Group Actions on Twisted Sums of Banach Spaces

We study bounded actions of groups and semigroups G on exact sequences of Banach spaces from the point of view of (generalized) quasilinear maps, characterize the actions on the twisted sum space by commutator estimates and introduce the associated notions of G-centralizer and G-equivariant map. We will show that when (A) G is an amenable group and (U) the target space is complemented in its bidual by a G-equivariant projection, then uniformly bounded compatible families of operators generate bounded actions on the twisted sum space; that compatible quasilinear maps are linear perturbations of G-centralizers; and that, under (A) and (U), G-centralizers are bounded perturbations of G-equivariant maps. The previous results are optimal. Several examples and counterexamples are presented involving the action of the isometry group of Lp(0,1),p≠2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_p(0,1), p\ne 2$$\end{document} on the Kalton–Peck space Zp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_p$$\end{document}, certain non-unitarizable triangular representations of the free group F∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {F}}_\infty $$\end{document} on the Hilbert space, the compatibility of complex structures on twisted sums, or bounded actions on the interpolation scale of Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_p$$\end{document}-spaces. In the penultimate section we consider the category of G-Banach spaces and study its exact sequences, showing that, under (A) and (U), G-splitting and usual splitting coincide. The purpose of the final section is to present some applications, showing that several previous result are optimal and to suggest further open lines of research.


Introduction
This paper emerges from the observation of similarities between different problems: (a) The construction of non-unitarizable, bounded, representations of the free group F ∞ on the Hilbert space.(b) The construction of operators on the Kalton-Peck space Z 2 .(c) The differential process associated to a complex interpolation scheme.(d) Actions of groups on exact sequences of Banach spaces.(e) The existence of certain bounded groups of isomorphisms on the space c 0 .In all cases, certain non-linear maps (including sometimes linear unbounded maps) and their compatibility with the action of some groups of operators through commutator estimates are at the core of the problem.In (a), a linear unbounded map used to define a non-inner derivation and therefore a non-unitarizable representation [42]; in (b) the Kalton-Peck map KP [36]; in (c) is the "Ω-operator" mentioned by several authors Cwikel et al. [23], Rochberg [43], Carro [14]...And in (d) we encounter the Banach version of the three-representation problem (see [37]).Another unexpected example (e) is a linear unbounded map used in [1] to define a nontrivial derivation in a study of bounded groups acting on c 0 .Connections between some of those elements had been observed before: for instance, Kalton observed [33,34] that while working on Köthe spaces, Ω-operators are a special type of quasilinear map, that he called L ∞ -centralizers, intimately connected with the complex interpolation scale.
To obtain a unified point of view we consider a group or semigroup G, two bounded actions u, v on two Banach spaces X, Y and introduce the notion of G-centralizer Ω: this allows us to construct an exact sequence 0 −→ X → X ⊕ Ω Y −→ Y −→ 0 of Banach spaces and connect possible actions of G on the twisted sum space X ⊕ Ω Y with commutator estimates involving Ω and derivations of the group.
Our results move at two levels, the theoretical and the examples/counterexamples.On the former side, the theory we display could be described as follows.Let (A) be the condition: G is amenable and let (U) be: X is G-complemented in its bidual (meaning, complemented by a G-equivariant projection).
• Triangular representations of groups on the Hilbert space H may be interpreted as diagonal representations on H seen as a twisted Hilbert space.• Under (A) and (U), a uniformly bounded family (T g ) g∈G of operators yielding commutative diagrams provides a compatible action of G on X ⊕ Ω Y .• Every Ω compatible with an action on X ⊕ Ω Y is a linear perturbation of a G-centralizer (possibly with values in a larger target space).• Under (A) and (U), every G-centralizer is a bounded perturbation of a G-equivariant map.• We introduce the category of G-Banach spaces and show that, under (A) and (U), a G-exact sequence of G-spaces G-splits if and only if it splits as an exact sequence of Banach spaces.
The results above are optimal because on the side of counterexamples: • We will use a construction of Pytlic and Szwarc [42] to show a centralizer that is not a bounded perturbation of an equivariant centralizer when G is non-amenable.We will provide another counterexample, inspired from [1], when X is not complemented in its bidual.
• We will show that the Kalton-Peck map is not a centralizer for the groups of isometries on L p , p = 2 or isometries preserving disjointness on L 2 .It is however compatible with the actions of those groups.• In the case of the group of isometries of L 2 , the Kalton-Peck map is not even compatible with the action of that group.
There are specific sections devoted to actions of groups on complex interpolation scales, on Kalton-Peck spaces and on higher order Rochberg spaces, as well as to the connections between G-centralizers and almost transitivity.

The Background
Let X, Y be Banach spaces.In what follows ∆ ⊂ Y represents a dense subspace of Y (sometimes called the intersection space), while Σ represents the ambient space, namely, a vector space containing X.When necessary, we will alternatively assume that there is an injective linear map  : X → Σ, in which case the subspace [X] will be normed with (x) = x X .A homogeneous map Ω : ∆ −→ Σ is a z-linear map ∆ X if there is a constant C such that for all finite sequences of elements y 1 , . . ., y In this paper we mainly use the notation Ω : ∆ X, although Ω : Y X can also be appear when the choice of ∆ is clear from the context or irrelevant.When condition (b) holds only for pairs of points then Ω is called quasilinear.A quasilinear map Ω : ∆ X with ambient space Σ is said to be trivial if there is a linear (not necessarily continuous) map L : ∆ −→ Σ such that Ω − L : ∆ → [X] is bounded, in the sense that Ω(y) − L(y) [X] ≤ M y Y for some constant M and all y ∈ ∆.Two quasilinear maps Φ, Ψ : ∆ X with ambient space Σ are said to be equivalent , and denoted Φ ∼ Ψ, (resp.boundedly equivalent) if Φ − Ψ is trivial (resp.Φ − Ψ : ∆ −→ X is bounded).The twisted sum generated by a quasilinear map Ω is the completion X ⊕ Ω Y of the space X ⊕ Ω ∆ := {(ω, y) ∈ Σ × ∆ : ω − Ωy ∈ [X]} endowed with the quasi-norm y Y + ω − Ωy [X] .From now on, except when in need, we shall omit the embedding .If Ω is z-linear then • Ω is equivalent to a norm, and thus X ⊕ Ω Y is a Banach space.Kalton showed [30] that quasilinear maps on B-convex Banach spaces are zlinear; therefore, twisted sums in which the quotient space is B-convex are Banach spaces.The map ı : X −→ X ⊕ Ω Y given by ı(y) = (y, 0) is an into isometry and the map π : X ⊕ Ω Y −→ Y given by π(ω, x) = x takes the unit ball of X ⊕ Ω Y onto that of X.These spaces and operators form a short exact sequence 0 / / X ı / / X ⊕ Ω Y π / / Y / / 0 that shall be referred to as the sequence generated by Ω.Two exact sequences of Banach spaces are called equivalent when there is an operator T making the diagram Given two maps S, T , its commutator is defined as [S, T ] = ST − T S provided this makes sense.We will need to use a generalized commutator for three maps defined as [u, Ω, v] = uΩ−Ωv, whenever this makes sense.

