DIFFERENTIALS OF COMPLEX INTERPOLATION PROCESSES (The research of geometric structures in quantum information based on Operator Theory and related topics)

DIFFERENTIALS OF COMPLEX INTERPOLATION PROCESSES

JESÚSM. F. CASTILLO

This paper contains an

expanded

version of _mytalk at the RIMS Conference

“The research of

_geometric

structures in

_Quantum

information based on

Operator

Theory

andrelated

_topics”.

I would liketothank the

_organizers

for

_giving

methe

opportunity

to

_participate

atthe conference and

_especially

toProf. MuneoCho for

making

it

_possible.

1. THECLASSICALTHEORY REVISITED

Let me

explain

first what adifferential_process

is,

how

they

_appearand what do

they

mean. Assumeonehasa _space3of Banach_spacevalued functions definedon

some

“parameter”

_{space F}; for whichwe can assumethat is a Banach_space itself when endowed withanaturalnorm

\Vert\cdot\Vert_{S} (say,

the_supremumonF

).

Evaluationat

a

_point

$\theta$\in F

produces

aBanach_space

X_{ $\theta$}=\{f( $\theta$) : f\in \mathfrak{F}\}

,whichisthe

quotient

spaceof3endowedwith the natural

quotient

norm

\displaystyle \Vert x\Vert_{ $\theta$}=\inf\{\Vert f\Vert : f( $\theta$)=x\}

. If

$\delta$_{ $\theta$}

denotes the evaluationmap then

X_{ $\theta$}=$\delta$_{ $\theta$}(\mathfrak{F})

. Since$\delta$_{ $\theta$} is anopen

mapping,

in

fact

_{$\delta$_{ $\theta$}(B_{\mathfrak{F}})}

_{=B_{X_{ $\theta$}}}

,

given

x \in X_{ $\theta$}, an $\epsilon$‐extremal

f_{x}

is anelement

f_{x}

\in S sothat

f_{x}( $\theta$)=x

and such that

_{\Vert f_{x}\Vert \leq(1+ $\epsilon$)\Vert x\Vert_{ $\theta$}.}

A differential process is a

correspondence

x \rightarrow

_{f_{x}'( $\theta$)}

. Of course that one has

assumed that function of3aredifferentiable. Much trickieristhe

point

of where

f_{x}'( $\theta$)

lies.

To

_{clarify this,}

let usconsider the

perfect

situationwhere a differential_process

appears:

complex interpolation.

Recall that an

interpolation

methodmeanstohave

a_{spaceparameter}Fsothat

given

two

_(in

the

_simplest

_version)

Banachspaces

A,

B

one can

_assign

toeach $\theta$\in F aBanach _space

X_{ $\theta$}

that isintermediate between A

and B inthesensethat whenan_operatorsendsA\rightarrow Aand B\rightarrow B

automatically

sends X_{ $\theta$} \rightarrow X_{ $\theta$}. Thereare many

interpolation

methods: real methods

(several),

minimal

_methods,

orbit

_method,

methods

_specific

for Köthe spaces We focus

now onthe

complex

method.

We describeit in the

_following

way: Assume that both

A,

B canbe embedded

into a

_larger

Banach_spaceand consideras

3

theso called Calderonspaceof holo‐

morphic

A+B valued functions defined onthe unit’

complex

_strip S=

\{z

: 0 \leq Rez\leq

1\}

such that

_f(it)

\in A and

_f(1+it)

\in B. A few technical

properties

are

usually

added.

Then,

for each inner _point $\theta$ \in S, thespace

X_{ $\theta$}

=$\delta$_{ $\theta$}(S)

is an

interpolation

spacebetween A and B.

Let us _present a

simple

and natural

example:

Pick

A=P_{1}

and

B=\ell_{\infty}

. Inter‐

polation

at

_z=1/2

_produces

thespace

p_{2} by

the Riesz‐Thorin theorem:

i.e.,

each

holomorphic

function that takes valuesin

_{\ell_{1}}

at0andtakes valuesin

_{\ell_{\infty}}

at1 takes values in _{P_{2}} at

_1/2.

What is the differential _{process here?}

_{Well, given}

x

\in\ell_{2}

an

extremal canbe

_given

by

f_{x}(z)

=x^{2(1-z)}

since—we

simplify

abit

just

picking

0

and 1 to _represeńt

_points

it and 1+it-

_{f_{x}(0)}

=x^{2} \in

_{P_{1}}

and

_{f_{x}(1)}

= 1 _\in

\ell_{\infty}.

