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Journal of Mathematical Analysis and
Applications
www.elsevier.com/locate/jmaa
Operator martingale decompositions and the Radon–Nikodým property
in Banach spaces
Coenraad C.A. Labuschagne
a,
∗
,
1, Valeria Marraffa
b,
2a_{Programme in Advanced Mathematics of Finance, School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, PO WITS 2050,} South Africa
b_{Dipartimento di Matematica ed Applicazioni, Università Degli Studi Di Palermo, Via Archiraﬁ, 34, 90123 Palermo, Italy}
a r t i c l e
i n f o
a b s t r a c t
Article history:
Received 29 January 2009 Available online 2 September 2009 Submitted by Richard M. Aron
Keywords:
Banach lattice Banach space Bochner norm
Cone absolutely summing operator Convergent martingale
Convergent submartingale Dinculeanu operator Radon–Nikodým property Uniform amart
We consider submartingales and uniform amarts of maps acting between a Banach lattice and a Banach lattice or a Banach space. In this measurefree setting of martingale theory, it is known that a Banach spaceY has the Radon–Nikodým property if and only if every uniformly norm bounded martingale deﬁned on the Chaney–Schaefer ltensor product
E_{⊗}lY, where Eis a suitable Banach lattice, is norm convergent. We present applications
of this result. Firstly, an analogues characterization for Banach latticesY with the Radon– Nikodým property is given in terms of a suitable set of submartingales (supermartingales) on E_{⊗}lY. Secondly, we derive a Riesz decomposition for uniform amarts of maps
acting between a Banach lattice and a Banach space. This result is used to characterize Banach spaces with the Radon–Nikodým property in terms of uniformly norm bounded uniform amarts of maps that are norm convergent. In the case 1<p<∞, our results yield Lp_{(}
μ
_{,}_{Y}_{)}_{space analogues of some of the wellknown results on uniform amarts}inL1_{(}
μ
_{,}_{Y}_{)}_{spaces.}©2009 Elsevier Inc. All rights reserved.
1. Introduction
Let
(Ω, Σ,μ)
be a ﬁnite measure space and letY be a Banach space. For 1p<
∞
, let Lp(μ,
Y)
denote the space of (classes of a.e. equal) Bochnerpintegrable functions f:
Ω
→
Y and denote the Bochner norm on Lp(μ,
_{Y})
_{by}_{}
p, i.e.
p
(
f)
=
Ω fp_{Y}dμ
1/p.
Martingale theory in Bochner spaces Lp
(μ,
Y)
provides an important link to the study of geometric properties of the Banach spaceY. For example,Y has the Radon–Nikodým property if and only if every martingale(
fi)
⊆
Lp(μ,
Y), which is
uniformly bounded, converges in norm for all ﬁnite measure spaces
(Ω, Σ,
μ)
and 1<
p<
∞
(see [10,12]). It is well known thatLp(μ,
Y)
is isometrically isomorphic to the norm completion Lp(μ)
⊗
pY ofLp
(μ)
⊗
pY, where
p denotes the induced Bochner norm (see [10]).
*
Corresponding author.Email addresses:[email protected](C.C.A. Labuschagne),[email protected](V. Marraffa).
1 _{This work is based on research supported by the National Research Foundation and the Università degli Studi di Palermo, CORI 2008.} 2 _{This work is based on research supported by the M.I.U.R. of Italy.}
0022247X/$ – see front matter ©2009 Elsevier Inc. All rights reserved.
Chaney [3] characterized the Radon–Nikodým property on Y in terms of equality ofLp
(μ)
⊗
pY and the spaceL
Dinc_{L}q(
μ
),
_{Y}=
_{T}_{:}_{L}q(
μ
)
→
_{Y}_{:} _{T}_{is a Dinculeanu operator}for 1
<
p<
∞
and 1_{p}+
1_{q}=
1. An operatorT:
Lp(μ)
→
_{Y}_{, for 1}<
_{p}<
∞
_{, is called a}_{Dinculeanu operator}_{if}
T
q:=
sup _{n} i=1α
iT(
χ
Ei)
Y<
∞
,
where the supremum is taken over all simple functions f
=
n
_{i}_{=}_{1}α
iχ
Ei withχ
E1, . . . ,
χ
En∈
Lq
(μ),
α1, . . . ,α
n
∈
R
and fq1 (see [8–10]).In [4–7,14,15,22–24], using ideas on measurefree martingale theory (in which the measure theoretic aspects are replaced by operator theoretic ones), geometric aspects of the Banach space Y are characterized by means of uniformly bounded martingales in the ltensor product and by equality of the ltensor product with the space of cone absolutely summing maps (see Theorems 2.1 and 2.2).
The aim of this paper is to derive analogous results for submartingales (supermartingales) and uniform amarts.
