Results on Lipschitz-free spaces, approximation properties, and colored graphs with a tree limit

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.

• Evaluations keep distances — Lemma 1.1. In the Lipschitz-free space F(M) of a pointed metric space, |delta_x - delta_y| = d(x, y).
• Linearization of Lipschitz maps — Lemma 2.1. A pointed Lambda-Lipschitz map u between pointed metric spaces induces exactly one bounded linear operator between the free spaces with delta_x -> delta_(u(x)); its norm is at most Lambda.
• From a dense set to compact sets — Lemma 3.1. If bounded operators S_m on a Banach space have norms at most lambda and S_m v -> v on a dense set, then S_m -> Id uniformly on every compact set.
• The free space of a bounded countable space with distances at least 1 — Proposition 4.1. For a countable pointed metric space N with distances >= 1 and diameter <= B, c -> sum c_x delta_x is a linear bijection of l_1(N minus o) onto F(N) with (1/2)|c|_1 <= |Jc| <= B|c|_1, and F(N) has 2B-BAP.
• A uniformly discrete space whose free space has AP and not BAP — Theorem 5.2. There is a countable pointed metric space M with distances >= 1 such that F(M) has the approximation property and has lambda-BAP for no lambda. Proved in the lesson on the wedge from the theorem on blocks. No hypothesis.
• Norms on l_1 without the metric approximation property — Corollary 5.3. For every integer p >= 1 the real space l_1 has a norm with (1/2)|c|_1 <= |c|_p <= B_p |c|_1, B_p = 300p^2 + 150p + 5, that has 2B_p-BAP and not p-BAP. For p = 1 it fails the metric approximation property. No hypothesis.
• The approximation property of a wedge — Proposition 4.1. If the free space of each of countably many pointed metric spaces has the bounded approximation property (with constants that may be unbounded), the free space of their wedge has the approximation property.
• Linearization of maps into a Banach space — Lemma 6.1. A Lambda-Lipschitz map gamma from a pointed metric space M to a Banach space X with gamma(o) = 0 extends to exactly one bounded linear operator T from F(M) to X with T delta_x = gamma(x); its norm is at most Lambda.
• A space with AP that is not approximable — Proposition 6.2. Assume (K) (a theorem of Kalton on nets; not proved in the programme). If M is countable with distances >= 1 and F(M) fails BAP, then F(M) is not approximable.
• Balls in permutation multigraphs are trees — Lemma 2.2. For Delta/2 independent uniform permutations of N vertices and two different vertices, the balls of radius r around them are disjoint tree balls with probability at least 1 - C^2/(N - C), C = 2(1 + Delta + ... + Delta^(r+1)).
• Colored regular multigraphs with a colored tree limit — Proposition 2.4. For even Delta >= 4 and k >= 1 there is a sequence of finite Delta-regular multigraphs with vertex colors in 1..k such that for every r and every type of colored ball of radius r the fraction of vertices with this type converges to its probability in the Delta-regular tree with independent uniform colors.
• The geometry of the block — Proposition 5.1. The block M_p: an integer-valued metric with diameter <= B_p; labels are 1-Lipschitz for the sup norm; every source vertex has distance R to its anchor and at least R to all anchors; closed balls of radius 2L have at most Q points; every label has at most H coordinates below D.
• The signed measures of an operator — Lemma 3.2. An operator on F(M_p) with range in the span of finitely many evaluations, norm <= 2p and anchor errors < 1 defines signed measures mu_x of mass 1 with the affine pairing, total variation <= 1 + 4pB and bump integral >= 3L/4.
• Long matched walks are rare — Proposition 5.1. For a finite set S of states and n >= 1, the fraction of vertices of G_m at which a matched walk of length n starts has limsup at most |S| (H/k) (Delta Q H / k)^n.
• Exceptional directions — Lemma 6.1. For an incidence e from x to y and states j, j' with x, j compatible, y, j' compatible and d(j, j') <= 2L: e is exceptional at (x, j) or its reverse is exceptional at (y, j').
• Extension of real Lipschitz functions from a finite set — Lemma 1.1. For 1-Lipschitz data v on a finite subset P of a metric space with values in [-L, L], z -> min over j of (v(j) + d(z, j)), cut off at L and -L, is a 1-Lipschitz extension with values in [-L, L].
• The anchor obstruction — Theorem 5.1. For every integer p >= 1, every bounded linear operator T_0 of finite rank on F(M_p) with norm <= p has |T_0 delta_a - delta_a| >= 1/2 for some anchor a. (Exercise 7.4: even >= 150p^2 + 7p.) No hypothesis.

Reading order

Five lessons. The first gives the definitions and the statements; the second proves the main theorem from the theorem on blocks; lessons 3 to 5 prove the theorem on blocks. The third lesson can be read without the first two. The theorems need no hypothesis; only the last section of the second lesson uses a named theorem of Kalton, which is not proved in the programme.