Results on strong Kadison–Kastler stability and its limits, 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.

• Strong Kadison–Kastler stability — Theorem 2.1. For every eps > 0 there is delta(eps) > 0 such that unital von Neumann algebras M, N on any Hilbert space with d(M, N) < delta(eps) satisfy uMu* = N for a unitary u with ||u - I|| < eps; delta depends on eps alone. Proved in lessons 2 to 17 (lesson 17, Theorem 2.1).
• Close separable C*-algebras without spatial conjugacy — Theorem 2.3. For every eps > 0 there are unital norm-separable C*-algebras A, B on a separable Hilbert space with common identity, d(A, B) < eps and A'' = B'', such that no unitary u satisfies uAu* = B. Proved in lessons 18 and 19.
• One-sided near inclusions without small spatial embeddings — Theorem 2.4. There are eps_0 > 0 and unital von Neumann algebras M_n, N_n on separable Hilbert spaces with gamma(M_n, N_n) -> 0 such that, for large n, unitaries u with uM_nu* in N_n exist and all satisfy ||u - I|| >= eps_0. Proved in lessons 20 and 21, with eps_0 = 1/20000.
• Commutants of close algebras — Corollary 3.2. If d_cb(A, B) <= gamma, then every h in A' satisfies dist(h, B') <= gamma ||h||, and d(A', B') <= 2 gamma.
• Close representations — Lemma 3.3. Normal representations rho, theta of a von Neumann algebra with ||rho - theta||_cb <= e < 2 are conjugate by a unitary u with ||u - I|| <= e.
• A common infinite family of isometries — Corollary 4.3. If unital von Neumann algebras M, N contain a common infinite family of isometries, then d_cb(M, N) = d(M, N).
• Christensen's theorem on injective near inclusions — Theorem 3.1. If R is injective and nearly included in Q within gamma < 1/100, a unitary v in the von Neumann algebra generated by R and Q satisfies vRv* in Q and ||v - I|| <= 150 gamma; with the reverse near inclusion and gamma < 1/101, vRv* = Q. Used without proof, as named result N1 (Cameron, Christensen, Sinclair, Smith, White and Wiggins, Theorem 3.3(i)).
• Multiplication perturbations (Ricard and Roydor) — Theorem 3.2. An associative bilinear map m on a von Neumann algebra P with ||m - m_P||_cb <= 1/11 is carried to the product of P by a completely bounded bijection L with ||L - id||_cb <= 10 ||m - m_P||_cb. Used without proof, as named result N2.
• The completely positive criterion — Proposition 4.1. If F: P -> Q is UCP with ||F - rho||_cb <= e < 10^-3 for a faithful normal representation rho, and every contraction of Q lies within g of rho(P)_1 with g + e < 1, then a normal *-isomorphism theta: P -> Q with ||theta - rho||_cb <= 32e is implemented by a unitary u with ||u - I|| <= 32e.
• Tensor gaps — Lemma 5.1. If d_cb(A, B) <= gamma, the same bound holds after tensoring both algebras with B(K), or with a von Neumann algebra D whose commutant is generated by an increasing sequence of finite-dimensional algebras.
• Boundary means — Lemma 2.1. Every countable group G acts nonsingularly on a standard probability space D with an equivariant positive unital conditional mean m from L^infty(G x D) to L^infty(D) that is a module map over L^infty(D); likewise for G x G on D x D.
• Correction of measurable cocycles — Lemma 3.1. A family of measurable unitary fields whose cocycle defect e is at most 1/64 lies within 2e of an exact measurable cocycle.
• Averaged intertwiners — Lemma 4.1. An exact cocycle within delta < 1 of a unitary representation U is intertwined with U by a measurable unitary field L with ||L - I|| <= 2 delta.
• Diagonal comparison — Lemma 1.1. For an exact cocycle W with values in a finite von Neumann algebra and r = sup ||W_g(x)W_g(y)* - I|| < 1 there is a measurable unitary field a on D x D with ||a - I|| <= 2r and a(x, y) = W_g(x) a(g^-1 x, g^-1 y) W_g(y)*.
• Double ergodicity with unitary coefficients — Lemma 2.1. A bounded measurable vector field on D x D that transforms under a unitary representation of G along the diagonal action is almost everywhere a constant invariant vector.
• Finite neighbours — Lemma 1.1. A unital von Neumann algebra at distance less than 1/10 from a finite factor is a finite factor.
• Popa's irreducible hyperfinite subfactor — Lemma 2.1. Every II_1 factor M with separable predual contains a hyperfinite II_1 subfactor R, generated by an increasing sequence of full matrix algebras, with R' n M = C1.
• The bimodular test — Lemma 3.1. An operator V on L^2(M) commuting with the left and right actions of R satisfies ||V|| <= 384 sup |<V a Omega, b Omega>| over contractions a, b of M.