G-centralizers
Definition 3.1.Let G be a semigroup.A G-space is a normed space X equipped with a bounded action G × X → X; namely, a morphism of semigroups u : G → L(X) such that γ(u) := sup{ u(g) : g ∈ G} < ∞.
Note that we do not require G to carry any topology and therefore there is no continuity involved with respect to G (alternatively we may think of G as discrete).Occasionally we will consider unbounded or even nonlinear actions, but that will be explicitly said.Paramount examples of bounded actions are (see the appropriate section in the paper for unexplained terms): (a) The action of the group of units U of L ∞ (S, µ) on either L ∞ -Banach modules or Köthe spaces.In particular, the action of the Cantor group 2 ω = {−1, +1} N on spaces with unconditional basis or that of the group 2 <ω of elements of 2 ω that are eventually 1 on c.(b) The action of the group generated by measure preserving rearrangements of the base space and change of signs on rearrangement invariant Köthe spaces.(c) The action of the group Isom(X) of isometries of X on X.(d) The action of the group Isom disj (L 2 ) of isometries that preserve disjointness on L 2 .(e) The natural left regular action of the free group F ∞ on the Hilbert space seen as ℓ 2 (F ∞ ).
Given an exact sequence 0 → X → Z → Y → 0 of G-spaces, we will agree for the rest of this paper that the action of G on X will be denoted u, that on Y will be denoted v and that on Z will be denoted λ.Definition 3.2.Let G be a semigroup.
G-operator: An operator (resp.a linear map)