In Section 3, Banach lattices F with the Radon–Nikodým property are characterized by means of submartingales (supermartingales) on theltensor product E
⊗
lY (Theorem 3.6).In Section 5, a Riesz decomposition for uniform amarts in this setting is obtained (Theorem 5.3). It is used to characterize Banach spaces with the Radon–Nikodým property (Theorem 5.6). These facts yield Lp
(μ,
_{Y})space results, for 1
<
_{p}<
∞
_{,}which are analogous of some of the wellknown results on uniform amarts in L1
(μ,
Y)spaces.
2. Preliminaries
We ﬁrst recall the following terminology from [4,6,24].
Let
(Ω, Σ,
μ)
be a ﬁnite measure space, 1p<
∞
and(Σ
i)
an increasing sequence of subσ
algebras ofΣ, let
E be aBanach lattice and Y a Banach space.
If
(
Ti)
is a commuting sequence (i.e., TiTj=
Ti=
TjTi for all i j) of contractive projections on Y, then(
Ti)
is a BSﬁltrationonY.It is well known that the sequence
(
E
(
·
Σ
i))
of conditional expectations on Lp(μ)
is aBSﬁltration onLp(μ).
If
(
Ti)
is aBSﬁltration onE such that eachTi0 and the rangeR
(
Ti)
ofTi, for eachi∈
N
, is a (closed) Riesz subspaceof E, then
(
Ti)
is aBLﬁltrationonE.In [21, p. 214] it is shown that, if T
:
E→
E is a projection which is strictly positive (i.e.,{
f∈
E: T(

f
)
=
0} = {
0}
) on a Banach lattice E, thenR
(
T)
is a Banach sublattice ofE. Thus, the sequence(E(
·
Σ
i))
of conditional expectations onLp(μ)
is aBLﬁltration on the Banach lattice Lp
(μ).
If
(
Ti)
is aBSﬁltration on Y and(
fi)
⊆
Y, then the pair(
fi,
Ti)
is amartingaleinY, if Tifj=
fi for allij. If(
fi,
Ti)
is a martingale in Y, then
(
fi,
Ti)
isﬁxedif there exists f∈
Y such that fi=
Tif for alli∈
N
. LetM
(
Y,
Ti)
=
(
fi,
Ti)
is a martingale inY: sup i fi<
∞
,
M
nc(Y,
Ti)
=
(
fi,
Ti)
∈
M
(
Y,
Ti)
:(
fi)
is norm convergent inY,
M
f(
Y,
Ti)
=
(
fi,
Ti)
∈
M
(
Y,
Ti)
:(
fi,
Ti)
is ﬁxed.
Let E be a Banach lattice and let Y be a Banach space. We recall from [21, Chapter IV, Section 3] that a linear map T
:
E→
Y is called cone absolutely summing if for every positive summable sequence(
xn)
in E, the sequence(
T xn)
isabsolutely summable in Y. Such maps generalize the Dinculeanu maps and are also known, in the terminology of Krivine, as 1concavemaps (see [18, p. 45]). The space
L
cas(
_{E},
_{Y})
= {
_{T}_{:}_{E}→
_{Y}_{:} _{T}_{is cone absolutely summing}}
is a Banach space with respect to the norm deﬁned by Tcas=
sup _{n} i=1 T xi: x1, . . . ,xn∈
E+,
n i=1 xi=
1,
n∈
N
for all T
∈
L
cas(
E,
Y).
It was noted in [6] that if
(
Ti)
is a BLﬁltration on E, then(
Ti), where each
Ti:
L
cas(
E,
Y)
→
L
cas(
E,
Y)
is deﬁned by TiF=
F◦
Tifor each F∈
L
cas(
E,
Y), is a
BSﬁltration onL
cas(
E,
Y).