• Common tracial position — Proposition 1.2. Close II_1 factors (mutual gaps g < 1.74 x 10^-13) sharing an injective subalgebra that contains its relative commutant have representations in common standard tracial position within beta < 50948 sqrt(g). Used without proof, as named result N3 (Cameron, Christensen, Sinclair, Smith, White and Wiggins, Lemma 4.10).
• The two conjugations are close — Lemma 2.2. For II_1 factors in common standard tracial position with a common irreducible hyperfinite subfactor acting alike on the right, d(M, N) < b implies ||J_M - J_N|| <= 768 b.
• Herstein's orientation lemma — Lemma 3.1. A surjective Jordan homomorphism from a unital complex algebra onto a factor is a homomorphism or an antihomomorphism.
• An isomorphism in common tracial position — Proposition 1.1. II_1 factors in common standard tracial position with a common irreducible hyperfinite subfactor and d(M, N) < b < 1/1564 admit a normal *-isomorphism theta with ||theta(x) - x|| <= 252 b on the unit ball.
• Strong stability of finite factors — Theorem 2.1. There are t_fin > 0 and eta_fin(t) -> 0 such that a finite factor M and a von Neumann algebra N on a separable Hilbert space with d(M, N) < t < t_fin are conjugate by a unitary within eta_fin(t) of I.
• Absorbing a fixed factor — Proposition 2.2. For every factor A with separable predual, the tensor product of A with the fixed factor D is a factor of type III_1.
• Irreducible hyperfinite subfactors in type III_1 factors — Proposition 3.1. A type III_1 factor P with separable predual has a faithful normal state phi and a hyperfinite II_1 factor R in the centralizer of phi, generated by full matrix algebras of sizes 2^j, with R' n P = C1.
• Preparing an infinite pair — Proposition 5.2. Infinite factors M_0, N_0 on a separable space with d(M_0, N_0) < t < 4 x 10^-9 become, after tensoring with D and unitaries within 3 x 10^7 t of I in total, a prepared pair with d_cb <= 6 x 10^7 t.
• Uniform graph comparison — Proposition 2.1. For a prepared pair with d_cb(M, N) <= gamma, the projections onto the rescaled graphs of the Tomita operators satisfy sup_c ||E_c^M - E_c^N|| <= 5 x 10^5 gamma.
• Logarithmic modular cuts — Corollary 3.1. For a prepared pair with d_cb(M, N) <= gamma and every real a, the spectral projections of log Delta_N for (-inf, a - 1/2] and of log Delta_M for [a + 1/2, inf) have product of norm at most 2.1 x 10^6 gamma, and symmetrically.
• Sakai's theorem — Theorem 0.1. Every derivation of a simple unital C*-algebra is inner. Used without proof, as named result N4.
• The uniform infinitesimal estimate — Lemma 2.1. There is an absolute constant K with dist(h, P + P') <= K sup over contractions x of P of dist([h, x], P), for every separably acting type III factor P and every self-adjoint h.
• Paths of approximate normalizers — Lemma 3.1. A norm-continuous path of unitaries starting at 1 that moves the unit ball of P within e < e_0 of P stays within C_path e of the group U(P)U(P'), on intervals of any length.
• Small exact normalizers — Lemma 4.1. A unitary normalizer v of a factor P with ||v - 1|| < 1/6 factors as v = ab' with a in U(P), b' in U(P') and ||a - 1|| + ||b' - 1|| <= 10 ||v - 1||.
• Arveson's distance formula — Theorem 2.1. For every nest and every operator T, the distance from T to the nest algebra is the supremum over the nest of ||(1 - N) T N||, and it is attained.
• Approximation by analytic multipliers — Proposition 4.2. Every bounded operator field F on the line is within ||(1 - P_0) M_F P_0|| of an analytic bounded field.
• Near normalizers at every time — Theorem 1.1. For a prepared pair with d_cb(M, N) <= gamma < gamma_0 and every real t there is a unitary n_t with n_t M n_t* = M and ||n_t - Delta_N^(it) Delta_M^(-it)|| <= C gamma, with absolute constants.
• Correction of normalizer representations — Lemma 2.1. Normalizers v_t of a factor, t real, with sup ||v_t v_s - v_(t+s)|| = e < 1/7208 lie within 4e of a representation of the discrete group of reals by normalizers.
• A common modular action — Theorem 3.1. For a prepared pair with d_cb(M, N) < gamma < c_mod there is a unitary q within C_mod gamma of 1 such that qMq* is normalized by the modular group of N and d_cb(qMq*, N) <= C_mod gamma.
• Conjugacy through a crossed product — Proposition 3.2. Von Neumann algebras P, Q normalized by one strongly continuous unitary group U, with d_cb(P, Q) < gamma < gamma_0 and C(Q, U) a factor of type II_infinity, are conjugate by a unitary within zeta(gamma) of 1, where zeta(gamma) -> 0.
• Strong stability of separably acting factors — Theorem 1.1. There are t_fac > 0 and eta_fac(t) -> 0 such that factors on a separable Hilbert space with d(M_0, N_0) < t < t_fac are conjugate by a unitary within eta_fac(t) of I, for factors of every type.