and let
Σ ⊃ X be a G-superspace of X.A quasilinear map Ω : ∆ X with ambient space Σ is said to be a G-centralizer if the family of maps [u(g), Ω, v(g)] takes values in X and is uniformly bounded, i.e., there exists a constant G(Ω) > 0 such that u(g)Ωy − Ωv(g)y X ≤ G(Ω) y Y for all g ∈ G and y ∈ ∆.
To avoid confusion, let us make explicit that in the above we use the same letter for an action on a G-space and for the action by restriction on a G-subspace; for example for any g ∈ G, u(g) extends to a map on Σ still denoted u(g).
It will spare us a few headaches to briefly discuss the roles of the "ambient" and "intersection" spaces Σ and ∆.Observe that Ω is in principle only defined on ∆, not in Y .It is well known [36] that every quasilinear map Ω : ∆ X can be extended to a quasilinear map Ω : Y → X, but replacing Ω by this "artificial" Ω may spoil the compatibility conditions with G, so this approach is useless for us.
When the spaces Σ, Ξ carry their own norms and X is a G-subspace of both Σ and Ξ then we must set PO = (Σ ⊕ Ξ)/∆ to make PO a G-superspace of X.However, once actions are involved, a situation appears: given an operator u : X → X and a quasilinear map Ω : ∆ → Σ the composition uΩ seems impossible.A way to overcome the difficulty is to assume that u : X → X is the (continuous) restriction of a linear map Σ → Σ.This is reasonable and, in most occasions, feasible; therefore we usually assume that Σ is a G-superspace of X, as in the definition of G-centralizer.
The issue of the dense subspace.In classical interpolation theory one considers choices of ∆ so that Ω : ∆ → X. Adapting their terminology, we can define the dominion of quasilinear map Ω : Y X as the space DomΩ = {y ∈ Y : Ωy ∈ X} endowed with the quasinorm y D = Ωy X + y Y .In this form DomΩ is isometric to the closed subspace {(0, y) ∈ X ⊕ Ω Y } of X ⊕ Ω Y .More often than not, DomΩ is dense in Y , as it is the case in the complex interpolation context (this explains why we impose the assumption on the interpolation couple (X 0 , X 1 ) of being regular, which means that X 0 ∩ X 1 is dense in both X 0 and X 1 ) and DomΩ = Y if and only if Ω : Y → X is bounded.On the other hand, it may well happen that DomΩ = {0}: see [6], Proposition 8.3 or the example after Proposition 3.10.
And again, when an action v of G on Y is involved, we need a sound meaning for Ωv(g), which is achieved by guaranteeing that v leaves ∆ invariant.Still a problem appears when one has two quasilinear maps Ω : ∆ X and Φ : ∆ ′ X defined on different dense subspaces ∆, ∆ ′ ⊂ Y .In this case we cannot consider them defined on the same dense subspace by making a simple intersection since it could well be that ∆ ∩ ∆ ′ = {0}.In most cases the choice of a common ∆ is natural, but, in general, one has to be careful with this point.
We are ready to start our study with a simple observation whose interest will be shown later: in the Banach ambient, the action of a group G induces an action of the semigroup ℓ 1 (G).Lemma 3.4.A z-linear G-centralizer between G-Banach spaces is an ℓ 1 (G)-centralizer between the associated ℓ 1 (G)-modules.
Our first objective is the three-representation problem that Kuchment considers in [37]: given an exact sequence 0 → X → Z → Y → 0 and some group G acting on Y, Z and X in a compatible way, to what extent the action on Z can be recovered from the actions on X and Y .Or else: given u, v, how to obtain a compatible action λ on X ⊕ Ω Y ?Definition 3.5.Let 0 → X → Z → Y → 0 be an exact sequence.Assume that X, Y are G-spaces.An action λ of G on Z will be called compatible with the sequence if for each g ∈ G there is a commutative diagram 0 Compatibility is a homological notion: G is compatible with a sequence if and only if its it compatible with any equivalent sequence.The existence of compatible actions and G-centralizers are connected: Proof.Observe that the action λ is defined first on X ⊕ Ω ∆ and then extended by density to X On the other hand, the best possible value of G(Ω) is at most γ(λ) since u(g)Ωy − Ωv(g)y X = λ(g)(Ωy, y) Ω ≤ λ(g) y Y .
. Lemma 3.7.Let 0 → X → X ⊕ Ω Y → Y → 0 be an exact sequence in which X, Y are G-spaces and Ω : ∆ X with ambient space Σ.If Ω is a G-centralizer then TFAE: (a) The quotient map admits a G-linear section L If, moreover, ∆ ⊂ Dom(Ω) then (a) and (b) are also equivalent to: Proof.Just set Ly = (ℓy, y).
The reason why the hypothesis "Ω is a G-centralizer" is needed is to be sure that λ(g) = u(g) 0 0 v(g) on X ⊕ Ω Y is a compatible bounded action on X ⊕ Ω Y .To describe the general situation we first need to develop a few ideas.The general version of Proposition 3.6 will be presented in Proposition 3.13 and that of Lemma 3.7 in Lemma 3.14.The fact that G-centralizers are, informally speaking, quasilinear maps having uniformly bounded commutators [u(g), Ω, v(g)] suggests to consider with special attention the case [u, Ω, v] = 0: In particular, G-equivariant linear maps (operators) are the G-linear maps (operators) of Definition 3.2.Since G-equivariant maps, as well as their bounded perturbations, are G-centralizers, it is natural to ask about the converse: Is a G-centralizer always a bounded pertubation of a G-equivariant map?And its "linear" version: is a linear G-centralizer always a bounded perturbation of a G-linear map?
Let us provide an optimal answer: yes when G is an amenable group and X is adequately complemented in its bidual.Recall that a Banach space is called an ultrasummand [32] when it is complemented in its bidual.We transplant this notion to G-Banach spaces.By G-projection we mean a G-operator which is a projection.Definition 3.9.A G-Banach space X is a G-ultrasummand if there exists a G-projection P : X * * → X.
Observe that if X is a G-space then also X * , hence X * * , is a G-space so that X is a G-subspace of X * * , so the above makes sense.One has: Proposition 3.10.Let G be an amenable group and let X, Y be G-spaces with X a Gultrasummand.(a) Any (linear) G-centralizer Ω : Y X is a bounded perturbation of a Gequivariant (linear) map.(b) A trivial G-centralizer Ω : Y X is boundedly equivalent to a G-linear map.
Proof.Proof of (a): since G is amenable, there is a left invariant measure µ on it, and since X is a G-ultrasummand there is a G-projection P : X * * → X.We define the bounded map B : Y → X By = P G (u(g −1 )Ωv(g)y − Ωy)dµ where we integrate in the weak* sense.If h ∈ G then The second part is clear: when Ω is linear, B is also linear.Proof of (b): if Ω = B + L with B bounded and L linear, L must also be a G-centralizer.Then apply (a).
Part (b) extends [11, Lemma 1]: a trivial L ∞ -centralizer is a bounded perturbation of a linear L ∞ -centralizer.As announced, the previous solution is optimal since the amenability condition is necessary.Let us put the counterexample in the proper context.As was proved by Day [25] and Dixmier [26], a bounded representation of a countable amenable group on the Hilbert space is unitarizable, meaning that it is a unitary representation in some equivalent Hilbert norm.Ehrenpreis and Mautner [27] provide a non-unitarizable bounded representation of a countable group on the Hilbert space.What is now known as the Dixmier problem asks whether unitarizability of all bounded representations of a countable group characterizes amenability.
Regarding the non-amenable free group F ∞ with countably infinitely many generators, Pytlic and Szwarc [42], see also [39,41], showed the existence of a bounded, non-unitarizable representation of F ∞ on the sum H ⊕ H of two copies of the Hilbert space.The authors of [28] used this example to investigate transitivity properties of bounded actions on the Hilbert space, and we now follow their lines with another perspective in mind.As in [28] we extend the action of F ∞ to Aut(T) where T is the Cayley graph of F ∞ with respect to its free generating set.Indeed, Aut(T) acts in a natural way on ℓ 2 (T) as well as on ℓ ∞ (T) or ℓ 1 (T), by the left regular unitary representation u: u Since [u(g), R] : ℓ 2 (T) → ℓ 2 (T) has norm at most 2 for all g ∈ Aut(T) [28], R is an Aut(T)centralizer ℓ 2 (T) ℓ 2 (T), which is moreover trivial since it is linear (note that we have chosen ∆ = ℓ 2 (T ) and Σ = ℓ ∞ (T ) here).We may obtain another Aut(T)-centralizer through the dual situation of the "left shift" operator L : ℓ 1 (T) → ℓ 1 (T) defined as L(e t ) = e t where t is the predecessor of t along T, and L(e ∅ ) = 0 (here we have chosen ∆ = ℓ 1 (T) and Σ = ℓ 2 (T)).Note that both R and L could also be defined as from ℓ 1 (T) to ℓ ∞ (T), in which setting L + R makes sense.Since L + R commutes with every g ∈ Aut(T), we have [u(g), L] = −[u(g), R] and so L is also an Aut(T)-centralizer L : ℓ 2 (T) ℓ 2 (T).One has: Proposition 3.11.The linear Aut(T)-centralizer R is not boundedly equivalent to a linear Aut(T)-equivariant map defined on the whole ℓ 2 (T).The linear Aut(T)-centralizer L is not boundedly equivalent to a linear Aut(T)-equivariant map defined on ∆ = ℓ 1 (T).
Proof.Since R takes values in ℓ ∞ (T) but not in ℓ 2 (T), such a a linear Aut(T)-equivariant map would have the same property.But it is proved in [28] that any linear (unbounded) map Aut(T)-equivariant map from ℓ 2 (T) to ℓ ∞ (T) must be homothetic, and in particular it must take value in ℓ 2 (T).Regarding L, the linear equivariant map ℓ would have to be continuous from ℓ 1 (T) to ℓ 2 (T).The dual map would then be continuous from ℓ 2 (T) to ℓ ∞ (T), and therefore would be homothetic, so ℓ itself would be homothetic.So L would be .ℓ 2 (T) − .ℓ 2 (T) bounded.This is false, since for x = t∈N e t , where N is a family of n elements of F ∞ of length 1, we have We now study the general case, namely, sequences 0 → X → X ⊕ Ω Y → Y → 0 in which there is a compatible action λ on X ⊕ Ω Y but it is not "diagonal".The first observation is that a compatible action λ has necessarily the form with d(g) a linear (not necessarily bounded, even when λ(g) is) map from ∆ to Σ. Observe that a compatible bounded nonlinear action on X ⊕ Ω Y always exists, and it is given by This sets the key idea of how d could be found: It could also occur that Ω and the actions u, v are so well coordinated as to make [u, Ω, v] linear: such is the case when Ω is the Kalton-Peck map, see Section 6.
Derivations are of course fundamental for the study of unitarizability of bounded representations on the Hilbert space, such as the above representation of Aut(T); we address the reader to Pisier's book [41].They also have been studied on direct sums of Banach spaces [28] but, as far as we know, not on twisted sums.To perform such an study we must begin relaxing the requirement that [u(g), Ω, v(g)] is linear to "being at uniform distance to a linear map", in the sense of the next definition: Definition 3.12.Let X, Y be G-spaces with respective actions u and v.We say that g → d(g) is a linear derivation of (u, v) if for all g ∈ G, d(g) : ∆ −→ Σ is a (possibly unbounded) linear map, and d(gh We are ready to obtain the general version of Proposition 3.6: Proposition 3.13.Let Ω : ∆ X be a quasi-linear map between two G-spaces.TFAE: Proof.The equality λ(gh The boundedness condition is a straightforward computation.
And, as promised, the general version of Lemma 3.7.
Lemma 3.14.Let Ω, Ω ′ : ∆ X be quasilinear maps between G-spaces Y and X, with ambient space Σ, and let L : ∆ −→ Σ be a linear map.Then In particular, homogeneous bounded maps admit associated derivation To avoid confusion let us make clear that all derivations in this lemma are meant to be derivations of the given pair of representations (u, v).
is clearly the general version of Lemma 3.7 (c) with a couple of delicate points to check: that (−Ly, y) ∈ X ⊕ Ω Y , which is true when y ∈ Dom(Ω + L) and that L is G-linear.To this end, observe that The example around Proposition 3.11 shows two essentially different bounded actions of Aut(T) on ℓ 2 (T) ⊕ ℓ 2 (T): one is the unitary action u(g) 0 0 u(g) and the other is . By the above discussion, this triangular action on ℓ 2 (T) ⊕ ℓ 2 (T) and the diagonal one on ℓ 2 (T) ⊕ L ℓ 2 (T) are "the same".Shifting the classical perspective, we can therefore reformulate this construction as the remarkable fact that Aut(T) with its diagonal action, is "centralized" by two essentially different quasilinear maps: 0 and L. Thus, all pieces are on the board, except one: how to obtain a linear derivation of a quasilinear G-compatible map (assuming it exists)?The context of interpolation will provide some answers, and this is the content of the next section.

