Results on free independence, polynomial flows, semicircular systems and free group factors, 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.

• Closures, groupings and moments of free families — Lemma 1.2. Freeness of unital *-subalgebras passes to the generated von Neumann algebras and to groupings, and mixed moments are determined by the restrictions of the trace.
• Conjugation by the other algebra — Lemma 1.3. If A and B are free and U in B is unitary, then UAU* and B are free, with the same mixed moments.
• Normal form in free groups — Lemma 2.1. Every element of F_d has exactly one reduced word, and homomorphisms from F_d may take arbitrary values on the generators.
• Free group factors — Lemma 2.2. F_d is ICC for d >= 2, so L(F_d) is a II_1 factor.
• Identification by a Haar tuple — Lemma 2.4. A freely generating Haar d-tuple U in (M, tau) gives a unique normal trace-preserving isomorphism L(F_d) -> M with lambda(x_j) -> U_j; the word algebra of U is dense in L^2(M).
• Changes of Haar tuples — Corollary 2.5. Two freely generating Haar tuples differ by a trace-preserving automorphism, automorphisms of F_d act on Haar tuples, and the components are free Haar unitaries.
• Free conjugates — Lemma 3.1. For a freely generating Haar tuple (A_1, ..., A_n, C) and S = arg C, the algebras A_g W*(C) A_g*, g in F_n, are free.
• Group labels — Lemma 3.2. Products of A_j, A_j* and S with nonidentity label have trace 0; those with identity label are products of conjugates S_g.
• Norm of a free sum — Lemma 4.1. For centred self-adjoint z_j in free von Neumann subalgebras, ||sum z_j|| <= 2 (sum ||z_j||_2^2)^(1/2) + max ||z_j||.
• Free sums of conjugates — Corollary 4.2. ||sum_g k(g) A_g S A_g*|| <= 3 pi ||k||_l2 for finitely supported real k.
• Cocycles on the free group — Lemma 1.1. Generator values h_1, ..., h_n extend uniquely to D with D_(gb) = D_g + lambda(g) D_b.
• The formal derivation — Lemma 2.1. delta z = 0 and delta a_j = i sigma(h_j) a_j define a *-derivation with delta a_g = i sigma(D_g) a_g and delta z_g = i [sigma(D_g), z_g], preserving labels.
• Trace identity at time zero — Lemma 3.1. tau(ev_0(delta^r p)) = 0 for every formal polynomial p and every r >= 1.
• Global unitary solutions — Lemma 4.2. The polynomial equation X_j' = i H_j(X) X_j has a unique global solution in every norm-closed *-subalgebra containing S, and it stays unitary.
• Moment invariance — Lemma 5.1. All moments of the moving tuple together with S are constant in time.
• Norm from moments — Lemma 6.1. ||b|| = lim tau((b*b)^k)^(1/2k) in a von Neumann algebra with faithful tracial state.
• Trace-preserving polynomial flows — Theorem 6.2. A one-parameter group of trace-preserving normal automorphisms fixing S and C, with ||beta_t(A_j) - A_j|| <= 3 pi |t| ||h_j|| and ||beta_t(A_w) - exp(itS) A_w|| <= 3 pi |t| ||D_w - delta_e||.
• Free prefixes — Lemma 1.1. p_j = (x_1 b_1)...(x_1 b_j), b_j = x_2^j x_3 x_2^(-j), freely generate a free subgroup of F_n (n >= 3).
• Cocycle of the prefix word — Lemma 1.2. D_(w_m) = T_m h for w_m = p_m x_1 and T_m = sum_(j<=m) lambda(p_j).
• Noncrossing reductions — Lemma 2.1. A word in which each free generator occurs twice with opposite exponents is trivial exactly when its matching is noncrossing.
• Catalan numbers — Lemma 2.2. There are (2r)!/(r!(r+1)!) noncrossing perfect matchings of 2r points.
• Moments of the prefix sums — Lemma 2.3. tau(Q_m^r) = Cat_r m(m-1)...(m-r+1) + O_r(m^(r-1)) for Q_m = T_m T_m*.
• Convergence to the limit law — Lemma 3.2. The distributions of Q_m/m converge on bounded continuous functions to the law with density (1/2pi) sqrt((4-x)/x) on (0,4).
• Little mass near zero — Corollary 3.3. For every kappa > 0 there are d > 0 and m_0 with mu_m([0,d]) < kappa for m >= m_0.
• A small cocycle with a prescribed word value — Theorem 4.1. For n >= 3 and eta > 0 there are a word w and finitely supported real h with ||h|| < eta and ||D^h_w - delta_e|| < eta.
• One-step perturbation — Proposition 1.1. A trace-preserving automorphism moves each A_j by less than eps while a word in the moved A_j comes within eps of C (n >= 3).
• Lipschitz bounds for words — Lemma 2.1. ||U_v - V_v||_2 <= |v| max_j ||U_j - V_j||_2 for unitary tuples, and the same in operator norm.
• Consecutive free group factors (OpenAI) — Theorem 3.1. L(F_n) and L(F_(n+1)) are isomorphic by a unital normal trace-preserving *-isomorphism for every n >= 3.
• Free group factors of rank at least three — Corollary 3.2. L(F_m) and L(F_n) are isomorphic for all integers 3 <= m <= n.
• Composition, direct sums and isomorphisms in the Borel calculus — Lemma 1.1. f(g(z)) = (f o g)(z) for bounded Borel f and g, the calculus of a direct sum is the direct sum, and normal isomorphisms commute with the calculus.
• Haar generators — Lemma 1.2. A self-adjoint element whose distribution function is continuous and strictly increasing on [a,b] generates the same von Neumann algebra as the Haar unitary exp(2 pi i F(y)).
• Free group factors from free generators — Lemma 1.3. Free von Neumann algebras each generated by a Haar unitary generate a copy of L(F_m).
• The semicircle law — Lemma 2.1. The Chebyshev polynomials U_n are orthonormal for the semicircle law; its image under t -> t^2 is the limit law nu with Catalan moments.