• Dixmier's approximation theorem — Theorem 1.2. For every element x of a von Neumann algebra and every eps > 0, an averaging map brings x within eps of the center; for self-adjoint x, the norm-closed convex hull of its unitary orbit meets the center.
• Close centers — Lemma 2.1. d(Z(M), Z(N)) <= 3 d(M, N); if d(M, N) < g < 1/303, a unitary within 450 g of 1 carries Z(M) onto Z(N).
• Measurable implementers — Lemma 4.1. For algebras with a common center, fiberwise implementing unitaries within a of 1 can be chosen measurably and assemble into one implementing unitary within a of 1.
• Stability on separable Hilbert spaces — Theorem 5.1. For every a > 0 there is delta_sep(a) > 0 such that unital von Neumann algebras on a separable Hilbert space with d(M, N) < delta_sep(a) are conjugate by a unitary within a of 1.
• Strong Kadison–Kastler stability on arbitrary Hilbert spaces — Theorem 2.1. For every eps > 0 there is delta > 0, depending on eps alone, such that unital von Neumann algebras on any Hilbert space with d(M, N) < delta are conjugate by a unitary within eps of 1.
• Matrix distances and commutants — Corollary 3.1. For every eps > 0 there is delta > 0 such that d(M, N) < delta implies d_cb(M, N) < eps and d(M', N') < eps.
• Algebras of convergent sequences — Proposition 1.1. For unital subalgebras P, Q of an algebra D with trivial center, the algebras of sequences in D converging into P, respectively Q, are at distance d(P, Q), have the same commutant, and every unitary conjugating one onto the other yields normalizers U_n of D with U_n p_b U_n* -> b for every b in Q.
• The obstruction — Corollary 2.3. If P has the spatial tensor-norm identity relative to D and Q does not, the two sequence algebras are not unitarily conjugate; if WPW* = Q for a unitary W, their distance is at most 2 ||W - I||.
• A uniformly small deformation — Proposition 2.2. For 0 < t < 1, the unitary representation pi_t of the free group on countably many generators with <pi_t(s_j) delta_e, delta_e> = t satisfies ||pi_t(g) - lambda(g)|| <= t + t^2 + t/(1 - t) for every group element g.
• Failure of the tensor-norm identity — Proposition 5.2. For 0 < t < 1, the subalgebra P_t fails the spatial tensor-norm identity relative to D: m generators with m > 4/t^2 give a product of norm at least mt and a tensor of norm at most 2 sqrt(m).
• The circle family — Proposition 1.2. There are unitary representations pi_z of the free group, z on the unit circle, with pi_z(g) = sum over k of z^k A_k(g), ||A_k(g)|| <= c_k uniformly in g, and pi_z(s_j) -> tz p_e weakly as j -> infinity.
• The near inclusion — Proposition 3.2. For unitary matrices U, V the von Neumann algebras M and N built from the circle family satisfy W*MW in N and gamma(M, N) <= C ||UV - VU||, with C = 61952 for t = 1/2.
• Voiculescu's matrices — Lemma 1.2. For n >= 3 the clock and shift matrices U_n, V_n have commutator norm nu_n = 2 sin(pi/n), and every commuting pair of unitaries is at total distance at least 1 - nu_n/2 from (U_n, V_n).
• A commuting corner — Lemma 2.2. Operators in two commuting von Neumann algebras within delta <= 1/400 in total of E_1 R_1 and E_2 R_2 give commuting unitaries within 305 delta of R_1 and R_2 on the finite-dimensional range of E_1 E_2.
• No small spatial embeddings — Theorem 3.1. For the algebras M_n, N_n built from the clock and shift matrices, gamma(M_n, N_n) <= C nu_n -> 0, M_n embeds spatially in N_n, and every unitary u with uM_nu* in N_n satisfies ||u - I|| >= 1/9760 - C nu_n/2.

Reading order

Twenty-one lessons, to be read in order. The first fixes the distances and states the three theorems. Lessons 2 to 7 prove strong stability for finite factors: averaging tools and the completely positive criterion, boundary means and measurable cocycles, double ergodicity, an irreducible hyperfinite subfactor, common tracial position, and an isomorphism close to the identity. Lessons 8 to 15 treat infinite factors: a fixed tensor factor of type III_1 and prepared pairs, rescaled Tomita graphs, infinitesimal normalizers of type III factors, frequency nests and analytic multipliers, near normalizers at every time, a common modular action and common crossed products. Lessons 16 and 17 pass to all von Neumann algebras, first on separable and then on arbitrary Hilbert spaces. Lessons 18 and 19 construct close separable C*-algebras that are not unitarily conjugate, and lessons 20 and 21 one-sided near inclusions without small spatial embeddings. Christensen's theorem on injective near inclusions, the perturbation theorem of Ricard and Roydor, the common-position lemma of Cameron, Christensen, Sinclair, Smith, White and Wiggins, and Sakai's theorem on derivations are used without proof, as the named results N1 to N4.