Actions on interpolation scales
We now consider exact sequences of G-spaces generated by complex interpolation of a scale on which G acts, in a way to be described.We refer to [2], [13] or [44] (see also [35] or [15] for specific details) for sounder information about the complex interpolation method for pairs and their associated differentials.An interpolation pair (X 0 , X 1 ) is a pair of Banach spaces, both of them linearly and continuously contained in a larger Hausdorff topological vector space Σ, which can be assumed to be Σ = X 0 + X 1 endowed with the norm x = inf{ x 0 0 + x 1 1 : x = x 0 + x 1 x j ∈ X j for j = 0, 1}.The pair will be called regular if, additionally, the intersection space X 0 ∩ X 1 is dense in both X 0 and X 1 .We denote by S the complex strip defined by 0 < Re(z) < 1.A Calderón space C is a certain Banach space of holomorphic functions F : S → X 0 + X 1 for which the evaluation maps δ z : C → Σ are continuous.This forces the evaluation of the derivatives δ ′ z : C → Σ to be continuous too.The interpolation spaces are defined to be X z = {x ∈ Σ : x = f (z) for some f ∈ C} endowed with natural quotient norm.There are various possible choices for C and we shall consider either C(S, X 0 + X 1 ), the classical Calderón space (see [2]), or F ∞ z , Daher's space (as in [24] or [15,Section 5]).If B z : X z → C is a homogeneous bounded selection for the evaluation map, the differential map of the process is An operator τ : Σ → Σ is said to act on the scale defined by the interpolation pair (X 0 , X 1 ) if it is a bounded operator X i → X i , i = 0, 1 [15].A generalized Riesz-Thorin theorem [45] yields that τ is automatically bounded from X θ → X θ for all 0 < θ < 1, with an estimate Definition 4.1.Let (X 0 , X 1 ) be a complex interpolation pair.A semigroup G acting on Σ is said to act on the scale if G acts boundedly on X i for i = 0, 1.
The interpolation estimate above implies that G also acts on X θ for all 0 < θ < 1 and that if G acts as an isometry group on the scale then it also acts as an isometry group on X θ , 0 < θ < 1, as well as on Σ and X 0 ∩ X 1 .The same holds for semigroups of contractions.Moreover, C is a G-Banach space defined by the action g C (f )(z) = g(f (z)) with estimate (in the standard situations described above) g C ≤ g .Note that the actions u and v in this setting are simply u(g) = v(g) = g.Where is our promised derivation?Here: 0.
2 admits an isometric version that we formulate now.A regular interpolation pair is optimal if for every 0 < θ < 1, every point in X θ admits a unique minimal function in F ∞ θ , see [15,Def. 5.7].Daher proved in [24,Prop. 3] that a regular pair of reflexive spaces with X 0 strictly convex is optimal.
Corollary 4.3.Let (X 0 , X 1 ) be an optimal interpolation pair.Then Ω θ is equivariant with respect to the semigroup of contractions on the scale which act as isometric embeddings on X θ .In particular, Ω θ is equivariant with respect to the group of isometries acting on the scale.
Proof.The map Ω θ is uniquely defined now since (B θ x)(θ) = x and B θ x = x θ .If g is a contraction on the scale, then g C also acts as a contraction on the space It is a bit disappointing that a zero derivative is all we got.There is a reason for that: the action of G on the scale (X z ) is constant: u z (g) = g, ∀z.To amend this, consider for each z a bounded action Assume one has a semigroup G with an action u on X θ (only for that fixed θ!) but in such a way that u = u θ for some analytic family (u z ) of actions.Thus, the compatible action of G on nonlinear, bounded action, what we need is to find linear bounded perturbations of [u(g), Ω θ ].
We use here some ideas of Carro [14]: Lemma 4.5.Let u = (u z ) z∈S be an analytic family of actions of G on the spaces of the scale (X z ) z∈S generated by a regular pair (X 0 , X 1 ).The following map is bounded Proof.The key observation is that for x ∈ X θ the function u z (g) (B θ x) (z)−B θ (u θ (g)x)(z) ∈ ker δ θ which implies that its derivative at θ must be in X θ .It only remains to compute This means that λ(g To obtain a bounded action we need that sup g λ(g) < +∞.Since ) is an analytic family of actions and we are using the standard Calderón space C = C(X 0 , X 1 ) endowed with the norm h ∞ := sup{ h(it) , h(1 + it) : t ∈ R} then we set γ(u) = sup g∈G sup t∈R { u it (g) , u 1+it (g) } (since X z+it = X z for real t, we may assume that u z+it (g) = u z (g)).We have: The interpolation inequality yields u θ (g) ≤ γ(u) and thus one has All this yields, Theorem 4.6.Let (X 0 , X 1 ) be a regular pair.Let u be an analytic family of actions of G on the scale (X z ) z∈S such that γ(u) < ∞.Then It is certainly satisfying that the term "derivation" agrees here both with the classical meaning and with Definition 3.12!Let us provide the first of a series of natural applications of these results.Let (X 0 , X 1 ) be an optimal interpolation pair of uniformly convex and uniformly smooth spaces and 0 < θ < 1.We claim that the semigroup of contractions on X θ is compatible with Ω θ : To prove it, we set a bounded action g d(g) and where Ω * θ denotes the quasi-linear map induced on X * θ by the identity X * θ = (X * 0 , X * 1 ) θ .If g(y) = φ, y x with φ, x is a contraction of rank 1 with norm at most 1 on X θ , let F be an extremal for x ∈ X θ and G an extremal for φ ∈ (X * 0 , X * 1 ) θ ).Then g z (y) = G(z), y F (z) defines an analytic family of contractions of rank 1 on the scale (X z ) such that g θ = g.Now apply Lemma 4.5 after calculating dg(y) dz | θ = Ω * θ (φ), y x + φ, y Ω θ (x).Further applications will be given in Sections 6 and 7.