There are two of the main results, noted in [6], that we use here. The ﬁrst is
Theorem 2.1.Let E be a Banach lattice with order continuous dual and Y a Banach space. Suppose that
(
Ti)
is a BLﬁltration on E. ThenM
f(
L
cas(
E,
Y),
Ti)
=
M
(
L
cas(
E,
Y),
Ti)
.It is easily veriﬁed that if
(
Ti)
is a BSﬁltration onE, then the sequence of adjoint maps(
Ti∗)
is aBSﬁltration on E∗,where E∗ denotes the continuous dual of E. However, it is not clear that
(
T∗_{i})
is a BLﬁltration whenever(
Ti)
is a BLﬁltration. If E is a Banach lattice with a nonempty quasiinterior Q_{+}, then aBSﬁltration
(
Ti)
on E is said to be quasiinterior preservingif TiQ+⊆
Q+ for eachi∈
N
. It was shown in [6] that, if E is a Banach lattice possessing nonemptyquasiinterior, then
(
T_{i}∗)
is aBLﬁltration onE∗ for any quasiinterior preservingBSﬁltration(
Ti)
onE.If
(
Ti)
is a BSﬁltration onY, then(
Ti)
iscomplementedin Y if there exists a contractive projection T∞:
Y→
Y withR
(
T_{∞})
=
_{i}∞_{=}_{1}R
(
Ti)
andTiT∞=
T_{∞}Ti=
Ti for alli∈
N
.It was noted in [6] that the BLﬁltration
(E(
·
Σ
i))
of conditional expectations on Lp(μ)
is complemented byE(
·
∞i=1Σ
i)
:
Lp(μ)
→
Lp(μ), where
_{∞}i=1
Σ
i denotes theσ
algebra generated by _{∞}i=1
Σ
i.Chaney and Schaefer extended the Bochner norm to the
·
l norm on the tensor product of a Banach lattice E and aBanach spaceY. The
·
lnorm is given by ul=
inf n i=1 yi
xi
: u=
n i=1 xi⊗
yifor allu
=
n
_{i}_{=}_{1}xi⊗
yi∈
E⊗
Y, and coincides with the Bochner norm onLp(μ)
⊗
Y for all ﬁnite measure spaces(Ω, Σ,μ)
and 1
p<
∞
(see [3,13,21]).Cone absolutely summing maps extend the Chaney–Schaeferltensor product in the following sense: The canonical map E∗
⊗
lY→
L
cas(
E,
Y), given by
n i=1 x∗_{i}⊗
yi=:
u→
Lu,
whereLux=
n i=1 x,
x∗_{i}yifor allx∈
E,
is an isometry (see [21, Chapter IV, Section 7] and [3,13,17]).
The following is the second main result from [6] that we use here:
Theorem 2.2.Let Y be a Banach space. Then the following statements are equivalent: (a) Y has the Radon–Nikodým property.
(b) E∗
⊗
lY=
L
cas(
E,
Y)
for all separable Banach lattices E with order continuous dual.(c)
M
(
E∗⊗
lY,
Ti∗⊗
lidY)
=
M
f(
E∗⊗
lY,
T∗i⊗
lidY)
for all separable Banach lattices E with order continuous dual and all BLﬁltrations(
Ti)
on E.(d)
M
(
E⊗
lY,
Ti⊗
lidY)
=
M
nc(E⊗
lY,
Ti⊗
lidY)
for all separable reﬂexive Banach lattices E and all complemented, quasiinterior preserving BLﬁltrations(
Ti)
on E.(e)
M
(
E⊗
lY,
Ti⊗
lidY)
=
M
(
E,
Ti)
⊗
lY for all separable reﬂexive Banach lattices E and all complemented, quasiinterior preserving BLﬁltrations(
Ti)
on E.3. Convergence ofltensor product submartingales
In this section, we establish a connection between convergence of submartingales (supermartingales) in E
⊗
l F, where E andF are Banach lattices, and the Radon–Nikodým property of the Banach latticeF.If E and F are Banach lattices, it is well known (see [3,13,16,17,21]) that E
⊗
lF is a Banach lattice, with positive conethelclosure of the projective cone
E_{+}
⊗
F_{+}:=
_{n} i=1 xi⊗
yi:n∈
N
,
xi∈
E+,
yi∈
F+;
moreover,u
∈
(
E⊗
lF)
+ if and only if Lu∈
L
cas E∗,
F_{+}.
We require Doob’s decomposition of submartingales (supermartingales) in our present situation.
We recall from [25] that E is called anordered vector spaceifE is a vector space with a partial ordering
such that, if f,
g,
h∈
E, fgand 0α
∈
R
, then f+
hg+
handα
fα
g.If E is an ordered vector space,
(
Ti)
is a commuting sequence of positive linear projections on E and(
fi)
⊆
E, then(
fi,
Ti)
is asubmartingale(supermartingale) inE, provided that fi∈
R
(
Ti)
for eachi∈
N
and fj()
Tj(
fi)
for all ji.The relevant parts of the proof of Doob’s decomposition of submartingales (supermartingales) presented in [14, Theorem 3.3], yield the following:
Theorem 3.1(Doob decomposition). Let E be an ordered vector space,
(
Ti)
a commuting sequence of positive linear projections on E and(
fi)
⊆
E. If(
fi,
Ti)
is a submartingale(supermartingale)andAi
=
i−1j=1
Tj
(
fj+1−
fj)
and Mi=
fi−
Ai for all i∈
N
,
then the decomposition, fi
=
Mi+
Aifor all i∈
N
, is the unique decomposition of(
fi,
Ti)
with(
Mi,
Ti)
a martingale,(
Ai)
positive and increasing(negative and decreasing), A1=
0and Ai+1∈
R
(
Ti)
for all i∈
N
.It was shown in [4] that, if
(
Ti)
is aBLﬁltration onE, then(
Ti⊗
lidF)
is aBLﬁltration on the Banach latticeE⊗
lF. Inparticular,
(
T⊗
lidF)
is a commuting sequence of linear projections on the ordered vector spaceE⊗
lF.Theorem 3.2 (Doob decomposition for maps). Let E and F be Banach lattices,
(
Ti)
a BLﬁltration on E and(
fi)
⊆
E⊗
l F . If(
fi,
Ti⊗
lidF)
is a submartingale(supermartingale)andAi
=
i−1j=1
(
Tj⊗
lidF)(
fj+1−
fj)
and Mi=
fi−
Ai for all i∈
N
,
then the decomposition, fi
=
Mi+
Aifor all i∈
N
, is the unique decomposition of(
fi,
Ti)
with(
Mi,
Ti)
a martingale,(
Ai)
positive and increasing(negative and decreasing), A1=
0and Ai+1∈
R
(
Ti⊗
lidF)
for all i∈
N
.Proof. It is a direct consequence of Theorem 3.1.
