On natural density, orthomodular lattices, measure algebras and non-distributive $L^p$ spaces

0.18 MB·English

by  Jarno Talponen

ON NATURAL DENSITY, ORTHOMODULAR LATTICES,
MEASURE ALGEBRAS AND NON-DISTRIBUTIVE Lp SPACES
5
1
JARNOTALPONEN
0
2
Abstract. Inthisnotewefirstshow,roughlyspeaking,thatifBisaBoolean
n
u algebra included in the natural way in the collection D{„ of all equivalence
classesofnaturaldensitysetsofthenaturalnumbers,modulonulldensity,then
J
Bextendstoaσ-algebraΣĂD{„ andthenaturaldensityisσ-additiveonΣ.
4
We prove the maintool employed inthe argument inamoregeneral setting,
2
involving a kind of quantum state function, more precisely, a group-valued

submeasureonanorthomodularlattice. Attheendwediscusstheconstruction
A] of‘non-distributiveLp spaces’bymeansofsubmeasuresonlattices.
F
.
h
t
a 1. Introduction
m
Thisarticledealswithabstractversionsofthemeasurealgebraandnaturalden-
[
sity notions. The considerations here are much in the spirit of Galois connections.

3 Let us denote by D the collection of all density sets, i.e. sets A Ă N such that
v the natural density
7 |AXt1,...,nu|
9 dpAq:“ lim
5 nÑ8 n
exists. We denote by N Ă D the collection of all null density sets, i.e. sets A
0
0 with dpAq“0. We denote by „ the equivalence relation on D given by K „M if
. the symmetric difference of K △ M is in N. Consider the set D{ of equivalence
1 „
0 classesrxsmodulonulldensity. IthasthenaturalpartialordergivenbyrxsĺN rys
5 if there is N PN such that xĂyYN.
1 The natural density is of course an important notion in number theory. For
:
v instance, recall Szemer´edi’s theorem in [15] which states that every positive (i.e.
i non-null) density set contains a k-arithmetic progressionfor every k PN.
X
Regarding the natural density as a content in the sense of measure theory is a
r
a classical theme, see e.g. [6, 7]. The following theorem is the main result in this
paper:
Theorem 1.1. Let F Ă D be a family closed under finite intersections. Then
there is a σ-algebra Σ order-isomorphically included in D{ such that F{ Ă Σ
„ „
and dˆ: ΣÑr0,1s, dˆpK{ q“dpKq, is σ-additive.
„
Morerover, if F{ is countable and the corresponding σ-generated measure alge-
„
bra pΣ,dˆq is atomless, then it is in fact isomorphic to the measure algebra on the
unit interval.
Date:June25,2015.
2010 Mathematics Subject Classification. 06C15,28A60,11B05,46E99.
Keywordsandphrases. Orthomodularlattice,Booleanalgebra,measurealgebra,group-valued
submeasure, natural density, density set, content extension, Banach space, function space, Lp
space.
1
2 JARNOTALPONEN
In the first part of the statement Σ can be viewed essentially as a family of
density sets closed under finite intersections and therefore the conditions in the
first part of the theorem are in fact equivalent. What is highly counter-intuitive
aboutthelatterpartoftheresultisthatthe algebraismodeledonN whichclearly
does not admit an atomless σ-additive probability measure.
In the classical papers on this topic [6, 7] equivalence classes were considered
essentially modulo finite subsets of the natural numbers, instead of taking equiva-
lenceclassesmodulonull-density. Presumably,theembeddingsofBooleanalgebras
in D{Fin behave differently compared to such embeddings in D{ . Also, the ques-
„
tion of embedding the measure algebra M into PpNq{Fin is very delicate, see e.g.
[5, 8, 13].