Hence differentiationon thepositive cone

produces

the_process:

The research has been supported in _{part by project} MTM2013‐45643‐C2‐1‐P and _Ayudaa

x\rightarrow-2x\log|x|.

This has to be understood as the _map that associated to the _sequence x the

sequence

(-2x(n)\log|x(n)|)_{n}

What is the

meaning

ofthis

map?

More

precisely,

Whatisthe

_meaning

of this

_{non‐linear, non‐continuous,}

andofcoursenotevenwell defined from_{l_{2}} intoP_{2}

map?

Well, given

a Banach _{space X},

non‐linear,

non‐continuous maps defined on X

but

_taking

values \mathrm{i}\mathrm{n}\backslash some

bigger

_space

(from

now on

simply

referred to as “the

cloud”)

have their

_place

in Banach _space

_{theory. Indeed,}

let $\Omega$ : X \rightarrow ₀ be one of those _maps. We will ask it a

couple

of conditions: to be

_homogeneous

( $\Omega$( $\lambda$ x) = $\lambda \Omega$(x))

and to

_verify

that for _{any two}

_points

_{x, y} \in X the

_Cauchy

difference

_{$\Omega$(x+y)- $\Omega$(x)- $\Omega$(y)\in X}

. Inother

words,

themap $\Omega$cantake values

whereveritwants, but the

_Cauchy

differences fallin X. With suchamapone can

form the derived _space

d_{ $\Omega$}X=\{(w, x) \in \mathrm{O}\times X : x\in X;w- $\Omega$ x\in X\}

endowed with the “norm”

_{\Vert(w, x)\Vert_{ $\Omega$}}

=

\Vert w- $\Omega$ x\Vert+\Vert x\Vert

.

Actually,

\Vert. \Vert_{ $\Omega$}

isnot

a_norm,

only

a

_quasi‐norm

since it fails the

_{triangle inequality.}

Butin most cases

it is

_equivalent

to a norm—the

one

having

as unit ball the closed convexhull of the unit ball of

_\Vert

.

\Vert_{ $\Omega$}-\mathrm{s}\mathrm{o}

we can

simplify

and

just

call it “a norm”’ The _space d_{ $\Omega$}X contains the

_subspace

_{\{(x, 0) x\in X\}}

isometric to Xand the

_quotient

space

d_{ $\Omega$}X/\{(x, 0) : x \in X\}

is

_again

isometric to X. This situation is well known in

homological algebra:

An exact _{sequence 0} \rightarrow \mathrm{Y} \rightarrow X\rightarrow Z\rightarrow 0, where

Y,

Z are

Banach_spaces and thearrows are

(bounded)

_operators is a

diagram

inwhich the kernel of eacharrow coincides with the

_image

of the

_preceding

one.

By

the _open

mapping

theorem this means that \mathrm{y} is

(isomorphic to)

a

_subspace

ofX in such a _waythat the

_{corresponding quotient}

X/Y

is

_isomorphic

to Z. The spaceX is

usually

calledatwistedsumofY and X.

Thus,

there is an_{exact sequence}of Banach_spaces

0 \rightarrow X \rightarrow d_{ $\Omega$}X \rightarrow X \rightarrow 0

and the _{space d_{ $\Omega$}X} is a twisted sum of X. Twisted sums have been studied

long

since. The

_{starting problem}

is

_probably

that of

_deciding

if a twisted sum

ofHilbert spaces must itself be a Hilbert _space. The first

_negative

answer came

from

_Enflo,

Lindenstrauss and Pisier

_[13]

_using

anad‐hoc

construction;

but it was Kalton

_[14]

who showed that all twisted sums of two

_{quasi‐Banach}

_spaces

_Y,

X

are

actually

induced

_by

a

quasi‐linear

_{map $\Omega$} : X \rightarrow Y. Recall that a _{map $\Omega$} : X\rightarrow Y iscalled

_{quasi‐linear}

ifit is

_homogeneous

and thereis aconstant Msuch

that

_{\Vert $\Omega$(u+v) - $\Omega$(u)}

_{- $\Omega$(v)\Vert}

_\leq

_{M\Vert u+v\Vert}

for all u,v \in X. In

fact,

there is

a

correspondence

(see

[11,

Theorem 1.5.\mathrm{c}, Section

1.6])

between exact sequences

0 \rightarrow

Y\rightarrow $\theta$\rightarrow X\rightarrow 0

of Banach spacesand a

special

kind of

_{quasi‐linear}

_maps $\omega$ : X \rightarrow \mathrm{Y}, called z‐linear maps, which are those

quasi‐linear

maps

satisfying

\displaystyle \Vert $\omega$(\sum_{i=1}^{n}u_{i})-\sum_{i=1}^{n} $\omega$(u_{i})\Vert