2
Our interest is in the case where the submartingales (supermartingales) in E
⊗
lF are bounded in a suitable way. Deﬁnition 3.3.LetE be a Banach lattice and(
Ti)
aBLﬁltration onE.(a) Denote by
B
(
E,
Ti)
the set of submartingales(
si,
Ti)
on E for which supi
s+_{i}<
∞
and sup i m−_{i}<
∞
,
wheresi
=
mi+
Uifor alli∈
N
is the unique Doob decomposition of(
si,
Ti)
into a martingale(
mi,
Ti)
and an increasingsequence
(
Ui)
for whichUi0 for alli∈
N
, andB
nc(
E,
Ti)
=
(
si,
Ti)
∈
B
(
E,
Ti)
:(
si)
is norm convergent inE.
(b) Denote by
S
(
E,
Ti)
the set of supermartingales(
si,
Ti)
onE for which sup is
− i<
∞
and sup im
+ i<
∞
,
wheresi
=
mi−
Lifor alli∈
N
is the unique Doob decomposition of(
si,
Ti)
into a martingale(
mi,
Ti)
and an increasingsequence
(
Li)
for which Li0 for alli∈
N
, andS
nc(E,
Ti)
=
(
si,
Ti)
∈
S
(
E,
Ti)
:(
si)
is norm convergent inE.
We recall from [19, p. 92] that a Banach lattice E is aKBspaceif every monotone sequence
(
xi)
in the unit ball ofE hasa limit in E.
Theorem 3.4.Let E be a KBspace and
(
Ti)
a BLﬁltration on E. If(
si,
Ti)
∈
B
(
E,
Ti)
, then, with the notation as in Deﬁnition3.3, there exists U∈
E such that Ui↑
U andUi↑
U.Proof. Since 0
Ui=
(
si−
mi)
+s+_{i}+
m−_{i} for alli∈
N
, supis+i<
∞
and supimi−<
∞
, it follows that supiUi<
∞
.As E is a KBspace andUi
↑
, there existsU∈
E such thatUi↑
UandUi↑
U.2
We recall the following result by Popa (see [20]):
Theorem 3.5(Popa). Let E and F be KBspaces. Then E
⊗
lF is a KBspace.Theorem 3.6.Let F be a Banach lattice. Then the following statements are equivalent: (a) F has the Radon–Nikodým property.
(b)
B
(
E⊗
lF,
Ti⊗
lidF)
=
B
nc(E⊗
lF,
Ti⊗
lidF)
for all separable reﬂexive Banach lattices E and all complemented, quasiinterior preserving BLﬁltrations(
Ti)
on E.(b∗)
S
(
E⊗
lF,
Ti⊗
lidF)
=
S
nc(E⊗
l F,
Ti⊗
lidF)
for all separable reﬂexive Banach lattices E and all complemented, quasiinterior preserving BLﬁltrations(
Ti)
on E.Proof. (a)
⇒
(b). Suppose F has the Radon–Nikodým property. Let E be a separable reﬂexive Banach lattice and(
Ti)
acomplemented, quasiinterior preservingBLﬁltration onE.
Let
(
si,
Ti⊗
lidF)
∈
B
(
E⊗
l F,
Ti⊗
lidF)
and let si=
mi+
Ui be the unique Doob decomposition, where(
mi,
Ti)
is amartingale and
(
Ui)
is increasing, Ui0 for all i∈
N
andmi,
Ui∈
E⊗
l F for all i∈
N
. Then misi for all i∈
N
. Butsupi
s+i<
∞
, and so supim+i<
∞
. Consequently, supimi<
∞
, as supim−i<
∞
by assumption. By Theorem 2.2,we get that
(
mi,
Ti)
is a convergent martingale; i.e., there existsm∈
E⊗
lF such that(
mi)
converges inlnorm tom.As Banach lattices with the Radon–Nikodým property and reﬂexive Banach lattices are KBspaces, both E and F are KBspaces. Then E
⊗
lF is also a KBspace, by Theorem 3.5. Therefore, by Theorem 3.4, there exists U∈
E⊗
lF such that(
Ui)
converges inlnorm toU.Consequently,
(
si)
converges inlnorm tom+
U; i.e.,B
(
E⊗
lF,
Ti⊗
lidF)
⊆
B
nc(E⊗
lF,
Ti⊗
lidF). As
B
(
E⊗
lF,
Ti⊗
lidF)
⊇
B
nc(E⊗
lF,
Ti⊗
lidF)
trivially holds, (b) follows.(b)
⇒
(a). If (b) holds, then (b) holds in particular for the corresponding spaces of martingales; i.e.,M
(
E⊗
lF,
Ti⊗
lidF)
=
M
nc(E⊗
lF,
Ti⊗
lidF). It follows from Theorem 2.2 that
F has the Radon–Nikodým property.(b
∗)
⇔
(a). The proof is similar to that of (b)⇔
(a).2
4. Stopping times and spaces of maps
Our next goal is to consider uniform amarts of maps. For this we need to consider stopping times.
