Twisted sums of Banach spaces generated by complex interpolation

Twisted sums of Banach spaces generated by complex interpolation

Manuel González

Departamento de Matemáticas Universidad de Cantabria

Santander, Spain

First Brazilian Workshop in Geometry of Banach Spaces Maresias, São Paulo, 25-29 August 2014

Joint work with Jesús M. F. Castillo and Valentin Ferenczi.

Exact sequences of Banach spaces

LetY andZ be Banach spaces. Anexact sequenceis a sequence 0 −−−−→ Y −−−−→^j X −−−−→^q Z −−−−→ 0,

j,q continuous operators,Kerj ={0},Ranj =Kerq, andRanq =Z. j(Y)is a closed subspace ofX andX/j(Y)≡Z.

X is aF-space; sometimes not (equivalent to) a Banach space.

The exact sequence istrivialifRanj is complemented: X ≡Y ×Z. Atwisted sum ofY andZ is a non-trivial exact sequence.

(i.e., the spaceX in a twisted sum ofY andZ).

A twisted sum ofY andZ is calledsingularifq is strictly singular.

(q|_M is an isomorphism for no inf. dim. subspaceMofX).

Construction of twisted sums

LetY andZ be (infinite dimensional) Banach spaces.

Aquasi-linear map fromZ toY is a mapF :Z →Y₀, Y₀⊃Y, with F(λu) =λF(u), F(u+v)−F(u)−F(v)∈Y ∀λ∈K;u,v ∈Z, andkF(u+v)−F(u)−F(v)k_Y ≤Mku+vk_Z for someM>0.

A quasi-linear mapF :Z →Y induces an exact sequence 0→Y →^j Y ⊕_F Z →^q Z →0,

where Y ⊕_F Z :={(y,z)∈Y₀×Z :y−F(z)∈Y} endowed with the quasi-norm k(y,z)k_F =ky−F(z)k_Y +kzk_Z.

The embedding isj(y) = (y,0)while the quotient map isq(y,z) =z. Each exact sequence can be obtained by means of a quasi-linear map.

F is calledsingularifq is strictly singular (i.e.,Y ⊕_F Z singular).

We can identify exact sequences and quasi-linear maps.

An example: the Kalton-Peck map K (·)

K:x = (x_n)∈`₂→

−x_nlog_kxk^|xn|

2

∈S forx 6=0, andK(0) =0.

is a quasi-linear map from`<sub>2</sub>to`₂, withSthe space of all sequences.

0→`<sub>2</sub>→<sup>j</sup> Z<sub>2</sub>:=`₂⊕_K`<sub>2</sub>→<sup>q</sup> `₂→0.

Z₂:=

(yn),(xn))∈S×`₂:P∞

n=1|x_n|²+

yn+xnlog_kxk^|xn|

2

2

<∞

. Z₂is a singular twisted sum of`₂with itself: qis strictly singular.

Z₂contains no complemented copies of`2.

Complex interpolation (two spaces)

(X₀,X₁)compatible pair of complex Banach spaces.

S:={λ∈C:0<Reλ <1}unit strip.

H(S): Bounded continuous functionsg:S→X₀+X₁ withg analytic onS,g(0+it)∈X₀andg(1+it)∈X₁; endowed with the supremum normk · k_∞.

Fix 0< θ <1. Forn=0,1,2, . . .,

δ_θ⁽ⁿ⁾:g ∈ H(S)→g⁽ⁿ⁾(θ)∈X₀+X₁; defines a bounded operator.

We write δθ =δ_θ⁽⁰⁾ andδ⁰_θ=δ⁽¹⁾_θ .

Complex interpolation spaces: X_θ:={g(θ) :g∈ H(S)} ≡ _ker^H(S_δ⁾

θ. kxk_θ:=inf{kgk_∞:g ∈ H(S),g(θ) =x}

Complex interpolation (family of spaces)

{X_(j,t):j =0,1;t∈R}compatible family of complex Banach spaces Σ(X_j,t)denote the algebraic sum of these spaces.

H(X_j,t): Bounded continuous functionsg:S→Σ(X_j,t), analytic onS, and satisfyingg(it)∈X_(0,t)andg(it+1)∈X_(1,t)fort∈R, endowed withkgk∞=sup{kg(j+it)k_(j,t) :j=0,1;t ∈R}.

Fixθ∈S. Forn=0,1,2, . . .,

δ⁽ⁿ⁾_θ :g ∈ H(X_j,t)→g⁽ⁿ⁾(θ)∈Σ(X_j,t) are bounded operators.

Complex interpolation spaces: X_θ:={g(θ) :g∈ H(X_j,t)} ≡ ^H(X_kerδ^j,t)

θ .

Complex interpolation and quasi-linear maps

We consider the quotient mapδθ :g∈ H(S)→g(θ)∈X_θ. We fix a homogeneous bounded selectionB_θ :X_θ→ H(S)ofδ_θ.

It satisfiesδ_θ◦B_θ =I_Xθ.

Then Ωθ:=δ⁰_θ◦B_θ:x ∈X_θ →B_θ(x)⁰(θ)∈X₀+X₁ defines a quasi-linear map fromX_θ toX_θand

X_θ⊕_Ωθ X_θ =n

g⁰(θ),g(θ)

:g ∈ H(S)o

≡ H(S)/ Kerδ_θ∩Kerδ_θ⁰ . QUESTIONS. Given the exact sequence

0→X_θ →^jθ X_θ⊕_Ωθ X_θ→^qθ X_θ →0, when isX_θ⊕_ΩθX_θ a twisted sum?

when isq_θ strictly singular?

which spaces appear as`<sub>2</sub>⊕<sub>Ωθ</sub> `₂(twisted Hilbert spaces)?

