Markushevich bases, twisted sums and linear extensions

David Yost

June 2019, Lviv, Banach Spaces and their Applications International conference dedicated to the 70th birthday of

Professor A. M. Plichko

AbstractMy random walk through mathematics has taken me from weaklycompactly generated spaces to polytopes, via C*-algebras,approximation theory, optimization, Lipschitz multifunctions andcombinatorial geometry. I will survey some of my steps on thisjourney, highlighting my interactions with Anatolij Plichko and hiswork.

All Mathematics is connected

The final year project of my Bachelor’s degreewas in Group Theory.A group G is called P-by-Q if it has a normal subgroup N withproperty P, and the quotient G/N has the property Q.For example, every finitely generated abelian-by-polycyclic group isresidually finite (Jategaonkar, 1974).Every finitely presented centre-by-metabelian group is residuallyfinite (Groves, 1978).This terminology can be handy in other branches of mathematics:

every separable-by-separable Banach space is separable;every reflexive-by-reflexive Banach space is reflexive.

However Hilbert-by-Hilbert does not imply Hilbert (Enflo,Lindenstrauss and Pisier, 1974).

My Masters thesiswas mainly about Weakly Compactly Generated (WCG) Banachspaces, i.e. those for which there is a weakly compact set withdense linear span, e.g.all separable Banach spaces, all reflexive Banach spaces.Some of their important properties: They• have Markushevich bases,• admit lots of projections (more precisely, enjoy projectionalresolutions of the identity, or PRI),• and can be renormed nicely.• Preservation of the WCG property is also an interesting question.The existence of a PRI together with a transfinite inductionargument establishes many results about WCG spaces (Amir andLindenstrauss, 1968).

A Markushevich basis (xγ , fγ)γ∈Γ for a Banach space X is abiorthogonal system for which (xγ)γ∈Γ generates a dense subspaceof X and (fγ)γ∈Γ generates a weak* dense subspace of X

∗.Any WCG space has an M-basis.An M-basis is called bounded if ‖xγ‖‖fγ‖ is bounded.It is called strong if every x ∈ X actually lies in the closed linearspan of {fγ(x)xγ : γ ∈ Γ}.Troyanski (1971) showed that every Banach space with a strongM-basis has an equivalent locally uniformly convex norm.Plichko (1977) proved the existence of a bounded M-basis in anyweakly compactly generated space.Later (1982) he showed that every Banach space with an M-basisactually has a bounded M-basis;and also that there is a Banach space with an M-basis whichcontains `∞, and hence cannot have a strong M-basis.

Being WCG is obviously preserved under quotients and directsums. It is not preserved under subspaces, althoughcounterexamples are rare.What about twisted sums: does WCG-by-WCG imply WCG?Johnson and Lindenstrauss (1974) showed that it does not ingeneral, but there are some straightforward partial results:

reflexive-by-WCG implies WCG, andWCG-by-separable implies WCG.

Things don’t work the other way: separable-by-reflexive does notimply WCG.The first counterexample boils down to the fact that c0 is notcomplemented in `∞.

This result is essentially due to Nakamura and Kakutani (1943).

TheoremIf Γ is an uncountable set, with cardinality no more than thecontinuum, then there is a subspace A of `∞ containing c0, suchthat A/c0 ∼= c0(Γ).Proof. A result of Sierpinski (rediscovered several times) says thatthere is a collection (Nγ)γ∈Γ of infinite subsets of N such thatNα ∩ Nβ is finite whenever α 6= β. Let A be the closure of thesubspace generated by c0 ∪ {χNγ : γ ∈ Γ}. It is evident that A is asubalgebra of `∞ and that c0 is an ideal in A. Puteγ = χNγ + c0 ∈ A/c0. Since χNαχNβ ∈ c0 whenever α 6= β, it isclear that eαeβ = 0 whenever α 6= β. A short calculation showsthat every n elements in {eγ : γ ∈ Γ} generate a Banach algebraisometric to `∞(n). From this, we obtain A/c0 ∼= c0(Γ).

Since A ⊂ `∞, A∗ is weak* separable and thus every weaklycompact set in A must be separable. This shows that A is notWCG, even though both c0 and c0(Γ) are.In particular A is not isomorphic to the WCG space c0 ⊕ c0(Γ),whence c0 cannot be complemented in A, let alone complementedin `∞.