Actions on Köthe spaces
When working with Köthe spaces with base measure space S, the ambient Σ is usually chosen as the space L 0 (S) of measurable functions on S, and ∆ as Y itself.A Köthe space is a vector subspace K of L 0 (S) endowed with a norm such that f ∈ K and |g| ≤ |f | then g ∈ K and g ≤ f .A r.i.Köthe space over [0, 1] is a Köthe space X such that f ∈ X ⇒ f σ ∈ X for every measure preserving map σ : [0, 1] → [0, 1].Köthe spaces are usually considered in their L ∞ -module and L ∞ -centralizer structures.The notion of L ∞ -centralizer can be subsumed in our notion of G-centralizer when the spaces are B-convex (e.g.superreflexive), which is actually the most interesting situation regarding interpolation.Indeed, if U will denote the group of units of L ∞ (µ), i.e. of unimodular functions in L 0 (S) then Proposition 5.1.Let Ω : Y X be a quasilinear map, with X, Y B-convex Köthe spaces.Then Ω is an U-centralizer if and only if it is an L ∞ -centralizer.
Proof.The set of convex combinations of units is dense in the ball of L ∞ .If Ω is a Ucentralizer then, by Proposition 3.4, it is a ℓ 1 (U)-centralizer.We can use density plus z-linear character of Ω to conclude that Ωa − aΩ ≤ K for all a in the unit ball of L ∞ .
U-actions on Köthe spaces have a somehow "rigid" nature, whose paradigm is Kalton's stability theorem [34]: the endpoint spaces of an interpolation scale of uniformly convex Köthe spaces X 0 , X 1 are uniquely determined, up to equivalence of norms, by the pair formed by the space X θ and the differential Ω θ , 0 < θ < 1.We additionally have: Theorem 5.2.Let (X 0 , X 1 ) be an interpolation pair of superreflexive Köthe spaces on a measure space S. Let G be a group containing the group of units U(S), acting boundely on X θ and acting on Σ. TFAE: (a) Ω θ is a G-centralizer.
(b) G acts on the scale.
Proof.One implication is Proposition 4.2.Assume that Ω θ is a G-centralizer.For g ∈ G, and i = 0, 1 let g −1 X i ⊂ Σ be endowed with the complete norm x g i = gx i .Form the new Calderón space C(g −1 X 0 , g −1 X 1 ) and define an isomorphism θ is boundedly equivalent to Ω θ , with a uniform constant on g.Since G contains the group U of units, Ω θ and Ω ′ θ are L ∞ -centralizers.Kalton's stability theorem will imply that the norms .i and .g i are equivalent, with a constant independent of g ∈ G, as soon as we amend in the next Lemma the required amalgamation.In conclusion, that G acts on the scale.
We will need to simultaneously consider differentials in various scales, so we will denote Ω W the differential generated by W := (W 0 , W 1 ).Lemma 5.3.There exists a function K(•) such that whenever X := (X 0 , X 1 ) and Y := (Y 0 , Y 1 ) are interpolation pairs of superreflexive Köthe spaces on the same measure space such that Y θ = X θ , with C-equivalence of norms, and Ω X θ and Ω Y θ are C-boundedly equivalent then the norms for which the conclusion of the theorem does not hold for C and K(n) = n.The pairs ℓ 2 (X n i ) and ℓ 2 (Y n i ) generate C-equivalent interpolation spaces with C-boundedly equivalent differentials while their norms are are not equivalent, in contradiction with Kalton's theorem [34] (in the version presented in [15,Thm. 3.4]).

Actions on Kalton-Peck spaces
Differentials obtained from complex interpolation of pairs (X 0 , X 1 ) of two Köthe spaces on the same base measure space are L ∞ -centralizers.The differential generated by the interpolation pair (L ∞ (µ), L 1 (µ)) deserves special attention.As it is well-known (L ∞ (µ), L 1 (µ)) 1/p = L p (µ); and if one picks positive normalized f then B(f )(z) = f pz is an extremal and thus for θ = 1/p one gets Ω θ (f ) = B(f ) ′ (θ) = pf log(f ).In what follows we will call Kalton-Peck map on L p the L ∞ -centralizer KP : L p L p defined by KP(f ) = pf log f f (the p is important for duality issues).The twisted sum space Z p (µ) = L p (µ) ⊕ KP L p (µ) will be called the Kalton-Peck space.Especially interesting is the case L ∞ (µ) = ℓ ∞ since Banach spaces with unconditional basis are ℓ ∞ -modules.
Fix 1 < p < ∞ and let us think now about compatible ℓ ∞ -actions on the Kalton-Peck space Z p .The Kalton-Peck map has a peculiarity: if w = (w n ) is an infinite sequence of successive normalized blocks in ℓ p and τ w : ℓ p → ℓ p is the operator τ w (x) = x n w n = w • x then the commutator [τ w , KP] is linear: Therefore, if we consider the semigroup BC p of the block contractions above on ℓ p then we get: Lemma 6.1.There is a compatible bounded action of BC p on Z p given by τ w τ KPw 0 τ w These operators were introduced by Kalton [31] in the case p = 2 to obtain isometric complemented copies of Z 2 inside Z 2 .In the next section we will generalize these results.

Actions on Rochberg spaces
We refer to [9] for possible unexplained definitions or facts.Given an interpolation pair (X 0 , X 1 ), the n th Rochberg space R n z is defined to be the space R n z = {( f (n−1) (z) (n−1)!, . . ., f ′ (z), f (z)) : f ∈ F } endowed with its natural quotient norm.It is clear that R 0 z = X z and R 1 z is isomorphic to X z ⊕ Ωz X z .It was shown in [9] that Rochberg spaces are connected forming natural exact sequences 0 z with ambient space Σ m .We are especially interested in the maps with short exact rows and columns (even if we omitted the 0's).We focus now on the situation in which the spaces of the scale have a common unconditional basis (e n ) and an additional property.For X with basis (e n ) we will call property (W ) the fact that for each normalized block sequence w = (w n ) of X, the map τ w : x −→ w • x is an operator of norm at most 1 (equivalently, λ n w n ≤ λ n e n ); and that the maps τ w form a semigroup for composition.Identifying w with τ w , this allows us to see the set of normalized block sequences w = (w n ) on X as a semigroup Block X acting on X. Assume the spaces of the scale have property (W), and given θ, an analytic family of actions of Block X θ can be defined as follows: let B θ be a homogeneous optimal selector for the evaluation map δ θ : F → X θ with the property that supp B θ (x)(z) ⊂ supp x for each finitely supported x.This makes that for w ∈ Block X θ and all z one has B θ (w)(z) ∈ Block Xz .We define the following analytic family of actions w z (x) = B θ (w)(z) • x and note that w θ (x) = w.x.
x and thus, by Theorem 4.6, there is an action on R 2 z given byλ 2 (w) = w Ω θ (w) 0 w in accordance with the previous result for Kalton-Peck maps.The result can now be iterated for higher order Rochberg spaces to obtain: Proposition 7.1.Let (X 0 , X 1 ) be an optimal interpolation pair of spaces such that X z has property (W) for each z.For fixed θ there is a bounded action of the semigroup Block X θ of normalized block sequences of X θ on R n z given by  θ (w) . . .
This applies in particular to the scale of ℓ p spaces, and provides new operators on Rochberg spaces that can be used to get insight on their properties.This research will appear elsewhere.

Actions and almost transitivity
An isometric action u of a group G on a space X is said to be almost transitive if the orbit u(G).x is dense in S X for some (and therefore for all) x ∈ S X , [40].A bounded action u of G on X is said to be almost transitive if there is some u(G)-invariant renorming of X for which the isometric action u is almost transitive.The definition is independent of the choice of an invariant renorming, since all these renormings are multiple of each other [22].Proof.The objective is to show that DomΩ = 0 ⇒ DomΩ = Y .Pick y ∈ S Y generating a dense orbit with respect to an equivalent invariant norm for the action v and use also a norm on X invariant under the action u.Then Ω(v(g)y) −u(g)Ωy X = [u(g), Ω, v(g)]y X ≤ C therefore Ωv(g)y Y ≤ C + Ωy X , so Ω is bounded on a dense subset of S Y .
L ∞ -centralizers acting on Köthe spaces have dense dominion [3].Consequently: Proposition 8.2.Let (X 0 , X 1 ) be an interpolation pair with a common Köthe space structure and let 0 < θ < 1.If Ω θ is unbounded then no bounded group acting on the scale can act almost transitively on X θ .
Proof.If an almost transitive group G acts boundedly on the scale, Ω θ would be a Gcentralizer by Proposition 4.2, therefore would be bounded by Proposition 8.1.
Recall from [46] (see also [18,Propositions 6.1 and 6.2]) that if X is a space with a shrinking basis then (X, X * ) 1/2 is a Hilbert space.Therefore, if X is either (a) a supereflexive Köthe space on a measure space S different from L 2 (S), or (b) a space with a shrinking basis such that the differential Ω 1/2 generated at (X, X * ) 1/2 is unbounded then no almost transitive bounded group of automorphisms on the Hilbert space H can act boundedly on the scale, i.e. it cannot induce a bounded action on both X and X * .
⋆ Consider the group Isom(L p ) of isometries of L p (0, 1), p = 2.One has: • KP is compatible with the natural action of Isom(L p ) on L p .
• KP is not an Isom(L p )-centralizer.
• KP is a linear perturbation of an Isom(L p ) centralizer with trivial domain.