It was shown in [14,15] that in the setting ofLp
(μ)spaces, there is a connection between stopping times and increasing
sequences
(
Pi)
of commuting linear band projection on Lp(μ). For the convenience of the reader, we recall this connection
from [14,15] (see also [5]).
Let
(Ω, Σ,μ)
be a probability space and 1p<
∞
. Astopping timeadapted to a ﬁltration(Σ
i)
is a mapτ
:Ω
→
N
∪{∞}
such that
τ
−1(
{
_{1, . . . ,}_{n}}
)
∈
Σ
n for eachn
∈
N
.Let
(Σ
i)
be a ﬁltration andτ
:Ω
→
N
∪ {∞}
a stopping time adapted to(Σ
i). For
f∈
Lp(μ), deﬁne
Pif
=
f·
χ
τ−1_{(}_{{}_{1}_{,...,}_{i}_{}}_{)} for eachi∈
N
.
It was observed in [14] (and it is easy to verify directly) that
(
Pi)
is an increasing, commuting sequence of linear projectionsonLp
(μ), 0
PiidLp_{(}_{μ}_{)}for alli∈
N
(i.e. each P_{i} is a band projection) andT_{j}P_{i}=
P_{i}T_{j}for allij, where T_{j}=
E
[·
Σ
_{j}]
.Conversely, let
(
Pi)
be an increasing, commuting sequence of linear projections on Lp(μ), 0
PiidLp_{(}_{μ}_{)} for alli∈
N
andTjPi
=
PiTj for alli j, whereTj=
E
[·
Σ
j]
. From the remarks at the end of [15], it follows that there is a stoppingtime
τ
adapted to(Σ
i)
such that Pif=
f·
χ
τ−1_{(}_{{}_{1}_{,...,}_{i}_{}}_{)}for eachi∈
N
.Let
T
denote the collection of all stopping times adapted to a ﬁltration(Σ
i). If
τ
,
σ
∈
T
, deﬁneτ
σ
if and only ifτ
(
x)
σ
(
x)
almost everywhere.
If
σ
is also a stopping time, and Si(
f)
=
f·
χ
σ−1_{(}_{{}_{1}_{,...,}_{i}_{}}_{)} for all f∈
Lp(μ), then, counterintuitively,
τ
σ
means that SiPi for alli∈
N
(see also [14]).Let P
=
(
Pi)
and S=
(
Si), where
(
Pi)
and(
Si)
are increasing, commuting sequences of linear projections on Lp(μ),
0
Pi,
SiidLp_{(}_{μ}_{)}for alli∈
N
,T_{j}P_{i}=
P_{i}T_{j}for allij, andT_{j}S_{i}=
S_{i}T_{j} for alli j. DeﬁneP
S if and only if SiPi for alli∈
N
.
IfPis in correspondence with
τ
andSis in correspondence withσ
, thenτ
σ
if and only if PS(see [14,15]).
We recall that
τ
is aboundedstopping time if there existsn∈
N
such thatτ
(
x)
nalmost everywhere onΩ; i.e., up
to measure zero,τ
−1(
{
_{1, . . . ,}_{n}}
)
=
Ω. Thus, if
τ
_{is a bounded stopping time such that}τ
(
_{x})
_{}
_{n}_{almost everywhere on}Ω
_{,} then, in terms of the Pi’s, it means thatPi=
idLp_{(}_{μ}_{)}for allin.We denote the collection of all bounded stopping times adapted to the ﬁltration
(Σ
i)
byT
∗.If
τ
is a bounded stopping time, let(
fi)
⊆
Lp(μ)
such that each fiisΣ
imeasurable. Then thestopped processis the pair(
fτ, Σ
τ), where
fτ=
i
which may be written as fτ
=
i(
Pi−
Pi−1)
fi andE
(
·
Σ
τ)
=
i(
Pi−
Pi−1)
E
(
·
Σ
i)
(see [14]). This gives us a description of the stopped process
(
fτ, Σ
τ)
in terms of maps and the description is independent of the underlying measure space.Deﬁnition 4.1.(See [14].) LetE be an ordered vector space and
(
Ti)
a commuting sequence of positive linear projections. If(
Pi)
is an increasing commuting sequence of linear projections such that(a) Pi
idE for alli∈
N
, and(b) PiTj
=
TjPifor all i j,then
(
Pi)
is astopping time adapted to(
Ti).
