Results on relative bicentralizers, binormal states and core commutants, with their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Structure of the relative bicentralizer — Theorem 2.3. B(N in M, phi) is a sigma-invariant von Neumann subalgebra containing N' cap M, contained in N_phi' cap M, meeting N in B(N, phi), with a unique phi-preserving faithful normal expectation.
• Uniform modulus — Lemma 2.4. For a in B(N in M, phi) and eps > 0 there is delta > 0 with ||[a, y]||# <= eps for contractions y in N with ||[y, phi]|| <= delta.
• Bottom factors of type III_1 — Corollary 2.6. If N is a type III_1 factor with separable predual, E(b) = phi(E(b)) 1 for b in B and E_B(x) = phi(x) 1 for x in N.
• Averaging and killing — Theorem 3.4. E_B is a limit of finite averages of conjugations by unitaries almost commuting with phi, and every z with E_B(z) = 0 is moved by a unitary of the asymptotic centralizer by at least ||z||.
• Transition isomorphisms — Theorem 2.1. Unitary sequences moving phi to psi conjugate B(phi) onto B(psi) by a state-compatible isomorphism independent of the sequence.
• The relative bicentralizer flow — Theorem 3.1. Admissible lambda-families give a u-continuous phi-preserving action beta of R_+^* on B(phi), commuting with the modular group and fixing N' cap M (Ando-Haagerup-Houdayer-Marrakchi).
• A state-independent algebra — Theorem 4.1. N v B(N in M, phi) does not depend on phi.
• Matrix amplifications — Proposition 5.1. B(N in M, tau_R (x) chi_1) = 1 (x) B(N_1 in M_1, chi_1) for N = R (x) N_1, M = R (x) M_1 with R a matrix algebra.
• Binormal states of a finite algebra — Theorem 2.4. A binormal state of B(K) for a finite algebra Q in its trace representation is a limit of vector states of vectors almost of the form Y(1 (x) eta) with uniformly bounded Y in A_F (x) Q.
• Bounded approximating vectors — Theorem 4.2. A binormal state of B(L^2(M)) at modular energy zero is a limit of vector states of vectors almost of the form y(1 (x) xi) with uniformly bounded y in A_F (x) M.
• The binormal identity (Marrakchi) — Theorem 5.1. A state with the two marginals of phi o E, Phi(e_N) = 1 and modular energy zero satisfies Phi(a rho(b)) = <a xi b, xi> for a in B(N in M, phi), b in M.
• Approximate eigenoperators — Proposition 1.2. ||sigma_t(v) - e^{its} v||_phi^2 + e^{-s} ||(sigma_t(v) - e^{its} v)*||_phi^2 <= 512 (1+|t|)^2 ||v|| ||v phi - e^{-s} phi v||.
• Modular frequency — Corollary 1.3. v phi = e^{-s} phi v if and only if sigma_t(v) = e^{its} v for all t.
• The zero-shift map (Marrakchi) — Proposition 3.1. A ucp map zeta of B(L^2(M)) fixing e_N, bimodular over N', with zeta = E_B on M and zeta(f(log Delta)) = f(log Delta).
• Spectral shift maps (Marrakchi) — Theorem 4.1. For every s a ucp map theta_s fixing e_N, bimodular over N', with theta_s = gamma_s o E_B on M and theta_s(f(log Delta)) = f(log Delta - s).
• Binormal identity at frequency s — Corollary 5.1. A state with the marginals of phi o E, Phi(e_N) = 1 and modular energy s satisfies Phi(gamma_s(a) rho(b)) = <a xi b, xi>.
• The two relative products — Proposition 2.2. x Omega (x) a Omega -> xa Omega and -> ax Omega extend to isometries R, L of L^2(N) (x) L^2(B) into L^2(M).
• Exact eigenoperators — Proposition 2.3. On an eigenoperator x of frequency h, U = L*R acts as the flow W_h on the second factor and agrees with D = exp(iX (x) Y).
• The iterated kernel (OpenAI) — Proposition 4.1. Every word in U and flow shifts has a ucp map Z with marginals phi, Z(e_N) = 1, Z(f(log Delta)) = f(S_n + d), whose coefficients give the word.
• Spectral transfer — Lemma 4.2. A band-limited element with bound L maps the spectral subspace of log Delta for an interval J into that for its closed L-neighbourhood.
• Uniform energy estimate (OpenAI) — Proposition 1.1. ||(C - D_all)(xi (x) eta)|| <= eps_eta(delta) ||xi|| for all words, shifts and xi with total modular energy in an interval of length delta.
• Hybrid and averaged words — Lemma 2.1. The same estimate holds when some U_j are replaced by D_j or all by (U_j + D_j)/2.
• Time averaging — Lemma 3.1. Integrated against a probability density p_delta in time, the squared error is at most eps_eta(delta)^2 ||xi||^2, with no spectral restriction.
• U = D (OpenAI) — Theorem 1.1. The comparison contraction L*R of the two relative products equals exp(iX (x) Y); equivalently R = LD.
• The algebra N v B — Proposition 1.1. L^2(N v B) = L(L^2(N) (x) L^2(B)); there B acts on the second factor and N as D(N (x) 1)D*.