We show that, once again, [KP, T ] is linear
An alternative form of finding this compatible action is considering the analytic family of To prove the second part note that the group Isom(L p ) contains the units of L ∞ and acts, linearly, on L 0 thus, from Theorem 5.2 we get that KP is an Isom(L p )-centralizer if and only if Isom(L p ) acts boundedly on the scale of L p -spaces.Since Isom(L p ) acts almost transitively on L p , see [29], and KP has dense domain, we get from Proposition 8.1 that if if Isom(L p ) acts boundedly on the scale of L p -spaces then KP must be bounded on L p , something it is not.
We know from Lemma 3.3 that there is a linear map L such that KP + L is an Isom(L p )centralizer, precisely, KP + L : L p −→ L p ⊕ KP L p given by (KP + L)(y) = (KPy, y).This map is an Isom(L p )-centralizer (Proposition 10.3) with trivial domain: We can provide additional information about this strange phenomenon; to ease notation we will call G = Isom(L p ). Proof.If y ∈ ∆ ∩ Dom(L) then (KP + L)y ∈ L p .Since g(KP + L) − (KP + L)g is bounded, then (KP + L)z belongs to L p for all z in the G-orbit of y; and since KPz ∈ L p because ∆ is G-invariant, we deduce that Lz ∈ L p on the G-orbit of y.Let ∆ ′ = span(Gy) ⊂ DomKP ∩ DomL.By quasi-transitivity of L p , Gy is dense on the unit sphere and z → (Lz, z) is a G-linear lifting for the quotient map L p ⊕ KP L p on ∆ ′ .Since (Lz, z) = (Lv(g)y, v(g)y) = λ(g)y ≤ C y we actually have a linear bounded lifting on a dense subspace, and KP should be trivial, which is not.
⋆ The case of the group Isom disj (L 2 ) of isometries of L 2 preserving disjointness is analogous: KP is compatible with the action of Isom disj (L 2 ), it is not an Isom disj (L 2 )-centralizer but it is a linear perturbation of an Isom disj (L 2 )-centralizer.
⋆ An even more stunning situation appears when considering the unitary group Isom(L 2 ) Proposition 8.5.KP is not compatible with the natural action of Isom(L 2 ) on L 2 .
Proof.Our starting point is the fact proved in [16] that some complex structure on H does not extend to a complex structure on Z 2 .Everything consists in proving that such pathological complex structure may be chosen to be a unitary map.Let Ψ be a quasilinear map on ℓ 2 and let [x i ] be a finite sequence of n normalized vectors.Following [16] we set where the average is taken over all the signs ±1.Assume that Isom(L 2 ) is compatible with KP and let g → d(g) be the associated derivation.The linearity of d(g) implies that, where D(g) = [g, KP, g] + d(g) as in Lemma 3.
Let g be some unitary operator such that g(x i ) = y i , i = 1, . . ., n, we get a contradiction for large n.The result translates to any infinite dimensional L 2 through the fact that the restriction of KP to an ℓ 2 -subspace generated by disjoint characteristic functions of intervals coincides, up to a linear term, with the own KP map on ℓ 2 [11].

G-equivariant maps
The purpose of this section is showing that if G-centralizers are connected with interpolation scales of G-spaces, G-equivariant maps are connected with rigid interpolation scales.Let us give a precise meaning to that word: Definition 9.1.An interpolation pair (X 0 , X 1 ) will be called θ-rigid if whenever Y 0 , Y 1 ⊂ X 0 + X 1 defines another regular pair of interpolation such that X θ = Y θ isometrically and Ω X θ = Ω Y θ , it follows that X t = Y t isometrically, for all 0 < t < 1.The pair is said to be rigid, if it is θ-rigid for all 0 < θ < 1.
As it was announced, typical examples of rigid scales are provided by p-convexifications of r.i.Köthe spaces: Lemma 9.2.When X is an r.i.Köthe space the pair (X, L ∞ ) is rigid.
Proof.In the case of discrete spaces we apply the previous proposition to the open set some neighborhood of any y ∈ U, so the local Lipschitz property will be satisfied.The same idea applies to the case of r.i.spaces on [0, 1].
A rigid pair is such that X i = Y i isometrically, i = 0, 1, as soon as x i = lim t→i x t , i = 0, 1 for x ∈ X 0 ∩ X 1 , a condition satisfied for most examples (see [38]).It is an open question of [15] whether optimal pairs of interpolation are rigid, even in the special case in which Ω X θ is bounded.A positive answer was presented in [15] under the assumption Ω X θ = 0, or even when Ω X θ is linear (under technical restrictions).We present a few additional partial answers: Proposition 9.3.Assume (X 0 , X 1 ) is an optimal interpolation pair such that either (a) X 0 and X 1 have a common monotone basis (e n ).In this case we set E n = [e 1 , . . ., e n ]; or (b) X 0 and X 1 are r.i.Köthe spaces on [0, 1].In this case we let E n be the subspace generated by the characteristic functions of the intervals (k − 1)/2 n , k/2 n , k = 1, . . ., 2 n .Assume the restriction of Ω θ to S X θ ∩ E n is locally Lipschitz on a dense open subset for each n.Then the pair (X 0 , X 1 ) is rigid.
According to [15,Theorem 5.11], the optimal analytic function B θ (x) satisfies the differential equation Claim.The equation has a unique holomorphic solution with values in S X θ in each of the cases (a) and (b) for x in the corresponding dense open subset.
Proof of the Claim.Since Ω θ is locally Lipschitz, if F and G satisfy the differential equation for x in the dense open subset of S X θ ∩ E n , then for some K and t close enough to 0. So max 0≤s≤t F (s) − G(s) ≤ Kt max 0≤s≤t F (s) − G(s) and thus F (s) = G(s) on some small enough interval [0, t].By holomorphy F = G.

admits a version for rigid pairs:
Theorem 9.4.Let (X 0 , X 1 ) be a rigid interpolation pair, and let G be a group of isometries on X θ .Then the following are equivalent: (a) Ω θ defined on X θ is G-equivariant.
(b) G acts as an isometry group on the interior of the scale.
Proof.(b) ⇒ (a) is Proposition 4.3.The prof of (a) ⇒ (b) goes as that of Theorem 5.2 until getting Ω ′ θ (x) = Ω θ (x), where the rigidity hypothesis applies to conclude that gx t = x g t = x t for 0 < t < 1 and all g ∈ G.
As an example the Kalton-Peck map KP when defined on a p-convex Köthe space is Uequivariant even if it is not equivariant in the associated L ∞ -structure.In general, equivariant quasi-linear maps with respect the the module structure seem to be only possible in trivial cases.It is different for linear maps: an U-linear map L : Y → X on spaces having unconditional bases is obviously diagonal since ge n = ±e n are the only options; if the bases are symmetric and G is the group of operators acting by change of signs and permutations of the vectors of a symmetric basis, G-linear maps are homotheties.However, combining Corollary 3.10 and Proposition 4.2 one gets: Corollary 9.5.Let (X 0 , X 1 ) be an interpolation pair.Assume X θ is reflexive and that G is an amenable group acting on the scale.Then (a) Ω θ is boundedly equivalent to a G-equivariant map.(b) If Ω θ is trivial then it is boundedly equivalent to a G-linear map.

