Results on similarity, derivations and free-product estimates, 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.

• Contractive homomorphisms — Lemma 2.1. A unital homomorphism of a unital C*-algebra into B(H) of norm at most 1 is a *-homomorphism.
• Unitary absorption — Lemma 2.2. Representations similar through an invertible V are unitarily equivalent through V|V|^-1, and |V| commutes with the first representation.
• Paulsen's similarity theorem — Theorem 3.1. A unital homomorphism is similar to a *-homomorphism iff it is completely bounded; the least condition number of a similarity is ||pi||_cb, attained by a positive similarity.
• Triangular homomorphisms — Lemma 4.2. [[lambda, c Delta],[0, sigma]] is a unital homomorphism with norm at most beta(c||Delta||) and cb norm at most beta(c||Delta||_cb).
• Implementation of derivations — Theorem 4.3. A completely bounded rectangular derivation is implemented by an operator of norm at most ||Delta||_cb / 2.
• Christensen's criterion — Corollary 4.4. A rectangular derivation is implemented by a bounded operator iff it is completely bounded.
• Arveson's distance formula — Corollary 4.5. For a C*-algebra A in B(H) containing 1 and x in B(H), 2 dist(x, A') = ||ad(x)|_A||_cb, and the distance is attained.
• Noncommutative little Grothendieck inequality (Haagerup) — Theorem 1.1. For T: A -> H bounded there are states f, g with ||Tx||^2 <= ||T||^2 (f(x*x) + g(xx*)).
• Row and column estimate (Christensen; Dickson) — Theorem 2.1. Every bounded unital homomorphism of a unital C*-algebra satisfies ||phi||_row, ||phi||_col <= sqrt2 ||phi||^2.
• Rows and columns of derivations — Proposition 2.2. Bounded rectangular derivations satisfy ||Delta||_row, ||Delta||_col <= 4 sqrt2 ||Delta||.
• Finite cyclic sets — Theorem 3.1. A bounded unital homomorphism with a cyclic set of m vectors satisfies ||pi||_cb <= m ||pi||_row ||pi||_col <= 2m ||pi||^4.
• Haagerup's cyclic similarity theorem — Corollary 3.2. Such a homomorphism is similar to a *-homomorphism with condition number at most 2m ||pi||^4.
• Column factorization — Lemma 4.1. In a von Neumann algebra with a cyclic vector every unit vector of K^h is v zeta with v a contractive column over the algebra and zeta a unit vector.
• Implementation on a cyclic domain — Theorem 4.2. A bounded rectangular derivation whose domain representation is cyclic has ||Delta||_cb <= 8 sqrt2 ||Delta|| and is implemented by an operator of norm at most 4 sqrt2 ||Delta||.
• Exponential form of a similarity — Lemma 1.1. A similarity of the identity representation is a unitary conjugate of x -> e^-t x e^t with ||r||_cb = ||e^t|| ||e^-t||.
• Three-lines bound — Lemma 2.1. For t self-adjoint and a unitary, ||ta - at|| <= log max(||e^-t a e^t||, ||e^-t a* e^t||).
• Integral formula — Lemma 3.1. If ||r_t||_cb = ||e^t|| ||e^-t||, then log ||r_t||_cb <= ||delta_t||_cb.
• Kirchberg's lemma — Theorem 4.1. If ||delta_t||_cb <= k ||delta_t|| for all self-adjoint t, then every similarity r of the identity representation satisfies ||r||_cb <= ||r||^(2k).
• Similarity from inner derivations — Theorem 5.1. A uniform bound ||Delta_t||_cb <= k ||Delta_t|| for inner derivations in all representations gives ||pi||_cb <= ||pi||^(2k) for all bounded unital homomorphisms.
• Inner products of words — Lemma 1.2. Centred alternating words of different types are orthogonal; for equal types the inner product is the product of the letterwise inner products.
• Word decomposition — Theorem 1.3. L^2 of the algebra generated by two free subalgebras is the direct sum of the tensor products of their centred L^2 spaces over alternating types.
• Left multiplication — Lemma 2.1. Left multiplication by Q_1 acts on the first factor of L^2(Q_1) (x) F_1.
• Conditional expectations — Lemma 2.2. The expectation onto Q_1 kills words containing a Q_2 letter, and agrees with the expectation onto Q_1' on W*(Q_1', Q_2).
• Letterwise automorphisms — Proposition 3.1. Trace-preserving automorphisms of the two free algebras extend to a trace-preserving automorphism implemented letter by letter.
• Conjugation by a free Haar unitary — Lemma 4.1. If W*(u) is free from P for a Haar unitary u, then u B u* is free from P for every subalgebra B of P.
