restricted action with $\|a\|\leq\mu/2$. Set $k=e^{-a},\qquad \psi=\phi_k, \qquad \beta=\sigma^\psi. \tag{CAM.28}$ The density is bounded and boundedly invertible. On $Q$, the two inner actions cancel and $\beta_t=\operatorname{id}$. In particular $Q\subset M_\psi$; centralizer cyclicity makes $\psi|_Q$ a finite faithful trace.

CAM-04 now gives the exact three spectral estimates $\begin{aligned} M^\alpha(({-\infty},-2\mu]) &\subseteq M^\beta(({-\infty},-\mu]),\\ Q&\subseteq M^\beta(\{0\}),\\ M^\alpha([2\mu,\infty)) &\subseteq M^\beta([\mu,\infty)). \end{aligned} \tag{CAM.29}$ We justify passing from those three spaces to all of $M$. Apply the compact filters $\chi_n$ of RSF-01 to any $x$. The compact set $\operatorname{supp}\chi_n\cap\operatorname{Sp}(\alpha)$ is covered by the three disjoint open regions separated at $\pm3\mu/2$, because (CAM.26) removes both separating points and their surrounding annuli. Choose a finite smooth partition equal to one near that compact set and subordinate to the three regions. The remainder of $\chi_n$ vanishes near its intersection with the action spectrum, so its filter is zero by the compact-support ideal lemma. Each of the three resulting filtered vectors belongs to the corresponding old spectral space in (CAM.29). Since $\alpha_{\chi_n}x\to x$ weak-star, their linear span is weak-star dense in $M$.

The closed-space criterion for $\beta$, applied to the closed union of the three sets on the right of (CAM.29), consequently proves $\operatorname{Sp}(\beta) \subseteq(-\infty,-\mu]\cup\{0\}\cup[\mu,\infty). \tag{CAM.30}$ Every vector with $\beta$-spectrum in $[-\mu/2,\mu/2]$ therefore has spectrum in $\{0\}$, so CAM-01 makes it fixed. Thus $\psi$ is lacunary. Choose a maximal abelian algebra of $M_\psi$ containing $a$. It is ambient maximal abelian by WC-06 and is contained in $M_\phi$, by the same cancellation argument as (CAM.5)–(CAM.6).

Now assume that $M$ is a factor and $\Gamma(\sigma^\phi)=\{0\}$, without presuming a global annular gap. Choose a maximal orthogonal family of nonzero projections $e_i\in Z(M_\phi)$ such that each $e_iM_\phi e_i$ contains an ambient maximal abelian algebra $A_i\subset e_iMe_i$. Zorn applies to families of projections with their chosen algebras. Let $e=\sum_i e_i$. If $r=1-e\ne0$, CAM-02 supplies a nonzero central fixed $f\leq r$ satisfying (CAM.14). The fixed-corner modular theorem CZ-09 identifies the restricted action with that of the faithful bounded functional $\phi_f$ on $fMf$. The construction (CAM.26)–(CAM.30) supplies an ambient maximal abelian algebra there inside $fM_\phi f$, contradicting maximality of the family. Hence $e=1$.

The bounded block-diagonal algebra $A=\bigoplus_i A_i$ belongs to $M_\phi$ and is maximal abelian in $M$. Indeed an element commuting with it commutes with every $e_i$, hence is block diagonal; each block belongs to $A_i'\cap e_iMe_i=A_i$. Its bounded finite partial sums converge strongly to the original element, which therefore belongs to $A$. This argument uses nets and allows any cardinality. We have proved $\Gamma(\sigma^\phi)=\{0\} \quad\Longrightarrow\quad \exists A\subset M_\phi,\qquad A'\cap M=A. \tag{CAM.31}$