TheoremLet Z be a nonseparable WCG space. Then there exists anextension of c0 by Z which is not WCG.

Proof. If (xγ , fγ)γ∈Γ is a (bounded) M-basis for Z then theoperator T : Z → c0(Γ), defined by Tx = (fγ(x)), has dense range.Passing to a subset, we may suppose that the cardinality of Γ is nomore than the continuum. The structure of the space A gives us ashort exact sequence (the first line in the following diagram). LetQ : A→ c0(Γ) be the quotient map and denote by X the pullbackof T and Q, i.e. X = {(a, z) ∈ A⊕ Z : Qa = Tz}. It is easy to seethat the natural map X → Z is a quotient map with kernelisomorphic to c0. Completing the diagram:

0 −→ c0 −→ A −→ c0(Γ) −→ 0∥∥∥ x x0 −→ c0 −→ X −→ Z −→ 0.

The map X → A has dense range, so X cannot be WCG.

Taking Z = `p(Γ), which is reflexive when 1 < p

Similar diagram chasing gives the following

TheoremIf P is a Banach space property for which P-by-reflexive impliesWCG, then P-by-WCG implies WCG.Cabello and Castillo (2000) showed that dual-by-reflexive impliesdual. This leads, amongst other things, to the conclusion that`1-by-WCG implies WCG.I presented these and related results at a conference in Trier in1997. Plichko was in the audience and soon afterwards produced asignificant improvement: if Y is isomorphic to a subspace ofL1(0, 1), then Y -by-WCG implies WCG.Outcome: J.M.F. Castillo, M. González, A.M. Plichko and D.T.Yost, Twisted properties of Banach spaces, Math. Scand. 89(2001), 217–244.If µ is a σ-finite measure, then L1(µ) is WCG. It remains unknownwhether L1(µ)-by-WCG implies WCG, in case L1(µ) is notseparable.

In 1998 Plichko and I were at a conference in Jarandilla. We endedup writing the semi-survey paperComplemented and uncomplemented subspaces of Banach spaces,Extracta Math. 15 (2000), 335–371.It aimed to systematically present what was then known aboutwhether a given infinite-dimensional Banach space has anynontrivial complemented subspaces. One central theme of thepaper was the existence of a Projectional Resolution of the Identityon any WCG space. This can be proved in several ways; they alluse weak compactness of some generating set in an essential way.Let us digress to discuss some related ideas.

In 1970, Tacon had the idea of working in a dual space, and usingthe weak* compactness of its unit ball instead. As a first step inthe process, he essentially proved the following.

TheoremIf M is a separable subspace of a Banach space X , and F is aseparable subspace of X ∗, then there is another separable subspaceN of X , which contains M, and a linear mapping L : N∗ → X ∗such that Lf is a norm preserving extension of f for each f ∈ N∗,and L(N∗) contains F .

The case F = {0} of this result is due to Heinrich and Mankiewicz(1982). A simpler proof was given later by Sims and Yost (1988),and subsequently generalised to involve the subspace of X ∗. Theonly essential difference between Tacon’s Lemma 5 and the resultabove is the extraneous hypothesis that X is smooth (i.e. that forevery norm one x ∈ X , there is a unique support functionalfx ∈ X ∗ with ‖fx‖ = fx(x) = 1).

In fact, the separability assumption can be weakened. Given anarbitrary subspace M, one can find a subspace N, containing bothM and a dense subset no larger than M, with the requiredextension property. A transfinite induction argument then gives thefollowing

TheoremLet X have the property that every separable subspace hasseparable dual (i.e. X is an Asplund space). Then there is a longsequence of subspaces Mα and linear extension mappingsLα : M

∗α → X ∗, indexed by ordinal numbers α, so that

(i) Mα ⊂ Mβ whenever α < β,(ii) Mα contains a dense subset no larger than α,(iii) Lα(M

∗α) ⊂ Lβ(M∗β) whenever α < β,

(iv) Mα = ∪β

Serious applications of the preceding theorem require an additionalcondition, namelyLα(M