IfS
=
(
Si)
andP=
(
Pi)
are stopping times adapted to(
Ti), deﬁne
PS⇔
SiPi for alli∈
N
.
We also recall from [14] that a stopping time
(
Pi)
adapted to(
Ti)
isboundedif there existsm∈
N
so that Pn=
idE foralln
m. We say(
Pi)
stopsatm∈
N
, ifm∈
N
is the least natural number for whichPm=
idE. We denote byT
∗the set ofall bounded stopping times with respect to
(
Ti).
If
(
Ti)
is a commuting sequence of positive linear projections onE, then so is(
TP)
P∈T∗, as may be readily veriﬁed (see also [14]).In order to consider uniform amarts in the measurefree setting, it is necessary to consider a suitable notion of a stopped process
(
FP,
TP)
for the case whereTPis deﬁned as before, but FP:
E→
Y has to be a linear map withY a vector space. Deﬁnition 4.2. Let E be an ordered vector space, P=
(
Pi)
a bounded stopping time which is adapted to a commutingsequence
(
Ti)
of positive linear projections onE,Y a vector space and Fi:
E→
Y a linear map for eachi∈
N
such that(
Fi)
is an adapted sequence (i.e., FiTi
=
Fifor alli∈
N
). Deﬁne the operator stopped process(
FP,
TP)
P∈T∗ byFP
:=
∞ i=1 Fi(
Pi−
Pi−1) and TP=
∞ i=1(
Pi−
Pi−1)Ti.
If Y is an ordered vector space and
(
Fi)
is increasing and positive (decreasing and negative) and P,S∈
T
∗ withPS,then it is easily veriﬁed that FP
FS (FPFS).The proof of our Riesz decomposition for uniform amarts of maps (Theorem 5.3 below) uses an operator version of Doob’s optional stopping theorem.
Theorem 4.3(Optional stopping theorem). Let E be an ordered vector space,P
=
(
Pi)
andS=
(
Si)
bounded stopping times adapted to the commuting sequence(
Ti)
of positive linear projections on E withSP, Y a vector space and Fi:
E→
Y a linear map for each i∈
N
. If(
Fi,
Ti)
i∈Nis a martingale, then FPTS=
FS.Proof. Letnbe a bound for both of the stopping timesSandP. Then
FPTS
=
_{n} i=1 Fi(
Pi−
Pi−1) _{n} j=1(
Sj−
Sj−1)Tj=
n i=1 n j=1 Fi(
Pi−
Pi−1)(Sj−
Sj−1)Tj.
AsS
P, it follows that PiSj=
SjPi=
Pifor all ji and(
Pi−
Pi−1)(
Sj−
Sj−1)
=
0 for allji+
1.
This enables us to deduce thatn
i=1 n j=1 Fi(
Pi−
Pi−1)(Sj−
Sj−1)Tj=
n i=1 i j=1 Fi(
Pi−
Pi−1)(Sj−
Sj−1)Tj.
As
(
Fi,
Ti)
is a martingale, Fi=
FnTi. The commutation of Ti and Pi−
Pi−1 and the commutation of Tj and Si−
Sj−1, together with TiTj=
Tjfor ji, yieldThus n
i=1 i j=1 Fi(
Pi−
Pi−1)(Sj−
Sj−1)Tj=
n i=1 i j=1 Fn(
Pi−
Pi−1)Tj(
Sj−
Sj−1)=
Fn _{n} i=1(
Pi−
Pi−1) _{n} j=1 Tj(
Sj−
Sj−1)=
Fn n j=1 Tj(
Sj−
Sj−1)
=
n j=1 Fj(
Sj−
Sj−1)
=
FS.
Hence, FPTS=
FS.
2
5. Riesz decomposition of uniform amarts of maps
We recall the following from [1,2,11,12]:
Deﬁnition 5.1.An adapted sequence
(
Fi, Σ
i)
is called auniform amartif, for bounded stopping timesτ
,
σ
, lim sup σ sup τσ ΩE
[
Fτ
Σ
σ] −
Fσdμ
=
0.
Motivated by this notion of a uniform amart inL1
(μ,
_{X}), we introduce
Deﬁnition 5.2.Suppose E is a Banach lattice,Y is a Banach space and
(
Ti)
is aBLﬁltration onE.(a) If
(
Fi)
⊆
L
cas(
E,
Y)
is adapted to(
Ti), we call
(
Fi,
Ti)
anL
cas(
E,
Y)
uniform amartprovided that, for bounded stoppingtimesSandP adapted to
(
Ti),
lim sup P sup SP FSTP−
FPcas=
0.