• Free corner estimate (OpenAI, 2026) — Theorem 1.1. An operator commuting with L_D whose commutators with a corner e L e of trace 1/N are at most eps almost commutes with trace-zero symmetries of A_0 on L^2(A_0), with constant 3 pi^2 (1 + 16 sqrt2); proof completed in the Hankel test lesson.
• Factoriality — Lemma 2.1. D' cap L = C1, so L is a II_1 factor.
• Symmetrization — Lemma 3.1. One may assume G commutes with the letterwise automorphisms induced by the matrix unitaries.
• Schur-Weyl for unitary groups — Corollary 5.2. Operators commuting with all u^(x)p are combinations of permutations, independent when m >= p.
• Invariant multilinear forms — Proposition 5.3. Unitarily invariant multilinear forms on M_m^p are unique combinations of cycle-trace forms when m >= p.
• Invariant intertwiners — Theorem 6.1. A bounded operator between word levels intertwining L_D and the matrix conjugations vanishes between different levels and is I (x) sum a_sigma U_sigma on equal levels.
• Word structure — Theorem 7.1. The symmetrized G preserves word levels and acts on repeated-letter tensors only through operators B_k on the A_0 letters.
• Full columns from a small corner — Lemma 1.1. On vectors with right support of trace at most t, G agrees up to c_0 eps with a right multiplication.
• Propagation — Lemma 2.1. ||B_k(s^(x)k (x) v) - s^(x)k (x) B_0 v|| <= 2 c_0 eps ||v||_2 for every k.
• Compressed commutator — Lemma 3.1. On the copies U_k L^2(A_0), ||[I (x) B_0, i* L_h i]|| <= (1 + 4 c_0) eps ||h|| for h in e L e.
• The dihedral corner — Lemma 1.1. w = ds is a Haar unitary, p C p is spanned densely by the p w^l p, and pDp, pCp are free.
• Even functions of w — Lemma 1.2. Trace formulas for g(c) w^l and g(c) w^l d with c = Re w.
• Free compression — Lemma 2.1. E_C(ebe) = u^2 b and ||ebe|| <= 2 sqrt(u) ||b||_2 + u ||b|| for centred self-adjoint b in pCp.
• Toeplitz and Hankel coefficients — Proposition 3.1. The compression of L_h to the copies is u (T_g (x) I + H_g (x) L_s), so the commutator with I (x) B_0 is u H_g (x) [B_0, L_s].
• Irreducible matrix subfactors (Popa) — Lemma 1.2. A II_1 factor with separable predual contains an irreducible subfactor generated by increasing full matrix algebras.
• Free position — Lemma 2.1. For Q in P irreducible there is a Haar unitary u in Q^omega free from P, and u Q u* is free from P.
• Finite standard multiplicities — Proposition 4.1. For a II_1 factor with separable predual on finitely many copies of L^2(M), ||[Y^(h), X]|| <= (3 c_1 + 2) g_M(Y) ||X||.
• Factors — Proposition 1.1. For a factor on a separable space, ||[Y^(h), X]|| <= C_fac g_P(Y) ||X||.
• Separable von Neumann algebras — Proposition 2.1. For a von Neumann algebra on a separable space, ||[Y^(h), X]|| <= C g_P(Y) ||X||.
• Uniform commutator estimate (OpenAI, 2026) — Theorem 3.1. For every von Neumann algebra P in B(K), Y in B(K) and X in M_h(P), ||[Y^(h), X]|| <= C g_P(Y) ||X|| with C < 6308.
• Derivations are inner — Theorem 1.2. Every bounded rectangular derivation of a unital C*-algebra has ||Delta||_cb <= C ||Delta|| and an implementing operator of norm at most C ||Delta|| / 2.
• Kadison's similarity theorem (OpenAI, 2026) — Theorem 2.1. Every bounded unital homomorphism pi of a unital C*-algebra satisfies ||pi||_cb <= ||pi||^(2C) and is similar to a *-homomorphism with condition number at most ||pi||^(2C).
• Universal hyperreflexivity (OpenAI, 2026) — Theorem 3.1. Every von Neumann algebra M in B(H) satisfies alpha_M(T) <= dist(T, M) <= 2C alpha_M(T).

Reading order

Read the lessons in order. The first three are classical: Paulsen's theorem and the implementation of derivations; the noncommutative little Grothendieck inequality with its row, column and cyclic consequences; and Kirchberg's passage from bounds for inner derivations to similarity. The fourth develops words in free subalgebras. The fifth to seventh prove the free corner estimate: invariant tensors and the word structure, propagation from small corners, and the Hankel test. The eighth transfers the estimate to II_1 factors, the ninth proves the uniform commutator estimate for all von Neumann algebras, and the tenth draws the consequences: derivations are inner, Kadison's similarity theorem and universal hyperreflexivity.