Singularity criterion for a pair with unconditional basis

For two functionsf,g:N→Rwe writef ∼g

if 0<lim inff(n)/g(n)≤lim supf(n)/g(n)<+∞.

A_X(n) :=sup{kx₁+. . .+xnk : kx_ik ≤1, n<x₁< . . . <xn}.

Proposition

Let(X₀,X₁)be a pair of spaces with a common1-unconditional basis and A_X0 6∼A_X1, and let0< θ <1.

Suppose A^1−θ_X

0 A^θ_X

1 ∼A_Xθ ∼A_Y for all subspaces Y ⊂X_θ. ThenΩ_θ is singular.

EXAMPLE:X_j reflexive, asymptotically`_pj,p₀6=p₁, with uncond. basis.

NOTE:(X,X^∗)_1/2≡`₂whenX is reflexive with uncond. basis.

In this way we get a family`₂⊕_Ωi

θ `₂(i ∈R) of pairwise non-isomorphic twisted Hilbert spaces.

Singularity criterion for a pair of Köthe function spaces

M_X(n) :=sup{kx₁+. . .+xnk : kx_ik ≤1, (x_i)disjoint inX}.

Ω_θ disjointly singular: the restriction ofΩ_θto a subspace ofX_θ generated by a disjoint sequence is never trivial.

Proposition

Let(X₀,X₁)be an admissible pair of Köthe function spaces with M_X0 6∼M_X1, and0< θ <1. Suppose M_X^1−θ

0 M_X^θ

1 ∼M_Xθ ∼M_Y for each Y ⊂X_θ generated by a disjoint sequence, and X_θ reflexive.

ThenΩ_θ is disjointly singular; hence non-trivial.

Corollary

Let X be a reflexive, p-convex Köthe function space with p >1.

Assume M_X ∼M_[xn]for every disjoint sequence(x_n)⊂X .

Then the Kalton-Peck mapK(f) =flog_kf^|f|_k is disjointly singular on X .

Singularity criterion for a family of spaces with a basis

Proposition (MON)

Let{X_(j,t) :j=0,1;t ∈R}be an admissible family of spaces with a common1-monotone basis.

Let1≤p₀6=p₁≤+∞, ¹_p = ^1−θ_p

0 +_p^θ

1, and0< θ <1.

Assume the spaces X_j,t satisfy an asymptotic upper`pj-estimate with uniform constant, and for every block-subspace W of X_θ, there exist a constant C and for each n, a C-unconditional finite block-sequence n<y₁< . . . <y_nin B_W such thatky₁+· · ·+y_nk ≥C⁻¹n^1/pand [y₁,· · ·,yn]is C-complemented in X_θ.

ThenΩ_θ is singular.

Twisting Ferenczi’s space F

For eacht ∈R, take X_(1,t) =`qwith 1<q<∞, and X_(0,t)a GM-like space (varies witht) with 1-monotone basis.

Fixθ∈S. Then

F₁={x ∈Σ(X_j,t) :x =g(θ)for someg ∈ H(X_j,t)} ≡ H(X_j,t)/kerδ_θ. is a uniformly convex H.I. Banach space (Ferenczi 1997).

Theorem

The spaces in the construction ofF₁satisfy the conditions of Proposition (MON). SoΩ_θ gives a singular twisted sum

0 −−−−→ F₁ −−−−→ F₂:=F₁⊕_Ωθ F₁ −−−−→ F^π2,1 ₁ −−−−→ 0.

Corollary

Sinceπ_2,1is strictly singular, the spaceF₂is H.I.

Iterated twisting of F

Recall thatF₂={ g⁰(θ),g(θ)

: g∈ H(X_j,t)}.

Giveng ∈ H(X_j,t)andk ∈N, we denoteg[kˆ ] :=g^(k−1)(θ)/(k−1)!.

Forn≥3 we define:

F_n :={ g[n], . . . ,ˆ g[2],ˆ g[1]ˆ

: g ∈ H(X_j,t)} ≡ H(X_j,t)/Tn−1

k=0kerδ_θ^(k). Proposition

Let m,n∈Nwith m>n.

1 π_m,n: (x_m, . . . ,x_n, . . . ,x₁)∈ F_m →(x_n, . . . ,x₁)∈ F_nis surjective.

2 in,m(xn, . . . ,x₁)∈ F_n→(xn, . . . ,x₁,0, . . . ,0)∈ F_m is an isomorphic embedding withRan(i_n,m) =Ker(πm,m−n).

3 The operatorπ_m,nis strictly singular.

Iterated twisting of F

(II)

Corollary

For m,n∈Nwith m>n, the sequence

0 −−−−→ F_n −−−−→ F^in,m _m −−−−→ F^πm,m−n _m−n −−−−→ 0 is exact and singular.

Hence all the spacesF_mare H.I.

Proposition

Let l,m,n∈Nwith l >n. Then the diagonal push-out sequence

0 −−−−→ F_l −−−−→ Fⁱ _n⊕ F_l+m −−−−→ F^π _m+n −−−−→ 0, where i(x) = (−π_l,nx,i_l,l+mx) and π(y,z) =in,m+ny+π_l+m,m+nz, is a twisted sum (nontrivial exact sequence) which is not H.I.

Thank you

for your attention.