The category of G-Banach spaces and its exact sequences
We shift now our point of view from "compatibility of group actions on twisted sums" to "equivalence of exact sequences of G-spaces".We thus introduce the category GBan of G-Banach spaces whose objects are Banach G-spaces, and whose arrows are G-operators.An exact sequence in GBan is an exact sequence formed by G-Banach spaces and G-operators.
An exact sequence of G-Banach spaces can be described by a pair (Ω, d), where Ω : Y X is quasi-linear and d is an associated derivation that determines the bounded action λ(g) = u(g) d(g) 0 v(g) on the twisted sum space X ⊕ Ω Y .Let us transplant Lemma 3.14 to this language: The following are objects of GBan: ) when L is linear.
• (B, T ) when B and T are bounded.
In order to consider maps Ω defined on a fixed dense G-subspace ∆ ⊂ Y (in particular, ∆ must be G-invariant), the role of this ∆ must be examined since the exact sequence of G-spaces does not depend on ∆ while the representation (Ω, d) does.We can assume that all the maps involved have a common ambient space Σ by the observations we made in 'The ambient issue' section.Observe the following definitions: Equivalence of sequences: The sequences generated by Ω 1 and Ω 2 are said to be Gequivalent if there is a G-operator T making the following diagram commute Let us check that the two definitions are equivalent.First observe commute and is a G-operator since Finally, τ can be extended to a G-operator T : by density: set (ω, y) = lim(ω n , δ n ) and define T (ω, y) = lim τ (ω n , δ n ).Since both actions are continuous, The other implication is immediate: the existence of T already implies equivalence of the exact sequences in the category of Banach spaces, so that Ω 1 − Ω 2 is boundedly equivalent to L. Thus, there is a vector space structure on the set of pairs (Ω, d) (defined on the same ∆) given by (Ω Proof.Let 0 → X → Z → Y → 0 be an exact sequence of G-spaces.Set Σ = Z as the ambient space equipped with λ(g) as the extension of u(g).Any homogeneous bounded selection B for the quotient map: B : Y → Z is a G-centralizer generating the same sequence; in particular, writing Z = X ⊕ Ω Y and Ω ′ y = (Ωy, y), its associated bounded action has derivation 0 since the associated diagonal maps λ ′ (g) are uniformly bounded on X ⊕ Ω ′ Y : λ ′ (g)((x, 0), y) Ω ′ = ((u(g)x, 0), v(g)y) Ω ′ = ((u(g)x, 0) − (Ωv(g)y, v(g)y), v(g)y) We perform the standard pushout from Lemma 3.3 .
It only remains to see that T is a G-operator, i.e.We now give two easy lemmas that will help us simplify some proofs later on.
X is any linear map for which Ω − L : ∆ → X is bounded, then d(g) + [u(g), L, v(g)] is a uniformly bounded family of operators.
where P : X * * → X is a G-projection.Let us show that the map R = Id M 0 Id is a G-operator commute.The only part that is not evident, that R is a G-operator means We show this: Call g ′ g −1 = h −1 so that g = hg ′ and thus = P h∈G u(h −1 )d(hg ′ )y dµ(h) Corollary 10.8.Let G be a group.Let 0 Both hypothesis are necessary: (a) The amenability of G is necessary since Ext Aut(T) (ℓ 2 (T), ℓ 2 (T) = 0: indeed, the sequence 0 → ℓ 2 (T) → ℓ 2 (T) → ℓ 2 (T) → 0 does not Aut(T)-splits since it has the form (L, 0) and, otherwise, (0, −[u, L]) would be trivial and then [u, L] would be an inner derivation, a contradiction with the fact that the corresponding representation of F ∞ from [42] is non-unitarizable.(b) The G-ultrasummand character of X is necessary since we will show in Section 11.4 that Ext 2 <ω (R, c 0 ) = 0.
Theorem 10.7 seen together with the result of Cabello mentioned before its statement may seem surprising, since the group U of units of the L ∞ -module structure is abelian and (for 1 < p < ∞) L p spaces are reflexive.Let us spell what these two results together actuallly imply: no non-trivial element of Ext(L p , L q ) can be compatible with the canonical actions of U on these two spaces.
A significant consequence of Theorem 10.7 is a kind of uniqueness result for the possible derivation associated to fixed actions.It will help us at this point to use the classical terminology, which calls inner a derivation for which there exists a bounded linear map A : Y → X such that d = [u, A, v].Thus, (Ω, d 1 ) ≃ (Ω, d 2 ) if and only if d 1 − d 2 is inner.In the particular case of a direct sum of two copies of a Hilbert space 0 → H 1 → H 1 ⊕ H 2 → H 2 → 0, we recover in this way the classical fact that the representation λ is similar to the diagonal unitary representation given by u, v if and only if d is inner.Proposition 10.6 may be seen as a generalization of this fact for triangular representations on twisted sums: Corollary 10.9.Assume that G is an amenable group, Y, X are G-spaces with X a Gultrasummand and Ω : Y X is a quasilinear map.All compatible actions of G on X ⊕ Ω Y are conjugate; namely, given two actions λ 1 , λ 2 there is A ∈ L(Y, X) such that for all g ∈ G,

Variations and comments
This final section contains a miscellanea of results and problems connected with the ideas in this paper.
11.1.From uniformly bounded extensions to actions.The following situation was mentioned in the abstract: to which extent the existence of a uniformly bounded family of operators on a twisted sum space compatible with a couple of actions on the subspace and the quotient space induces an action on the twisted sum.We have: Proposition 11.1.Let Ω : Y X be quasi-linear between two G-spaces.Assume that there is a uniformly bounded family of operators (T g ) g∈G such that each T If G is an amenable group and X is a G-ultrasummand then there is a compatible action of G on X ⊕ Ω Y .
Proof.Each operator T g has the form T g = u(g) τ g 0 v(g) .We may assume wlog that τ e = 0 and from that T g −1 = u(g) τ g 0 v(g) .
It is easy to check that the family {[u(g), Ω, v(g)] + τ g } g∈G is uniformly bounded, which implies that the family {u(g)τ h + τ g v(h) − τ gh } h∈G is uniformly bounded, and thus after we check that also {u(gh } h∈G s uniformly bounded: We can therefore define where P is a G-operator X * * → X, and compute that d is a derivation: given It seems to be open whether the amenability and G-ultrasummand hypotheses are necessary.