Infactwewillprovethemaintoolappliedintheabovearesultinamoregeneral
form. In this general version we will replace Boolean algebras with orthomodular
lattices andthe measuresarereplacedby kindof quantumprobabilitymeasuresor
group-valuedsubmeasures,cf. [12]. Ortomodular lattices are closely related to the
formulation of quantum logic in physics. In this connection the above mentioned
measureissometimes calledthe state function. This mapping alsoappearsto have
a connection to graded semi-modular lattices. From the functional analysis point
of view the orthomodular lattices model a system of closed subspaces of a Hilbert
space with natural interpretations involving the complemented lattice operations.
The end of the paper is devoted to studying Lp spaces over ‘non-distributive
measure algebras’, or rather LppL,ϕq where L is a bounded lattice and ϕ is an
order-preserving subadditive mapping ϕ: L Ñ r0,1s with ϕp0q “ 0 and ϕp1q “ 1.
The non-commutative Lp spaces are under active investigation (cf. [14]) and the
above generalization of the classical Lp spaces goes to another direction. We will
adopt an approach which somewhat resembles the construction of some Banach
tensor products, cf. [10], [16].
2. Group-valued submeasures on lattices and extensions of the
natural density
We refer to the monographs in the references for suitable background informa-
tion. As mentioned above, our aim is to generalize the main tool applied in the
proofofTheorem1.1simultaneouslyto severaldirections. We list the assumptions
and conventions imposed in this section:
‚ In what follows L is a bounded countably complete orthomodular lattice.
The minimal and maximal elements are denoted by 0 and 1, respectively.
‚ Let Gbe a partiallyorderedlocallycompacttopologicalgroup. Its neutral
element is denoted by e. We impose the following conditions:
(1) If xďy then zxďzy and xz ďyz,
(2) If xďy then y´1 ďx´1.
(3) The topology is stronger than the order topology.
(4) The topology of G has countable character.
‚ Let I be an infinite index set.
‚ LetF beafilteronI whichiscountablyincompleteinthe sensethatthere
is a sequence pF qĂF with F “H.
n n n
‚ For each i P I we let m : L Ñ G be a mapping satisfying the following
i
Ş
properties:
(i) m p0q“e.
i
GENERALIZED MEASURE ALGEBRAS 3
(ii) m is order-preserving.
i
(iii) m px_yqďm pxqm pyq for x,y PL.
i i i
(iv) If xďy then m pyq“m pxqm py^xKq.
i i i
‚ We assumeΛĂListhemaximalsubsetsuchthatthe mappingm: ΛÑG
given by
mpxq“limm pxq, xPΛ,
i
i,F
is defined.
‚ This yields a natural ideal N “txPL: mpxq“euĂL.
‚ WedefineapreorderonLbyxĎN yifthereisN PN suchthatxďy_N.
‚ We define an equivalence relation on L by x„y if xĎN y and y ĎN x.
‚ Then„isa_-semilatticecongruenceandwehaveanaturalquotientmap-
ping LÑL{N, xÞÑrxs, which is a _-semilattice epimorphism.
‚ We define mˆ : L{N Ñ G by mˆprxsq “ mpxq. (This is well-defined by the
assumptions on m.)
‚ We assume that for each m there is a support s PL in the sense that
i i
m pxq“m px^s q @xPL.
i i i
‚ We assume that if ΓĂ I is such that IzΓPF then s P L exists and
iPΓ i
is included in N.
‚ We assume that if x,y PΛ, xĎN y, then x^y PΛ aŽnd rx^ys“rxs.
‚ If x,y PΛ, xĎN y, mpxq“mpyq, then rxs“rys.
Admittedly, this list of assumptions long and appears technical. It is not very
restrictive, for example the density function d fits easily to this framework. The
point here is stretching the generality in which our main tool holds to the limit, as
there appears to be applications for such structures, see [3, 12, 13].
Theproofofthefollowingauxiliaryfactissubsequentlyappliedinthe argument
of the main result.