(b) If
(
Fi)
⊆
E⊗
lY is adapted to(
Ti⊗
lidY), we call
(
Fi,
Ti⊗
lidY)
an E⊗
lYuniform amart provided that, for boundedstopping timesSandPadapted to
(
Ti),
lim sup P sup SP(
TP⊗
lidY)(
FS−
FP)
_{l}=
0.
The following result gives a connection between uniform amarts and martingales. It is motivated by a result of Alexandra Bellow (see [1,2,11,12]) for uniform amarts inL1
(μ,
_{X}). We ﬁrst introduce some notation.
For anyn
∈
N
, letS(n):=
(
_{S}i
), where
Si
:=
0
,
i<
n,
idE,
in.
ThenS(n)is a bounded stopping time with the property that ifP
∈
T
∗stops atn∈
N
, thenPS(n).Theorem 5.3(Riesz decomposition). Suppose E is a Banach lattice, Y is a Banach space and
(
Ti)
is a BLﬁltration on E.(a) If
(
Fi)
⊆
L
cas(
E,
Y)
and(
Fi,
Ti)
is anL
cas(
E,
Y)
uniform amart, then there exists a unique decomposition Fi=
Mi+
Zifor all i∈
N
, where(
Mi), (
Zi)
⊆
L
cas(
E,
Y)
,(
Mi,
Ti)
is a martingale andlimP∈T∗ZPcas=
0.(b) If
(
Fi)
⊆
E⊗
lY and(
Fi,
Ti⊗
lidY)
is an E⊗
lY uniform amart, then there exists a unique decomposition Fi=
Mi+
Zifor all i∈
N
, where(
Mi), (
Zi)
⊆
E⊗
lY ,(
Mi,
Ti⊗
lidY)
is a martingale andlimP∈T∗ZPl=
0.Proof. (a) We ﬁrst prove existence of such a decomposition. Let
>
0 be given. Selectn∈
N
such that PS(n) _{implies} supSPFSTP−
FPcas<
. As
Tn0, we have for all bounded stopping timesSPS(n)that(
FS−
FP)
Tn_{cas}=
(
FS−
FP)
TPTn_{cas}FSTP−
FPcasTnFSTP−
FPcas<
.
Thus, the net(
FPTn)
P∈T∗ is Cauchy in the Banach spaceL
cas(
_{E},
_{Y}). Let
_{M}i
:=
limP∈T∗FPTi. Then, forij, we haveTjTi=
TiTj=
Ti; consequently,(
Mj−
Mi)
Ticas=
limr→∞FrTiTj−
FrTicas=
0. Thus,(
Mi,
Ti)
i∈Nis anL
cas(
_{E},
_{Y})martingale. By
Theorem 4.3, the net
(
MP,
TP)
P∈T∗ is also anL
cas(
E,
Y)martingale. Moreover,
MS=
limP∈T∗FPTSfor any S∈
T
∗. Indeed,ifS
∈
T
∗, then there existsn∈
N
such thatSS(n)_{. As}(
_{M}P
,
TP)
is a martingale, we get thatMS=
MnTS=
(lim
PFPTn)
TS=
limP
(
FPTnTS)
=
limPFPTS.
Let Zi
=
Fi−
Mi for all i∈
N
. For bounded stopping times SP, it follows from MS=
limP∈T∗FPTS and ZSTP=
(
FS−
MS)
TP=
(
FS−
MP)
TP that limSPZSTPcas=
0. As ZPcasZP−
ZSTPcas+
ZSTPcas=
FP−
FSTPcas+
ZSTPcas, it follows that limP∈τZPcas=
0.We prove uniqueness of such a decomposition. Suppose Fi
=
mi+
zi for alli∈
N
, where(
mi), (
zi)
⊆
L
cas(
E,
Y),
(
mi,
Ti)
is a martingale and limP∈T∗
zPcas=
0. Then, limi→∞Mi−
micas=
0. Since(
Mi−
mi,
Ti)
is a martingale,Mi=
mifor all i∈
N
, from which we also get Zi=
zifor alli∈
N
.(b) The proof is similar to that of (a).