Recovering an inner period from the center flow

We state the exact earlier programme input that connects these spectral arguments with the type of a factor. The continuous-core spectrum theorem, from OA-FLOW lesson 139, equations (1)–(7), (13)–(18), supplies, for a type III factor $P$, a trace-scaling decomposition $\begin{gathered} P=N\rtimes_\theta\mathbb R, \qquad u(s)xu(s)^*=\theta_s(x),\\ \tau\theta_s=e^{-s}\tau,\qquad Z(N)^\theta=\mathbb C1,\\ \sigma_t^\Phi(x)=x\ (x\in N),\qquad \sigma_t^\Phi(u(s))=e^{-its}u(s), \end{gathered} \tag{CAM.32}$ where $\Phi$ is the dual weight. For every faithful normal semifinite weight $\rho$ on $P$, its theorem identifies $\begin{aligned} \Gamma(\sigma^\rho) &=\log\bigl(S(P)\setminus\{0\}\bigr)\\ &=\{s:\theta_s|_{Z(N)}=\operatorname{id}\}. \end{aligned} \tag{CAM.33}$ The proof of the all-weight equality is specifically (18a)–(18c) and (18), using the invariant-weight spectrum theorem, the weight-cocycle converse and the cocycle spectral intersection. MSP-01–02 supply the exact GNS domain, implementation continuity, Fourier filters and modular spectral equality. MSP-03 proves the factor cocycle invariance from CAM-02. The dual-center kernel and trace-scaling decomposition are its other exact inputs. These are programme proofs, distinct from the scalar spectral arguments in this lesson; the zero point of $S(P)$ is omitted before taking its logarithm. The argument below supplies the periodic-center step without assuming a compact-group representation theorem.

Let an ergodic normal action $\theta$ on an abelian von Neumann algebra $C$ have least positive period $L$. Put $P_0=2\pi/L$ and define its circle coefficients $E_n(c)=\frac1L\int_0^L e^{-inP_0s}\theta_s(c)\,ds. \tag{CAM.34}$ Changing the integration variable, using periodicity, gives $\theta_t(E_n(c))=e^{inP_0t}E_n(c)$. These normal integrals have the preadjoint construction of RSF-01 on a compact interval. The Fejér sums are $\begin{gathered} F_m(c)=\sum_{|n|<m}(1-|n|/m)E_n(c),\\ K_m(s)=\frac1m\left|\sum_{j=0}^{m-1}e^{ijP_0s}\right|^2. \end{gathered} \tag{CAM.35}$ Expanding the finite square shows that $F_m(c)=L^{-1}\int_0^LK_m(s)\theta_s(c)\,ds$, that $K_m\geq0$, and that its integral is $L$. On a closed subset of the circle away from zero, the geometric-sum formula bounds $K_m$ by a constant divided by $m$. A continuous scalar orbit coefficient is uniformly close to its value at zero on a small circle neighborhood of zero, and is uniformly bounded elsewhere. Splitting the integral over these two sets proves $F_m(c)\to c$ weak-star.

Every nonzero eigenvector $v$ in this abelian algebra has $v^*v$ fixed. Ergodicity gives $v^*v=d1$ with $d>0$, so after scalar normalization it is a unitary. Products and adjoints of these unitaries show that the set of occurring integers $H=\{n:E_n\ne0\}\subseteq\mathbb Z \tag{CAM.36}$ is a subgroup. An eigenvector at degree $n$ is fixed by $E_n$, so the definition agrees with occurrence of that degree. Every subgroup of $\mathbb Z$ is either $\{0\}$ or $d\mathbb Z$: take its least positive element and use integer division. If $H=\{0\}$, the Fejér convergence makes the action trivial. If $H=d\mathbb Z$ with $d>1$, time $L/d$ fixes every Fejér sum and then every element by normality, contradicting the least period. Thus $H=\mathbb Z$. In particular there is a unitary $a\in C$ with $\theta_s(a)=e^{iP_0s}a. \tag{CAM.37}$

Apply this to $C=Z(N)$ in (CAM.32), when the kernel in (CAM.33) is $L\mathbb Z$. Covariance gives $a u(s)a^*=e^{-iP_0s}u(s). \tag{CAM.38}$ Conjugation by $a$ fixes $N$; (CAM.32) and (CAM.38) therefore prove $\operatorname{Ad}(a)=\sigma_{P_0}^\Phi$ on the generators and hence on $P$. For any other faithful normal semifinite weight $\rho$, cocycle intertwining gives $\sigma_{P_0}^\rho =\operatorname{Ad}\bigl([D\rho:D\Phi]_{P_0}a\bigr). \tag{CAM.39}$ The cocycle is the one constructed and identified in CX-10; its covariance is unchanged by the normal isomorphism in (CAM.32). Thus a lattice kernel $L\mathbb Z$ supplies the inner modular period $2\pi/L$, with its sign and normalization determined explicitly.