• Relative commutants in the core — Lemma 2.2. For Q_0 fixed by the modular group, Q_0' cap c(M) = (Q_0' cap M) rtimes R.
• The core flow — Lemma 3.1. The relative bicentralizer flow extends to B rtimes R by gamma_s(lambda(t)) = e^{-ist} lambda(t).
• Core fixed point theorem (OpenAI) — Theorem 4.1. Fixed points of the core flow on B rtimes R commute with N.
• Criterion for subalgebras — Theorem 5.1. A modular-invariant P in N whose centralizer has relative commutant in M inside B and which contains complete eigen-rows for a dense set of frequencies has P' cap c(M) = N' cap c(M).
• Induced expectations and state changes — Lemma 1.1. For a matrix subalgebra F with matrix units diagonalizing psi_F: E_F is a faithful normal expectation onto F' cap M; rho o E_F = rho iff f_ij rho = (h_j/h_i) rho f_ij; ||rho o E_F - rho|| <= k max ||f_ij rho - (h_j/h_i) rho f_ij||.
• One unitary reduces the remainder — Lemma 1.3. If rho o E_T = rho for a matrix subalgebra T with its trace and u in T is unitary, then ||E_T(z)||_rho^2 <= ||z||_rho^2 - ||uzu* - z||_rho^2 / 4.
• Completing a partial isometry — Lemma 2.2. In a countably decomposable type III factor, a partial isometry under nonzero projections p, q is close to one from p onto q, with the distance controlled by the defects.
• Matrix units with a prescribed ratio — Proposition 3.1. In a type III_1 factor with separable predual, for lambda > 0 there are 2x2 matrix units with ||[e, phi]|| <= delta and ||x phi - lambda phi x|| <= delta; replacing phi by its product with omega_lambda moves it by at most 2 max(1, 1/lambda) delta.
• Lifting matrix units — Proposition 4.4. A unital *-homomorphism M_k -> N_{phi,omega}, N a countably decomposable type III factor, lifts to unital *-homomorphisms M_k -> N whose sequences asymptotically commute with phi.
• Dyadic approximation — Lemma 5.1. Every unitary of a II_1 factor is within delta in 2-norm of a unitary of a subalgebra isomorphic to M_{2^p}.
• One step — Proposition 2.1. Given z with E_B(z) = 0, a matrix subalgebra M_{2^p} with its trace and an eigen part (M_2, omega_mu) (x) (M_2, omega_nu) move the state by at most eps and give ||E_T(z)||^2 <= 15/16 ||z||^2 + eps.
• The tracial part and N v B — Proposition 4.3. The relative commutant in M of the tracial part T is contained in N v B(N in M, phi).
• Exchangeability — Lemma 4.5. The centralizer of the product state on the infinite tensor power G has trivial relative commutant in G (an operator-algebraic Hewitt-Savage law).
• Injective subalgebras preserving core commutants (OpenAI; bottom factors of type III_1) — Theorem 5.2. N in M with separable preduals and expectation, N a factor of type III_1: for every faithful normal state phi and eta in (0, 1] there are a faithful normal state psi with ||psi - phi|| <= eta and an approximately finite-dimensional, sigma^psi-invariant P in N with P_psi' cap N = C, P_psi' cap M in B(N in M, psi) and P' cap c(M) = N' cap c(M).
• Irreducible centralizers (Haagerup) — Corollary 5.3. In a factor of type III_1 with separable predual the faithful normal states psi with N_psi' cap N = C are norm dense in the normal states.
• Centralizer criterion — Lemma 1.1. If P in N is sigma^phi-invariant, P_phi' cap M is contained in N_phi' cap M, and N is generated by N_phi and P, then P' cap c(M) = N' cap c(M).
• Hyperfinite subfactors with the same relative commutant (Popa) — Theorem 2.2. N a II_1 factor, N in M with separable preduals and expectation: commuting matrix subalgebras T_n of N generate T with T' cap M = N' cap M.
• Type II_infinity bottom factors — Theorem 3.2. For N of type II_infinity and the state phi = sum_i 2^-i tau_0(x_ii) there is a sigma^phi-invariant AFD P in N with P' cap M = N' cap M and P' cap c(M) = N' cap c(M).
• Injective subalgebras preserving core commutants (semifinite bottom factors) — Theorem 4.1. N in M with separable preduals and expectation, N a semifinite factor: there is an AFD P in N with expectation, P' cap c(M) = N' cap c(M) and P' cap M = N' cap M.

Reading order

Read the lessons in order. The first two develop the relative bicentralizer, Connes' transition isomorphisms and the relative bicentralizer flow. The third proves Marrakchi's binormal identity through a tracial corner of the continuous core, and the fourth builds the spectral shift maps that move it to every modular energy. The fifth to seventh prove OpenAI's identity U = D: the two relative products and the iterated kernel, the estimate uniform in total modular energy, and the repeated tensor tests. The eighth draws the consequences in the continuous core. The ninth collects tools on finite-dimensional subfactors and lifting from the asymptotic centralizer, and the tenth constructs the injective subalgebra and proves the main theorem for bottom factors of type III_1. The eleventh treats semifinite bottom factors.