2
Corollary 5.4.Let E be a Banach lattice with order continuous dual,
(
Ti)
a BLﬁltration on E and Y a Banach space. Then every uniformly norm boundedL
cas(
E,
Y)
uniform amart(
Fi,
Ti)
, where(
Fi)
⊆
L
cas(
E,
Y)
, has a unique decomposition Fi=
Mi+
Zifor all i∈
N
, where(
Mi), (
Zi)
⊆
L
cas(
E,
Y)
,(
Mi,
Ti)
is a ﬁxed martingale andlimP∈T∗ZPcas=
0. If, in addition, E is separable and Y has the Radon–Nikodým property, then(
Mi)
⊆
E∗⊗
lY .Proof. By Theorem 5.3, we may decompose Fi as Fi
=
Mi+
Zi for all i∈
N
, where(
Mi), (
Zi)
⊆
L
cas(
E,
Y),
(
Mi,
Ti)
is amartingale and limP∈T∗
ZPcas=
0. Since(
ZP)
is uniformly norm bounded, we have that(
Zi)
is uniformly norm bounded.But then
(
Mi)
is uniformly norm bounded. It follows from Theorem 2.1 that(
Mi,
Ti)
is ﬁxed.If, in addition, E is separable andY has the Radon–Nikodým property, then it follows from Theorem 2.2 thatE∗
⊗
lY=
L
cas(
_{E},
_{Y}); thus,
(
_{M}i
)
⊆
E∗⊗
lY.2
Deﬁnition 5.5.SupposeY is a Banach space and
(
Ti)
is aBSﬁltration onY. LetU
(
Y,
Ti)
=
(
fi,
Ti)
is a uniform amart inY: sup i fi<
∞
,
U
nc(Y,
Ti)
=
(
fi,
Ti)
∈
U
(
Y,
Ti)
:(
fi)
is norm convergent inY.
As a consequence of the Riesz decomposition of uniform amarts of maps, we get the following description of Banach spaces with the Radon–Nikodým property:
Theorem 5.6.Let Y be a Banach space. Then the following statements are equivalent: (a) Y has the Radon–Nikodým property.
(b)
U
(
E⊗
lY,
Ti⊗
lidY)
=
U
nc(E⊗
lY,
Ti⊗
lidY)
for all separable reﬂexive Banach lattices E and all complemented quasiinterior preserving BLﬁltration(
Ti)
on E.(c)
U
(
Lp(μ,
Y), Σ
i)
=
U
nc(Lp(μ,
Y), Σ
i)
for all probability spaces(Ω, Σ,
μ)
, all ﬁltrations(Σ
i)
ofΣ
and all1<
p<
∞
. Proof. (a)⇒
(b). Suppose Y has the Radon–Nikodým property. Let E be a separable reﬂexive Banach lattice and(
Ti)
acomplemented quasiinterior preservingBLﬁltration onE.
If
(
Fi,
Ti⊗
lidY)
∈
U
(
E⊗
lY,
Ti⊗
lidY), then by Theorem 5.3, we may decompose
FiasFi=
Mi+
Zifor alli∈
N
, where(
Mi), (
Zi)
⊆
E⊗
lY,(
Mi,
Ti⊗
lidY)
is a martingale and limP∈T∗ZPl=
0. Since(
ZP)
is uniformly norm bounded, we havethat
(
Zn)
is uniformly norm bounded. But then(
Mi)
is uniformly norm bounded. As Y has the Radon–Nikodým property,it follows from Theorem 2.2 that
(
Mi)
is convergent in lnorm. Consequently,(
Fi)
is convergent in lnorm, and we getthat
(
Fi,
Ti⊗
lidY)
∈
U
nc(E⊗
lY,
Ti⊗
lidY). Thus,
U
(
E⊗
lY,
Ti⊗
lidY)
⊆
U
nc(E⊗
lY,
Ti⊗
lidY). As
U
(
E⊗
lY,
Ti⊗
lidY)
⊇
U
nc(E⊗
lY,
Ti⊗
lidY)
holds trivially, the statement in (b) is proved.(b)
⇒
(c). Suppose (b) holds. If(Ω, Σ,
μ)
is a probability space,(Σ
i)
is a ﬁltration ofΣ
and 1<
p<
∞
, then Lp(μ)
isa reﬂexive Banach lattice and the sequence of conditional expectation operators
(
E
(
·
Σ
i))
is a complemented quasiinteriorpreservingBLﬁltration onLp
(μ).
Under identiﬁcation ofLp
(μ,
Y)
andLp(μ)
⊗
pY, the conditional expectation operatorE
(
·
Σ
i)
onLp
(μ,
_{Y})
_{is}E
(
·
Σ
i)
⊗
lidY for alli
∈
N
(see [4]). By the remarks in Section 4 that, onLp(μ)spaces, stopping times in the classical setting are in
onetoone correspondence with the notion of stopping times in terms of band projections, we get that (c) is a special case of (b).
(c)
⇒
(a). Suppose (c) holds. Let(
Fi, Σ
i)
be a uniformly norm bounded martingale inLp(μ,
Y). Then, under identiﬁcation
of Lp
(μ,
_{Y})
_{and}_{L}p(μ)
⊗
_{}
pY,
(
Fi,
E(
·
Σ
i)
⊗
lidY)
is a uniformly norm bounded martingale inLp
(μ)
_{}
⊗
pY, and therefore
trivially, a uniformly norm bounded uniform amart in Lp
(μ)
_{}
⊗
pY. By the assumption (c),
(
Fi,
E(
·
Σ
i)
⊗
lidY)
is convergentAcknowledgments
The ﬁrst named author wishes to express his sincere gratitude to the second named author for her warm hospitality during his stay at Dipartimento di Matematica ed Applicazioni, Università degli Studi di Palermo, in January and June 2009.
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