Results on Haar unitaries, free independence and generators of II₁ 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.

• Trace inequalities — Lemma 1.1. In a tracial von Neumann algebra: |tau(ab)| <= ||a||_2 ||b||_2, ||abc||_2 <= ||a|| ||b||_2 ||c||, |tau(ab)| <= ||a|| ||b||_1, and ||b||_1 <= tau(q)^(1/2) ||b||_2 if b = bq.
• Haar unitaries in corners — Corollary 2.4. Every nonzero corner qQq of a II_1 factor contains a Haar unitary, and Q has projections of every trace.
• Decay of correlations — Lemma 2.5. For a Haar unitary h of qMq and b in M, sum_j |tau(h^j b)|^2 <= tau(q) ||b||_2^2.
• Freeness with arbitrary outer letters — Lemma 3.2. A Haar unitary t is free from a unital C*-algebra D iff tau(d_0 t^(n_1) d_1 ... t^(n_j) d_j) = 0 for nonzero n_i, centred interior d_i and arbitrary outer d_0, d_j.
• Absorption — Lemma 3.3. If t is a Haar unitary free from D and a, b are unitaries of D, then t* and atb are Haar unitaries free from D.
• Unitaries lift — Lemma 4.2. A unitary of an ultraproduct of finite factors has a representing sequence of unitaries.
• Local quantization in corners — Lemma 1.2. For an irreducible inclusion Q in M, a projection p of Q and finitely many y, there is a projection 0 != q <= p in Q with ||qyq - c_y q||_1 <= alpha tau(q), c_y = tau(pyp)/tau(p).
• Approximate independence — Theorem 3.1. For an irreducible inclusion Q in M, finitely many centred letters, a word length n and eps > 0, there is a normal partial isometry v in Q with tau(v*v) > 1/4 and all alternating words of length <= n of trace at most eps.
• Popa's free independence theorem — Theorem 4.2. For irreducible inclusions Q_n in M_n and countably many centred elements of the ultraproduct, there is a Haar unitary u in the ultraproduct of the Q_n with all alternating words in these elements and nonzero powers of u of trace zero.
• Free Haar unitaries in ultrapowers — Corollary 4.4. For an irreducible inclusion P in M of II_1 factors and a norm-separable unital C*-algebra D in M^omega, there is a Haar unitary of P^omega, liftable to unitaries of P, free from D.
• Haagerup's inequality for words of length one — Lemma 1.1. ||sum b_gamma lambda(gamma)|| <= 2 (sum |b_gamma|^2)^(1/2) over the free generators and their inverses.
• Perturbation with uniform correlation — Proposition 3.1. For unitaries u, z, xi in M, 0 < alpha < 1/2 and l >= 1 there are a unitary u' with ||u' - u||_2 <= alpha/sqrt(2l) + 1/l and a word W' in u' and unitaries of P with |tau(xi* W') - (alpha/2) tau(xi* z)| <= ||xi||_2 4 alpha^2/(1 - 2 alpha) + 1/l.
• Relative generation — Theorem 4.4. For an irreducible inclusion P in M of II_1 factors with M of separable predual, the unitaries u with W*(P, u) = M form a dense G_delta set for the trace norm (OpenAI, 2026).
• Irreducible hyperfinite subfactors — Theorem 1.1. Every irreducible inclusion N in M of II_1 factors with M of separable predual has a copy R_0 of the hyperfinite factor in N with R_0' cap M = C.
• Commuting projections from one element — Lemma 2.1. Commuting projections e_m are spectral projections of functions of b = sum 2 3^(-m) e_m.
• Four generators — Proposition 3.1. Every II_1 factor with separable predual is generated by four self-adjoint elements.
• Two by two encoding — Lemma 4.1. If N is generated by four self-adjoint elements, M_2(N) is generated by two.
• Single generation of II_1 factors — Theorem 5.1. Every II_1 factor with separable predual is generated by one operator (OpenAI, 2026).
• Operations on generators — Lemma 1.1. W*(a+ib) = W*(a,b); spectral projections of a diagonal operator with distinct eigenvalues tending to 0; merging a central self-adjoint generator; two central summands, on a separable Hilbert space.
• Separability — Lemma 1.2. A von Neumann algebra on a separable Hilbert space is generated by a sequence of self-adjoint contractions, is sigma-finite, and has separable predual.
• Properly infinite algebras (Wogen) — Theorem 2.1. A properly infinite von Neumann algebra on a separable Hilbert space is generated by two self-adjoint elements.
• Finite factors — Lemma 3.1. A nonzero finite factor on a separable Hilbert space is M_n(C) or a II_1 factor with separable predual, and is generated by one operator.
• Direct integrals (Willig) — Theorem 4.1. If almost every fibre of a direct integral of von Neumann algebras over a standard Borel space is singly generated, the direct integral is generated by the diagonal algebra and one decomposable operator, and is singly generated.
• The generator problem — Theorem 5.1. Every von Neumann algebra on a separable Hilbert space is generated by one operator, equivalently by two self-adjoint operators.
• Separable predual — Corollary 5.2. Every von Neumann algebra with separable predual is generated by one operator.
• An invertible positive generator contains the unit — Lemma 1.1. If c is self-adjoint with spectrum in (0, infinity), the non-unital von Neumann algebra generated by c contains 1.
• Few self-adjoint generators (Dykema-Sinclair-Smith-White) — Lemma 2.1. A II_1 factor generated by n self-adjoint elements has G_sa(M) <= n - 1.
• Compression (Dykema-Sinclair-Smith-White) — Lemma 2.2. G_sa(M) <= t^2 G_sa(pMp) for a projection p of trace t in (0,1) in a II_1 factor M.
• The generator invariant vanishes (OpenAI) — Theorem 3.1. G(M) = G_sa(M) = 0 for every II_1 factor with separable predual.

Reading order

Read the first lesson for the tools: trace inequalities, Haar unitaries, freeness and ultraproducts. The second proves Popa's free independence theorem for irreducible inclusions. The third combines it with Haagerup's inequality to prove that one unitary generates over an irreducible subfactor for most unitaries, and the fourth deduces that every II₁ factor with separable predual is singly generated. The fifth extends this to every von Neumann algebra with separable predual, and the sixth shows that the generator invariant of every II₁ factor with separable predual is 0.