Complex structures
. We now answer now a question complex structures (i.e.operators of square −Id) on real twisted sum spaces posed in [16].
Proposition 11.2.Let X, Y be Banach spaces admitting complex structures u, v and let Ω : Y X be a quasilinear map.If there exists a bounded operator Proof.We use the commutative, hence amenable, group G = {i, −1, −i, 1} for which no G-complementation is required since one performs just a finite average, providing that if This proof shows that complex structures exist in X ⊕ Ω Y as long as [u, Ω, v] is the sum of a bounded plus a linear map.The result had been proved in [16,Corollary 2.2] assuming that [u, Ω, v] was either bounded or linear.11.3.Why singular centralizers on L p do not exist.Singular quasilinear maps (those whose restriction to infinite dimensional subspaces is never trivial) are somewhat mysterious objects with many potential applications, whose paramount example is the Kalton-Peck map on ℓ p spaces (but not the Kalton-Peck map on L p spaces).The key result [5] is that no singular L ∞ -centralizer exists on L p , a result generalized in [17] to superreflexive Köthe space over a non-atomic base and the proof consists in showing that there is a copy of ℓ 2 contained in the domain of the centralizer.In the case of L p (0, 1), it is not hard to see any Rademacher function is in the domain of KP and then use almost-transitivity plus Proposition 8.1 to get that KP is not singular on L p .11.4.Actions of the Cantor group 2 ω and of 2 <ω .The Cantor group is the group of units of ℓ ∞ and thus 2 ω -centralizers are just ℓ ∞ -centralizers.Its diagonal action on ℓ ∞ restricted to c 0 is again the diagonal action, and thus it generates an action on ℓ ∞ /c 0 .We do not have any reasonable idea about a linear derivation d : ℓ ∞ /c 0 → c 0 of the Cantor group.The subgroup 2 <ω of elements of 2 ω that are eventually 1 is much more manageable.The space c is the living example that 2 <ω -groups are not 2 ω -groups.The natural diagonal action of 2 <ω on c and c 0 , that is therefore a 2 <ω -subspace, induces the identity action on the quotient R.This implies that the exact sequence 0 → c 0 → c → R → 0 of 2 <ω -spaces, which splits as a Banach space sequence, does not split as a 2 <ω -sequence since no 2 <ω -lifting R → c is possible, and thus Ext 2 <ω (R, c 0 ) = 0, which shows that G-complementation is necessary in Theorem 10.7.Since the triangular action on c has the form λ(g) = u(g) d(g) 0 Id R with u the diagonal action and d(g) : R → c 0 is d(g)(r) = r g i =−1 e i , this d(g) is a linear derivation on 2 <ω .We do not know if all actions of 2 <ω on c have the form u(g) xd(g) 0 Id R for x ∈ c since Corollary 10.9 does not apply here.Thus, all elements of L(R, c) = c define 2 <ω -centralizers and solving the equation [u(g), L, g]r = d(g)r yields that all L(r) = − r 2 x, for x ∈ c, are equivariant 2 <ω -centralizers.There is a general formulation for this situation: let X be a separable Banach space that we write as X = n F n for an increasing sequence of finite dimensional spaces F n .The space c 0 (F n ) admits a natural "diagonal" action g(f n ) = (g(n)f n ) that naturally extends to the space c X (F n ) = {(f n ) : ∃ lim f n }.What is interesting here is that the exact sequence 0 → c 0 (F n ) → c X (F n ) lim → X → 0 splits if and only if X has the Bounded Approximation Property although it never 2 <ω -splits since the action induced on X is the identity.All this was based on some ideas of [1], where an example of an SOT-discrete bounded group of operators on c 0 without discrete orbits was provided; the relation with twisted sums was not observed there.
The difficulty of obtaining derivations ℓ ∞ /c 0 → c 0 for 2 ω can be confronted with how easily one obtains derivations for 2 <ω on the subspace (here c is the cardinal of the continuum) c 0 (c) of ℓ ∞ /c 0 .Consider to this end the Nakamura-Kakutani (see [8]) sequences 0 −→ c 0 −→ C(∆ A ) −→ c 0 (|A|) −→ 0 also provide natural examples of 2 <ω -centraizers: pick A an almost disjoint family of subsets of N (i.e., |A ∩ B| < ∞ for all A, B ∈ A) containing the singletons.The cardinal of the family must be ℵ 1 ≤ |A| ≤ c since when |A| = ℵ 0 the sequence splits by Sobczyk's theorem.We will assume without loss of generality that it is the continuum.Let ∆ A be the one-point compactification of the locally compact space having N as isolated points and A ∈ A as the only accumulation point of {n : n ∈ A}.There is a natural action of 2 <ω on C(∆ A ): (gf )(n) = g(n)f (n) that yields the diagonal action on c 0 and induces the identity action on c 0 (c).Let c 00 (c) be the dense subspace of all finitely supported sequences.A quasilinear map Ω : c 00 (c) c 0 corresponding to this sequence can be easily described: fix a well-order on c and then for x ∈ c 00 (c) write it as x = λ i e i with the e i well ordered and define Ω( λ i e i ) = . This is a bounded map c 00 (c) → ℓ ∞ and therefore a 2 <ω -centralizer (with derivation 0).On the other hand, C(∆ A ) is a subspace of ℓ ∞ but the natural action of 2 ω does not respect C(∆ A ). 11.5.Groups and symmetries.To fix ideas, let us focus on N and ℓ ∞ -centralizers on Banach spaces with symmetric basis.A centralizer is symmetric if (Ωx)σ − Ω(xσ) ≤ C x for every permutation σ of N. For instance KP maps are symmetric.Symmetric centralizers live their own lives (see [34,4]) and there is a great difference between working with symmetric and non-symmetric centralizers.But the ideas in this paper allow us, once the action of a group G on a space with unconditional basis has been established, to also consider the action of the group G Θ = G × Θ, where Θ is a certain group of permutations of N in the form (g, σ)x = g(xσ).When Θ is the whole group of permutations we will call the group G sym .Thus, symmetric centralizers are 2 ω sym -centralizers.But other groups of permutations are also important: let (A n ) be a partition N = ∪A n of N into finite sets, A n < A n+1 , and let Θ be the group of permutations σ of N such that σA n = A n for all n.It turns out that G Θ -symmetric centralizers are sometimes useful: as the authors of [12] dismayingly recall, the first author has frequently asked about "how many" "different" exact sequences 0 → ℓ 1 → Z → c 0 → 0 exist.The same problem is addressed in [8].Let us lodge the problem in the theory developed in this paper.) is not, and can never be, a 2 ωcentralizer: otherwise there would be a compatible action of 2 ω on X and picking any extension T : X → R of the sum functional ℓ 1 → R we can form the 2 ω -invariant functional Λ(x) = 2 ω ε −1 T (εx)dµ.The road is now paved to define Q : X → ℓ * * 1 in the form Q(x)(ε) = Λ(εx) for ε an unit of ℓ ∞ and extend it linearly to a functional on ℓ ∞ .Finally, compose with a 2 ω -projection ℓ * * 1 → ℓ 1 .It is however perfectly reasonable to have a G-centralizer Ω and two operators α, γ so that αΩγ is a G ′ -centralizer for two different groups G, G ′ .Researchers willing to travel this road are advised to do so crossing through the horn gate of [12].11.6.Additional structures.Other additional structures than group structures may be considered on Banach spaces.See for example the work of Corrêa [19] on exact sequences of operator spaces and a solution to 3-space problem for OH spaces.It seems to be open whether a relevant theory of groups acting completely boundedly on extension sequences of operator spaces may be developed.
Proof.(a) and (b) are clear.(c) is a simple consequence of the fact that d + d ′ is linear and [

Proposition 8 . 1 .
Assume Ω : Y X is a G-centralizer and that G acts almost transitively on Y .If DomΩ = 0 then Ω is bounded.
to get ξΩ ′ = σΩ + L. Just before that lemma it was observed that T :=