Lemma 2.1. With the above notations the mapping mˆ is countably additive in the
following sense: Suppose that prx sq Ă Λ{N is an increasing sequence such that
n
mˆprx sq exists. Then rx sPΛ{N exists and mˆp rx sq“ mˆprx sq.
n n n n n n n n
PŽroof. Let G Ą U1 Ą U2ŽĄ ... Ą Un Ą ..., n P N, beŽa countablŽe neighborhood
basis of e. Let pF q Ă F be a decreasing sequence such that F “ H. Let
n nPN n
Γ1 ĂF1 be the subset of all indices i such that
Ş
mipx1qPmpx1qU1.
Next, we let Γ2 ĂΓ1XF2 be the subset of all indices i such that
mipx2qPmpx2qU2.
We proceed recursively in this fashion: for each n`1 P N we let Γn`1 Ă ΓnXFn
be the subset of all indices i such that
mipxn`1qPmpxn`1qUn`1.
Note that by the construction of m we have that Γ PF for each nPN.
n
Let s be the supports of the mappings m . We put
i i
z :“ s , nPN.
n i
iPłIzΓn
4 JARNOTALPONEN
Note that the orthomodular law yields that x _z “ z _pzK ^ px_ z qq for
n n n n
each x P L. Since mpz q “ e by the assumptions, we may modify recursively the
n
selection of the sets Γ to obtain a decreasing sequence pΓ1 q with Γ ĄΓ1 PF in
n n n n
such a way that
mipznK^pxn`1_znqqPmipxn`1qUn`1
and
mipxn`1qPmpxn`1qUn`1
for all iPΓ1 , nPN.
n`1
Put
wn “ zjK^pxj`1_zjq, nPN.
1ďjďn
ł
By using the countable completeness of L we may put
w “ w .
n
n
ł
Notethatifwechooseasequencepi qwithi PΓ thenmpx q´1m pwqÑeas
n n n n in
n Ñ 8 and that M “ mpx q exists by assumption. Thus by the assumptions
n n
involving the topology of G we obtain that pm px qqĂG contains a subsequence
Ž in n
converging to M. From the arbitrary nature of the selection of pi q we conclude
n
that limi,Fmipwq“M. In particular, wPΛ.
Note that
rxn`1s“rzn_pznK^pzn_xn`1qqs,
so that
rws“rz _wsěrz _w sěrx s.
n n n n
This means that rws is an upper bound for the sequence prx sq.
n
Assume next that rys P Λ{N is some upper bound for the sequence prx sq.
n
Then according to the assumptions y ^ w exists because rw s “ rx s and it
n n n
defines an increasing sequence. By the countable completeness of L we may define
w0 :“ ny^wn. Clearly w0 ĎN y and w0 ĎN w. Moreover, mpw0q “ M, since
w0 ď w and mpy ^wnq Ñ M as n Ñ 8. It follows from the assumptions that
Ž
rw0s“rws. Thus rws is the least upper bound for the sequence prxnsq.
(cid:3)
3. The main result
In this section we will give the proof for Theorem 1.1, the main result.
3.1. Pointless Lp spaces. We will introduce subsequently a way of defining Lp
spaces over a general lattice. This construction can be specialized to obtain Lp
spaces over measure algebra, i.e. LppΣ,µq (cf. [11, Ch. 36]). These spaces can be
built by defining a suitable norm for ‘simple functions’ a bM where a P R
k k k k
and M PΣ are essentially pairwise disjoint. Then the norm is given by
k
ř
1
p
a bM “ |a |pµpM q .
k k k k
› › ˜ ¸
›ÿk › ÿk
This space is then com››pleted to get››the required space LppΣ,µq. (The case with a
› ›
lattice, instead of a Boolean algebra is more complicated.) One can see that this
space has a natural Banachlattice structure and it admits a Lebesgue integrallike
functional.
GENERALIZED MEASURE ALGEBRAS 5
3.2. The proof.
Proof of Theorem 1.1. Letusresumethenotationsofthetheorem. Itiswell-known
that if A is a density set and N a null-density set, then AYN and AzN are also
density sets and have the same density as A. Therefore dˆin the theorem is well-
defined.