Strictly semifinite weights on type III factors

Let $M$ be a factor with separable predual of type $\mathrm{III}_\lambda$, where $0\leq\lambda<1$, and let $\phi$ be a faithful strictly semifinite normal weight. We prove that its centralizer contains an ambient maximal abelian algebra.

By WC-05, choose mutually orthogonal nonzero projections $e_i\in M_\phi$ summing to one, with $\phi(e_i)<\infty$. On $M_i=e_iMe_i$, the restriction $\phi_i$ is a faithful bounded normal functional; CZ-09 gives $\sigma^{\phi_i}=\sigma^\phi|_{M_i}, \qquad (M_i)_{\phi_i}=e_iM_\phi e_i. \tag{CAM.40}$ Each $M_i$ is a factor. Indeed a nontrivial central projection $z$ in $e_iMe_i$ would give $zM(e_i-z)=0$, contradicting the ambient carrier test in the factor $M$. A center with no nontrivial projections consists of scalars by the spectral calculus. Also $M_i$ is type III: finiteness of a projection $p\leq e_i$ is tested in its corner $pMp=pM_ip$, so it cannot become finite by passing to $M_i$.

The type convention and (CAM.33) give $\Gamma(\sigma^\phi)= \begin{cases} \{0\},&\lambda=0,\\ (\log\lambda)\mathbb Z,&0<\lambda<1. \end{cases} \tag{CAM.41}$ Here $S(M)=\{0,1\}$ in the first case and $S(M)=\{0\}\cup\lambda^{\mathbb Z}$ in the second. The corner-invariance proof (CAM.15), with (CAM.40), gives exactly the same Connes spectrum for $\sigma^{\phi_i}$; no subtype-invariance theorem for corners is silently imported.

If $\lambda=0$, CAM-05 applied to $\phi_i$ gives an ambient maximal abelian algebra $A_i\subseteq(M_i)_{\phi_i}$. If $0<\lambda<1$, apply the continuous-core theorem (CAM.32)–(CAM.33) to the type III factor $M_i$. Its center-flow kernel is $L\mathbb Z$, with $L=-\log\lambda>0$, by (CAM.15) and (CAM.41). CAM-06 supplies an inner modular period for $\phi_i$, and CAM-01 again gives an ambient maximal abelian $A_i\subseteq(M_i)_{\phi_i}$.

The block argument in CAM-05 now assembles $A=\bigoplus_iA_i\subseteq M_\phi$ and proves $A'\cap M=A$. We have established the desired theorem and its relative-commutant consequence: $\begin{gathered} \exists A\subseteq M_\phi,\qquad A'\cap M=A,\\ M_\phi'\cap M=Z(M_\phi). \end{gathered} \tag{CAM.42}$ For the second equality, an element commuting with $M_\phi$ commutes with $A$, hence belongs to $A\subseteq M_\phi$, and is therefore central there. The reverse inclusion is immediate. This is the exact type-dependent input to WC-07, which then proves the central-density characterization of centralizer inclusion.

The mathematical antecedent is M. Takesaki, Theory of Operator Algebras II, Lemma XII.4.7(ii), printed pages 408–409. The source comparison also uses the independently written OA-FLOW lessons 111, 121, 139, 140 and 148. The annular compactness argument, sharp bounded implementer, GNS frequency estimate and circle coefficient argument above retain the precise hypotheses needed by the two type branches. The negative spectral interval in (CAM.29) ends at $-\mu$; an endpoint $+\mu$ would not yield a gap.