∗α) = ∪β

Diestel and Uhl (1976) asked how much of this holds for a Banachspace with the Radon-Nikodým Property, but which is not a dualspace. The answer is: not much, in general. Building on someideas in our survey paper, Plichko and I showed thatThe Radon-Nikodým Property does not imply the SeparableComplementation Property, J. Funct. Anal. 180 (2001), 481–487.That is, there is a Banach space with the Radon-NikodýmProperty, and a separable subspace which is not contained in anycomplemented separable subspace. As far as we know, thefollowing questions are still open.Does there exist a Banach space with Radon-Nikodým Propertybut without any complemented infinite-dimensional separablesubspaces, or without an equivalent locally uniformly convex norm?

Going back in time again: my PhD thesis was supposed to beabout C*-algebras. Much of my work stems from the followingresult of Alfsen and Effros (1973).

TheoremLet A be a C*-algebra, J a closed subspace of A. Then J is anideal in A if and only if, whenever B1,B2,B3 are open balls in Awith J ∩ Bi 6= ∅ for each i , and B1 ∩ B2 ∩ B3 6= ∅, thenJ ∩ B1 ∩ B2 ∩ B3 6= ∅.This so-called 3-ball property has some consequences forapproximation theory (Holmes, Scranton and Ward, 1975).

TheoremLet M be a closed subspace of a Banach space X , with the 2-ballproperty. Then M is proximinal in X , and for every x ∈ X , the setof best approximants P = M ∩ B(x , d(x ,M)) is a set of constantwidth 2d(x ,M) in M, i.e. P − P ∼ BM(0, 2d(x ,M)).By ∼ we mean that the two sets have the same closure and thesame interior.

I am still thinking about sets of constant width, in finite andinfinite dimensions:E. Maluta and D. Yost, Thin sets of constant width, J. Math.Anal. Appl. 469 (2019), 1080–1087.Sets of constant width with a finite number of extreme points arealso interesting. Mostly I study polytopes now. Everything whichfollows is recent joint work with Guillermo Pineda-Villavicencio andJulien Ugon.For fixed v and d , what is the minimum number of edges, of alld-dimensional polytopes with v vertices?When v 6 2d , the answer is as follows.

TheoremFor k 6 d, a (d − k)-fold pyramid over a k-dimensional simplicialprism has d + k vertices and

(d+k2

)− 2(k

2

)edges; every other

d-polytope with d + k vertices has strictly more edges.

This verifies a conjecture of Grünbaum (1967).What about d-polytopes with more than 2d vertices?We will define the pentasm in dimension d as the Minkowski sumof a simplex and a line segment which is parallel to one 2-face, butnot parallel to any edge, of the simplex; or any polytopecombinatorially equivalent to it. In one concrete realisation, it isthe convex hull of 0, ei for 1 6 i 6 d and e1 + e2 + ei for1 6 i 6 d , where ei are the standard unit vectors in Rd .

TheoremConsider d-dimensional polytopes with exactly 2d + 1 vertices.The examples with d2 + d − 1 or fewer edges are as follows.(i) For d = 3, there are exactly two polyhedra with 7 vertices and11 edges; the pentasm, and a certain sum of two triangles whichwe call Σ3. None have fewer edges.(ii) For d = 4, a sum of two triangles ∆2,2 is the unique polytopewith 18 edges, and the pentasm is the unique polytope with 19edges. None have fewer edges.(iii) For d > 5, the pentasm is the unique d-polytope withd2 + d − 1 edges. None have fewer edges.We have a similar characterisation of polytopes with 2d + 2vertices and minimal number of edges. The case of 2d + 3 verticesseems to be difficult.

We define the excess degree ξ(P), or simply excess, of ad-polytope P as the sum of the excess degrees of its vertices, i.e.

ξ(P) = 2e − dv =∑

u∈VertP(degu − d).

Here as usual degu denotes the degree of a vertex u, i.e thenumber of edges of P incident with the vertex; VertP denotes theset of all vertices of P; and v and e denote the number of verticesand edges of P. Clearly a polytope is simple if, and only if, itsexcess degree is 0.

TheoremAny nonsimple d-polytope has excess degree at least d − 2.Moreover, if it has excess degree strictly less than d, then thenonsimple vertices all have the same degree, and they form a face.