We will apply the argument of the Dynkin-Sierpinski π-λ-lemma. We call a
subset ∆ĂD{ a d-system if it satisfies the following conditions:
„
(i) rNsP∆;
(ii) For each rAs,rBsP∆, rAsĺN rBs, we have rBzAsP∆;
(iii) For each rAs,rBsP∆, AXB “H, we have rAYBsP∆;
(iv) If pA q Ă ∆ is an increasing sequence in the order inherited from D{ ,
n „
thentheleastupperboundAforthissequenceexistsinD{ andmoreover
„
AP∆.
We claimthatD{ isad-systemitself. Indeed,the conditions(i)-(iii)followim-
„
mediatelyfromthewell-knownpropertiesofthenaturaldensity. Thelastcondition
follows from the proof of Lemma 2.1.
Let ∆ be the intersection of all d-systems in D{ containing F{ . Put
„ „
A1 “trAsP∆: rAXFsP∆ @F PFu.
WeclaimthatA1 isad-system. Indeed,sinceF isclosedunderfiniteintersections,
we obtain that F{„ Ă A1. Therefore, A1 satisfies (i). It is easy to see that A1
satisfies (iii)-(iv). Note that if rAXFs P ∆ for a given A P ∆ and F P F, then
rpNzAqXFs “ rFzpAXFqs P ∆ according to (ii). Thus, by using (iii) we obtain
that A1 satisfies (ii). Consequently, A1 is a d-system and it follows that A1 “∆.
Put
A2 “trAsP∆: rAXCsP∆ @C P∆u.
By studying A1 “ ∆ we obtain that F{„ Ă A2. We easily see again that A2 is
a d-system. It follows that A2 “ ∆. Thus ∆ satisfies that rAXBs P ∆ for any
rAs,rBsP∆.
Now we may define a Boolean algebra structure on ∆ as follows: rAs_rBs “
rAYBs “ rpAzBqYpAXBqYpBzAqs, rAs^rBs “ rAXBs and rAs “ rNzAs.
These operations satisfy the axioms of a Boolean algebra since the corresponding
operations on the subsets of N satisfy them. It follows from the last condition of
the d-system that ∆ forms in fact a σ-algebra.
Now, it is a basic well-known fact that dˆis finitely additive on ∆ in the sense
that if A,B P D are disjoint, then dˆprAs_rBsq “ dˆprAYBsq “ dˆprAsq`dˆprBsq .
It follows from the proof of Lemma 2.1 that µ“dˆis σ-additive on Σ“∆.
It remains to show the latter part of the statement. Let us study the space
L1pΣ,µq. Since any measure algebra can be represented by using the Stone space
asameasurealgebraofameasurespace(see[11,321J]),theabovespaceisactually
isometrically order-isomorphically a genuine function space L1pΩ,Σ,µq where µ is
a probabilitymeasure. If F{ is countable andΣ is σ-generatedby F{ , then it is
„ „
clearthatL1pΣ,µqisseparableasthesimplefunctionsgeneratedbyF{ aredense
„
(e.g. by the Martingale Convergence Theorem). Note that we assumed the mea-
sure algebra is atomless. Therefore Maharam’s classification of measure algebras
implies that pΣ,µq is isomorphic to M, the Maharam’s space corresponding to the
unit interval with the completed Lebesgue measure. Indeed, if pΣ,µq contained a
6 JARNOTALPONEN
part isomorphic to the measure space on t0,1uω1, generated by ℵ1-many i.i.d. fair
Bernoulli trials (i.e. coin tosses), then L1pΣ,µq would be non-separable. (cid:3)
4. Non-distributive Lp spaces
LetusconsideraboundedlatticeL“pL,_,^,0,1q(whichneednotbemodular
or countably complete). Let ϕ: L Ñ r0,1s be an order-preserving mapping such
that ϕp0q“0, ϕp1q“1 and
ϕpA_BqďϕpAq`ϕpBq @A,B PL.