\leq

_{M\displaystyle \sum_{i=1}^{n}\Vert u_{i}\Vert}

for all finite sets u_{1},..._,u_{n} \in X. \mathrm{A}

quasi‐linear

map $\Omega$ : X\rightarrow Y induces theexactsequence

0\rightarrow Y\rightarrow jY\oplus_{ $\Omega$}X\rightarrow^{p}X\rightarrow

0 inwhichY\oplus_{F}X denotes thevector space Y\times Xendowed with the

_quasi‐norm

\Vert(y, x)\Vert_{ $\Omega$}

=

\Vert y- $\Omega$(x)\Vert+\Vert x\Vert

. The

embedding

is

j(y)

=

(y, 0)

while the

quotient

map is

p(y, x)

=x. When $\Omega$ is z

‐linear,

this

quasi‐norm

is

equivalent

to a norm

[11,

Chapter

1].

On the other

_hand,

theprocess toobtaina z‐linear_{map out} from anexact _{sequence 0\rightarrow}

\mathrm{Y}\rightarrow i

_$\theta$

\rightarrow^{\mathrm{q}}X

\rightarrow 0 ofBanach spaces is the

following:

get a

homogeneous

bounded selection

b:X\rightarrow $\theta$

for the

_quotient

_{map q,} and then a

linear

_{P:X\rightarrow $\phi$}

selection for the

_quotient

_map. Then

_{$\omega$=i^{-1}(b-P)}

isa z‐linear

map X\rightarrow \mathrm{Y}. Thecommutative

diagram

obtained

_{by taking}

as T:

$\theta$\rightarrow \mathrm{Y}\oplus_{ $\Omega$}X

the _operator

T(x)

=

(x-\ell qx, qx)

shows

that the upper and lower exact sequencesare

_equivalent.

Two

_{quasi‐linear}

maps

F,

G : X\rightarrow Y aresaid tobe

_equivalent,

denoted F\equiv G, ifthe difference F-G

canbewrittenas B+L,whereB : X\rightarrow Y isa

homogeneous

bounded map

(not

necessarily

linear)

and L:X\rightarrow Y isa linear_map

(not

necessarily

bounded).

In

_[18]

Kalton and Peck show that in the

_particular

case of

\ell_{p}

_spaces

(more

generally,

Banach _spaces with an unconditional

basis)

it is

possible

to

_give

an

explicit quasi‐linear

map

$\Omega$_{p}

:

p_{p}\rightarrow P_{\mathrm{p}}

by

meansof

$\Omega$_{p}(x)=x\displaystyle \log\frac{|x|}{||x\Vert_{p}}

whenxisa

finitely supported

_sequence, and with the

_{understanding}

that

_{\log 0=0.}

And thismeans that on theunit

sphere

\Vert x\Vert_{2}

= 1the Kalton‐Peck

map$\Omega$_{2} is, up to a -2

_factor,

the differential process

corresponding

to the natural

_{interpolation}

scale

_{(l_{1}, \ell_{2})}

_by

the

_complex

method.

Tohave a better

understanding

of this connection between

_{interpolation}

scales and differential processes it will

help

us to

_display

a few elements of

_homological

algebra

inBanach_space

_theory.

Twoexact sequences0\rightarrow \mathrm{Y}\rightarrow X_{1}\rightarrow Z\rightarrow 0 and 0\rightarrow Y\rightarrow X_{2}\rightarrow Z\rightarrow 0 are

equivalent

if there existsan_{operator T:X_{1}\rightarrow X_{2}} such that the

_{following diagram}

commutes:

The classical 3‐lemma

_{(see [11,}

_p.

_3])

shows that Tmust be an

isomorphism.

An exact_{sequence is}trivial ifand

_only

ifitis

_equivalent

to

_{0\rightarrow Y\rightarrow Y\times Z\rightarrow Z\rightarrow 0,}

where\mathrm{Y}\times Z isendowedwiththe

_product

norm. In thiscase we_saythat theexact

sequence

splits.

Ofcoursetwoexact _sequences

induced

_by

two

_{quasi‐linear}

_maps

_$\Omega$,

$\Psi$ are

equivalent

if and

only

if $\Omega$ and $\Psi$ are

equivalent.

