Results on uniformly bounded representations, sparse assignments, frames and the unitarizability theorem, 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.

• Unitarizable representations — Proposition 1.1. A uniformly bounded representation is similar to a unitary one iff a bounded Q with a <= Q <= b (a > 0) satisfies pi(g)^* Q pi(g) = Q for all g, iff it preserves an equivalent inner product.
• Amenable groups are unitarizable (Day, Dixmier) — Theorem 1.2. Every uniformly bounded representation of an amenable discrete group on a Hilbert space is similar to a unitary representation.
• Triangular representations — Proposition 2.2. A bounded operator cocycle D for a unitary representation U gives the representation [[U, DU], [0, U]] with norms at most 1 + ||D(g)||; if it is unitarizable, D is inner.
• Induction from a subgroup — Lemma 3.1. A uniformly bounded representation of a subgroup K induces one of G on l^2(G/K; E) with the same uniform bound; if the induced representation is unitarizable, so is the original.
• Finitely generated subgroups and amenability — Lemma 3.3. If every finitely generated subgroup is amenable, the group is amenable; a nonamenable group has a finitely generated nonamenable subgroup.
• A contracting average — Lemma 1.2. A countable nonamenable group has a finite S containing e with ||(1/|S|) sum over t in S of R_t|| < 1 on l^2(G).
• Hall's theorem, countable form — Lemma 2.2. If X is countable, every N(x) is finite and |N(F)| >= |F| for every finite F, there is an injective choice phi(x) in N(x) on all of X (the finite case, Lemma 2.1, by augmenting paths).
• Sparse assignment — Proposition 3.2. The d^l products s_i of words of length l in S admit a: G x [n] -> {0,1} with at most r = ceil(n rho^l) ones at every row x and at most r zeros among the edges (y s_i^(-1), i) at every column y.
• Sums of random signs — Lemma 1.2. For independent uniform signs, P(|sum c_i eps_i| > u) <= 2 exp(-u^2 / (2 sum c_i^2)).
• Nets on spheres — Lemma 1.3. The unit sphere of R^m has a 1/4-net with at most 9^m points (volume comparison).
• Random sign frames — Proposition 1.5. There are n unit vectors in C^k, k = ceil(2r log 2n), such that every subfamily of at most min(2r, n) of them has synthesis operator of norm at most 10.
• Masked operator estimate — Lemma 2.1. If every row and every column of a mask carries at most p labels and all synthesis operators of at most p of the vectors have norm at most b_0, the masked operator has norm at most b_0^2.
• Uniform conjugation differences — Proposition 3.2. For the operator T of a sparse assignment, ||T - U(g) T U(g)^(-1)|| <= 100 for every g; the block rows of T and the block columns of A - T also have norm at most 100 (Corollary 3.1).
• Kernel energy — Lemma 3.3. The translation-invariant operator A has kernel W(s) = sum of P_i over s_i = s, with Hilbert-Schmidt energy at least n.
• Distance from the translation commutant — Lemma 1.1. If A commutes with translations and has kernel energy E, T has row at e and A - T column at e of norm at most K, then every C in the commutant satisfies E <= 4k(K + ||T - C||)^2.
• Direct-sum obstruction — Proposition 2.1. Uniform bounds K on rows, columns and conjugation differences with E_j / k_j -> infinity give a non-unitarizable representation with norms at most 1 + K on a separable space.
• Unitarizable groups are amenable (OpenAI; Dixmier's question) — Theorem 4.1. A discrete group is amenable iff unitarizable; a nonamenable group has non-unitarizable representations with all norms at most 1 + eps, on a separable space if the group is countable.
• Thompson's group F — Corollary 5.1. Thompson's group F has non-unitarizable representations with all norms at most 1 + eps on a separable space.
• Group C*-algebras — Theorem 5.2. For a non-unitarizable uniformly bounded representation, the bounded homomorphism T_pi of l^1(G) has no bounded unital homomorphic extension to C*(G) or C*_r(G) (via Kadison's similarity theorem).

Reading order

Read the lessons in order. The first collects the general facts on uniformly bounded representations and proves the Day–Dixmier theorem. The second turns nonamenability into a sparse assignment of edges. The third turns the assignment into operators with uniformly bounded rows, columns and conjugation differences. The fourth proves that their cocycle is not inner and deduces the theorem.