• Associativity of freeness — Lemma 3.1. Free families inside the members of a free family of groups form one free family.
• A criterion for freeness — Theorem 3.2. If L*dL = omega(d) for d in D, L* xi = 0 and L* D xi = 0, then L + L* is semicircular and free from D for the vector state of xi.
• Creation operators and the other variables — Lemma 4.1. l(g)* b l(g') = phi(b) <g', g> for b in the von Neumann algebra of the s(e), e orthogonal to g and g'.
• Free semicircular families — Theorem 4.2. s(e_1), ..., s(e_m) for an orthonormal basis are free semicircular for the vacuum, which is a faithful trace on their von Neumann algebra, and generate a copy of L(F_m).
• Semicircular matrices — Theorem 2.1. A 2x2 matrix with free semicircular diagonal and circular off-diagonal entries is semicircular for tr_2 (x) phi and free from M_2 of the algebra of the other variables.
• Several semicircular matrices — Corollary 2.2. Semicircular matrices on disjoint variables and M_2 of the remaining variables form a free family.
• Circular elements — Proposition 3.1. A circular element generates a copy of L(F_2), gamma* gamma has the limit law nu, the kernel is zero and the polar part is unitary.
• Isomorphisms from equal distributions — Lemma 1.1. Elements with the same *-distribution generate isomorphic von Neumann algebras, and the isomorphism carries polar decompositions.
• Conjugation by a free Haar unitary — Lemma 2.1. If v is a Haar unitary free from A, then P and vQv* are free for von Neumann subalgebras P, Q of A.
• A free Haar unitary absorbs a unitary — Lemma 2.2. If w is a Haar unitary free from A and v in A is unitary, then wv is a Haar unitary free from A.
• Changing a free Haar basis — Lemma 2.3. v_1, v_2 v_1*, ..., v_r v_1* are free Haar unitaries generating W*(v_1, ..., v_r).
• Polar decomposition of a circular element (Voiculescu) — Theorem 3.1. The polar part of a circular element is a Haar unitary free from its modulus.
• Free families from semicircular matrices — Corollary 3.2. Diagonal entries, polar parts and moduli of the circular entries of semicircular matrices on disjoint variables, together with the algebra of the remaining variables, form a free family.
• Generators of a corner — Lemma 1.1. pNp is generated by the elements a y b with y in a generating set, a in {p, W} and b in {p, W*}, for a partial isometry W from 1 - p to p.
• Corners of trace one half — Lemma 1.2. Corners by projections of equal trace are isomorphic, and R = M_2 (x) pRp when tau(p) = 1/2.
• Corner formula (Voiculescu) — Theorem 2.1. p L(F_n) p and L(F_(4n-3)) are isomorphic for a projection p of trace 1/2 and n >= 2.
• Isomorphism of the free group factors (OpenAI) — Theorem 3.1. L(F_2) and L(F_3) are isomorphic; hence L(F_m) and L(F_n) are isomorphic for all integers m, n >= 2.
• Corners and amplifications of L(F_2) — Corollary 3.2. p L(F_2) p and M_2 (x) L(F_2) are isomorphic to L(F_2); 1/2 and 2 lie in its fundamental group.
• Semicircular matrices of size k — Theorem 1.1. A k x k matrix with free semicircular diagonal entries and circular entries above the diagonal, scaled by k^(-1/2), is semicircular for tr_k (x) phi and free from M_k of the algebra of the other variables.
• Several semicircular matrices of size k — Corollary 1.2. Semicircular matrices on disjoint blocks and M_k of the remaining variables form a free family; diagonal entries, polar parts and moduli of the circular entries, with the remaining variables, form a free family.
• Generators of a corner with k matrix units — Lemma 2.1. P_1 N P_1 is generated by W_p y W_q* for y in a generating set, given partial isometries W_q from P_q to P_1 with sum of the P_q equal to 1.
• Matrix units in a II_1 factor — Lemma 2.2. pRp is M_m(C) (x) q_1Rq_1 for m orthogonal equal-trace projections with sum p, and every projection is such a sum for every m.
• Conjugation by several free Haar unitaries — Lemma 2.3. If A, W*(v_2), ..., W*(v_k) are free with Haar v_q, then B_1, v_2R_2v_2*, ..., v_kR_kv_k* are free for subalgebras of A.
• Corner formula for trace 1/k (Voiculescu) — Theorem 3.1. p L(F_n) p and L(F_(1 + k^2(n-1))) are isomorphic for a projection p of trace 1/k, n >= 2, k >= 2.
• Rational corners of L(F_2) — Corollary 4.1. Corners of L(F_2) by projections of rational trace, and M_m(C) (x) L(F_2), are isomorphic to L(F_2); every positive rational lies in its fundamental group.

Reading order

Read the lessons in order. The first collects free independence, freely generating Haar tuples and a norm bound for free sums. The second constructs the trace-preserving polynomial flows, and the third finds generator coefficients, small in l^2, whose cocycle value at a suitable word is close to the unit vector at the identity. The fourth combines them into the isomorphism of L(F_n) and L(F_(n+1)) for n ≥ 3. The fifth to seventh develop the free probability needed for the corner formula: semicircular elements on the full Fock space, matrices with semicircular and circular entries, and the polar decomposition of a circular element. The eighth computes the corner of trace ½ and proves that L(F_2) and L(F_3) are isomorphic. The ninth extends the corner formula to every trace 1/k and deduces that all rational corners of L(F_2) are isomorphic to it.