Given an _{exact sequence 0} \rightarrow \mathrm{Y} \rightarrow X \rightarrow Z \rightarrow 0 with associated

_{quasi‐linear}

map Fandan_operator $\alpha$ : Y\rightarrow Y',thereisa commutative

diagram

The lower _{sequence is} called the

_push‐out

_{sequence, its} associated

_{quasi‐linear}

map is

equivalent

to $\alpha$\circ F, and the space POiscalled the

push‐out

space. When

F is z

‐linear,

sois $\alpha$\circ F.

The

_{push‐out technique}

will

_clearly

showusthe connection betweendifferential

processesand

quasi‐linear

maps.

Indeed,

the

complex interpolation

method

_works,

asweindicated

_{above, by considering}

_{the exact sequence}

0 \rightarrow \mathrm{k}\mathrm{e}\mathrm{r}$\delta$_{ $\theta$} \rightarrow \mathfrak{F} \rightarrow^{$\delta$_{ $\theta$}} X_{ $\theta$} \rightarrow 0

Now,

abasic

principle

in

[5,

Theorem

4.1]

establishes

_that,

evenif functions of Scanhave their derivatives

_taking

valuesin the

_cloud,

the functions of

_{\mathrm{k}\mathrm{e}\mathrm{r}$\delta$_{ $\theta$}}

take their valueat $\theta$

_precisely

in

X_{ $\theta$}

;

thus,

if

$\delta$_{ $\theta$}'

denotestheoperator

f\rightarrow f'( $\theta$)

,theresult

above_saysthat

_{$\delta$_{ $\theta$}'}

:

\mathrm{k}\mathrm{e}\mathrm{r}$\delta$_{ $\theta$}\rightarrow X_{ $\theta$}

isalinear continuous_operator. To formulate this

inthe

_{homological language,}

observe thatwehave

already

the

setting

fora

push‐out

diagram

(2)

So,

it turnsoutthat ifonehasa

complex interpolation

scale

(X_{0}, X_{1})

andinter‐

polates

at $\theta$ and obtainsthe space X_{ $\theta$}then the method

_yields

atwistedsumof

X_{ $\theta$}.

To know

_exactly

whichone one

just

hastofollowthe

_diagram:

if

_{B_{ $\theta$}, L_{ $\theta$}}

arehomo‐

geneusbounded and linear selctions for

$\delta$_{ $\theta$}

then the

_{quasi‐linear}

mapassociated to

theupper sequence is

B_{ $\theta$}-L_{ $\theta$}

and

consequently

the

_{quasi‐linear}

_mapassociatedto

the lower

_pushLout

_sequenceis

$\delta$_{ $\theta$}'(B_{ $\theta$}-L_{ $\theta$})

.

In other

_words,

_{up to}alinear _{map it is}the differentialprocess

$\delta$_{ $\theta$}'B_{ $\theta$}.

Kalton

_{[15, 16]}

started the

_study

of differentialprocessesand dida

tremendously

deep

work in the contextof \mathrm{K}\ddot {}\mathrm{t}\mathrm{h}\mathrm{e} functionspaces. X over a measure_space

( $\Sigma$, $\mu$)

.

As a

particular

case of which are the Banach spaces with a 1‐unconditional ba‐

sis. We denote

_{by L_{0}}

the space ofall $\mu$‐measiưable functions and this is

_going

to beour cloud in this context. A centralizer on X is a

homogeneous

L_{0}‐valued

map $\Omega$ : X \rightarrow

_{L_{0}}

such that

_{\Vert $\Omega$(ax)-a $\Omega$(x)\Vert_{X}}

\leq

_{C\Vert x\Vert_{X}\Vert a\Vert_{\infty}}

for all a \in L_{\infty}

and x \in X. This notion coincides with Kalton’s notion of

“strong

centralizer”’

introduced in

_[15].

Centralizers arise

_naturally

ina

complex interpolation

scheme,

in whichthe

_{interpolation}

scale ofspaces share a common

_{L_{\infty}}

‐module structure:

in such\cdot

case, the space\mathcal{H} also

enjoys

thesameL_{\infty}‐module structure inthe form

(u\cdot f)(z)

=

u\cdot f(z)

. In thisway, the fundamental sequence of the

interpolation

scheme 0\rightarrow

_{\mathrm{k}\mathrm{e}\mathrm{r}$\delta$_{ $\theta$}}

\rightarrow \mathcal{H} \rightarrow _{X_{ $\theta$}} \rightarrow 0 is an _{exact sequence in the category} of L_{\infty}-modules. In an

interpolation

scheme

_starting

with a

_couple

(X_{0}, X_{1})

of Köthe function_{spaces, the map}

_{$\Omega$_{ $\theta$}=$\delta$_{ $\theta$}'B_{ $\theta$}}

isacentralizeron

X_{ $\theta$}.

Thus,

to some _extent, it can be said that the centralizer is the core _part of the

_{quasi‐linear}

_map, orthat the

_{quasi‐linear}

_{map is} alinear

perturbation

of the centralizer.

_Moreover,

centralizerson Köthe function_spaces are

quasi‐linear

_maps

[15,

Lemma

_4.2].

In additionto

_this,

Kalton

_proved

in

_[15,

Section

_4]

that every

self‐extension ofaKöthefunction_spaceXis

_{(equivalent to)}

theextensioninduced

by

acentralizeronX. Ofcoursethat twocentralizers

$\Omega$_{1},

$\Omega$_{2}onXare

equivalent

if

and

_only

iftheinduced exact sequencesare

equivalent,

which

happens

ifand

only

therefore sense to define a centralizer $\Omega$ on X to be bounded when there exists

a constant C > 0 so that

\Vert $\Omega$(x)\Vert_{X}

_\leq

C\Vert x\Vert_{X}

for all x \in X; which in

particular

meansthat

$\Omega$(x)\in X

for all x\in X. Two centralizers

$\Omega$_{1},

$\Omega$_{2}aresaidtobe

boundedly

equivalent

when

_{$\Omega$_{1}-$\Omega$_{2}}

isbounded.

We have the

_{following outstanding}

result of Kalton

_[16,

Theorem

_7.6]:

Theorem 1.1. LetX be a

separable superreflexive

Köthe

function

_space. Then

there exists a constantc

(depending

on the _concavity

of

a_q‐concave

renorming

of

X)

such that

_if

$\Omega$ \uparrow s areal centralizeronX with

$\rho$( $\Omega$)\leq c

, then

(1)

There is a_pair

of

Köthe

function

_spaces

_{X_{0},}

_{X_{1}} such that

X=\{X_{0},

_{X_{1})_{1/2}}

and

_{$\Omega-\Omega$_{1/2}}

is bounded.

(2)

The spaces

X_{0},

X_{1} are

uniquely

determined_{up to}

equivalent renorming.

An

_example

is inorder:

_taking

the

_couple

_{(P_{1}, \ell_{\infty})}

, the map

B(x)

=x^{2(1-z)}

is

a

homogeneous

bounded selection for the evaluation _map

_{$\delta$_{1/2}}

:

\mathcal{H}\rightarrow P_{2}

_;hence the

interpolation procedure yields

thecentralizer-2$\Omega$_{2};while the

couple

(\ell_{p}, l_{p^{*}})

yields

-2(\displaystyle \frac{1}{p}-\frac{1}{p^{*}})$\Omega$_{2}

. Aswe seethetwocentralizersarethesameuptothe scalarfactor.

That scalar factor howevercannot beoverlooked since it

_actually

determines the end

_{points X_{0}, X_{1}}

inthe

_{interpolation}

scale.

2. PERSPECTIVES

The research program in which \mathrm{I}'\mathrm{m} involved

_contemplates

to advance in the

understanding

of Kalton;s theorem. The basic

_question

canbestatedas:

Problem 1. Determine towhat.extent Kaltonis theorem works for

_general

inter‐

polation

scales.

A second line of research focusesonthe

underlying

connectionbetween Kalton\dot{J}\mathrm{s} theorem andthe

_theory

oftwistedsums. Observe that it iscontained in Kalton’s theorem that the associated centralizer isbounded if and

_only

if the two extremes

of the scale coincide. In other

_words,

the

_equality

notion that

_corresponds

to

differential _processes seems to be that of bounded

equivalence

because one _gets thattheinducedexact sequence is “trivial” ifand

_only

if the associated centralizer

isbounded. But that is not the notionthat appears via

homology:

the induced

sequence istrivial if and

only

if the associated exact sequence is

equivalent

to0;

i.e., thesum ofabounded

plus

alinear_map. So the

_question

is

Problem 2. Does thereexistaKalton\mathrm{s}theorem valid for standard

equivalence

of

maps

(bounded

plus

linear)

instead of bounded

equivalence?

A sentence in

_[6,

p.

364]

suggests a

positive

answer for Banach _spaces with

unconditional basis:

_If

_{(Z_{0}, Z_{1})}

are two

super‐reflexive

_{sequence spaces} and

Z_{ $\theta$}

=

[Z_{0}, Z_{1}]_{ $\theta$}

for

0 < $\theta$ < 1 is the usual

_{interpolation}

space

by

the Calderon

method,

one can

define

a derivative

_{dX_{ $\theta$}}

which is a twisted sum

X_{ $\theta$}\oplus_{ $\Omega$}X_{ $\theta$}

which

splits

if

and

_{only if}

_{Z_{1}} =

wZ_{0} for

_some

weight

_{sequence w} =

(w(n))

where

w(n)

_>

0

_for

all n. These remarks

follow easily

from

the methods

of

[16].

In

fact,

an

appeal

to the methods of

_[4]

allows one _{to prove it. Here is} an sketch of

proof.

Let

_{(X_{0}, X_{1})}

be an

interpolation couple

of

superreflexive

Banach_spaces

having

a commonunconditional basis

(e_{n})

and let 0< $\theta$<1. If$\Omega$_{ $\theta$}istrivial then

X_{1}=wX_{0}

forsome

_weight

_sequence.

Proof.

We need to show first that ifthe associated centralizer $\Omega$_{ $\theta$} is trivial then there is a function

f

\in

_{P_{\infty}}

sothat

$\Omega$_{ $\theta$}(x)

—fx

is bounded.

Indeed,

if there is a linear_{map L}sothat$\Omega$_{ $\theta$}-Ltakesvalues inX_{ $\theta$}andisboundedthere. The

_techniques

in

_[8]

show that aftersome

averaging

it is

possible

to_getalinear _map $\Lambda$ such that

every unit ofP_{\infty}. From whereit follows that

$\Lambda$(ax) =a $\Lambda$(x)

forevery a\in P_{\infty}. It

isthenastandard fact that $\Lambda$must have the form

_{$\Lambda$(x)=fx}

forsome function

f.

Andsinceone canassume

$\Omega$_{ $\theta$}(e_{n})=0

,

necessarily f\in\ell_{\infty}

. Therestis

simple, just

pick

w=e^{f} and observe that the centralizerassociated tothe

_couple

_{(X_{0}, wX_{0})}

is

boundedly equivalent

to

_{$\Omega$_{ $\theta$}}

.

Thus, by

Kaltonis

uniqueness

theoremX_{1} =wX_{0}. \square

Turinng

to

_{specific properties,}

the

_general

_questionis:

Problem 3. Obtain conditions to detect that the differential _{process $\Omega$_{ $\theta$}} has a

certain_property.

The first _property to look at is

_triviality,

and this

_brings

us back to

_problem

2. The next

_interesting

_property is

_singularity

_(a

_{quasi‐linear}

_{map is said} to be

singular

when no restriction to an infinite dimensional

subspace

is

trivial);

the Kalton‐Peckmap$\Omega$_{2} is

singular.

A

thorough study

on

_singularity

canbe found in

[8].

Super‐singularity

has however been consideredin

_[10].

After Problem 3

_they

come

questions involving

the

stability

of the

_properties

of the differential process, a

problem

for which several

_partial

results have been

obtainedin

_{[12, 9].}

_Precisely:

Problem4. Local

_stability.

Assume that forsome $\theta$the differential process

$\Omega$_{ $\theta$}

istrivial

_(in

some

sense).

Does thereexista

neighborhood

V of $\theta$sothat $\Omega$_{ $\nu$}is also trivial for all $\nu$\in V?

Problem 5. Global

_stability.

Assume that for some $\theta$ the differential_process

$\Omega$_{ $\theta$}

istrivial

_(in

some

sense),

Doesthereit follows that _{$\Omega$_{ $\nu$}}is trivial for all\mathrm{v}? Since

_{interpolation}

methodsthere are_many, the next set of

_questions

is

Problem6. Other

_{interpolation}

methods. Set and solve the

_problems

above for other

_{interpolation}

methods.

In

_particular,

_{the paper}

_[3]

_clearly

shows that

_{homological techniques provide}

a natural frameto iterate differential

_complex

_{processes. In}

_particular,

to obtain twistedsumsof twistedsums, and so on. It is natural toask

Problem6. Iteration. Doesasimilariteration process existfor other

interpola‐

tionmethods?

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DEPARTAMENTO DE _{MATEMÁTICAS,} UNIVERSIDAD DE _EXTREMADURA, AVDA. DE ELVAS _{\mathrm{s}/\mathrm{N},} 06011BADAJOZ