Consider the space c00pLq and denote its canonical Hamel basis unit vectors by
eA, APL. Let ∆Ăc00pLq be the linear subspace given by
∆“re0s`spanppeA`eBq´peA_B `eA^Bq: A,B PLu.
We let
q: c00pLqÑc00pLq{∆
be the canonical quotient mapping. Write X “c00pLq{∆ and we denote by
abA:“qpae qPX, aPR, APL.
A
Note that X is the space of vectors of the form
a bA , a PR, A PL, I finite.
i i i i
iPI
ÿ
Proposition 4.1. Let L be an orthomodular lattice and X be the corresponding
space defined as above. Then for any abA,bbB PX it holds that
abA`bbB “abpA^pA^BqKq`pa`bqbpA^Bq`bbpB^pA^BqKq.
Any element xPX can be represented as
x“ a bA
i i
i
ÿ
where A ^A “0 for i‰j. Moreover, if pΩ,Σ,µq is a measure space and L“Σ,
i j
then the space S of simple functions on pΩ,Σ,µq can be linearly identified with X
as follows:
a 1 ÞÝÑ a bA .
i Ai i i
i i
ÿ ÿ
Proof. Note that the orthomodularity condition yields
A“pA^pA^BqKq_pA^Bq, B “pB^pA^BqKq_pA^Bq.
It follows from the construction of X that
abA“abpA^pA^BqKq`abpA^Bq,
bbB “bbpB^pA^BqKq`bbpA^Bq.
It follows from the construction of X that
abpA^Bq`bbpA^Bq“pa`bqbpA^Bq
and the first part of the claim follows.
The second part of the statement thus follows by constructing a refined system
ofelements A where one uses recursivelythe abovetype decompositions. The last
i
partofthestatementistheneasytoseebystudyingamaximallinearlyindependent
system t1 u ĂS. (cid:3)
Aβ β
GENERALIZED MEASURE ALGEBRAS 7
Let Ď be the preorder on X generated by the following conditions:
(i) Ď satisfies the axioms of a partial order of a vector lattice, except possibly
anti-symmetry;
(ii) abAĎbbA whenever aďb;
(iii) 1bAĎ1bB whenever AďB in the intrinsic partial order of L.
We define a semi-norm on X by
1
p
a bA “inf |b |pϕpB q : ˘ a bA Ď |b |bB .
i i k k i i k k
› › $˜ ¸ ,
›ÿi › & ÿk ÿi ÿk .
› ›
Indee›d, it is eas›y to see that this is a semi-norm; the triangle inequality follows
› › % -
from the condition that x`y Ďv`w whenever xĎv and y Ďw.
WeletX “X{Ker}¨}andwestilldenoteby}¨}theobviousnorminducedbythe
abovesemi-normon this quotient space. We also take the quotientof the elements
abAandcontinueusingthe same notation. Dividing by the kernelcorrespondsto
theoperationofidentificationoffunctionsthatcoincidea.e. intheclassicalsetting.
However, we do not know if there is an additional construction-specific necessity
for performing this operation.
We let LppL,ϕq be the Banach space obtained as the completion of the normed
space pX,}¨}q. We define a relation ď on LppL,ϕq by means of the closure of the
positive cone given by the Ď relation.
Proposition 4.2. The space LppL,ϕq endowed with the order ď is an ordered
vector space with the property that if 0ďxďy then }x}ď}y}.
Proof. First note that the relation Ď is given by a convex cone, since we imposed
that Ď satisfies the vectorlattice conditions. Typically there exist elements xPX,
x‰0, such that ˘xĎ0.
If x ď y, then by the definition of the ď order, y ´x P C where the convex
`
cone C is the closure
`
C :“txPX: 0ĎxuĂLppL,ϕq.
`
It is clear that the relation ď is a vector lattice order if ď is an anti-symmetric
relation. This is the case exactly when the positive cone C is salient.
`
Note that in any case for 0ďxď y it holds that 0ďy´x and therefore there
is a sequence pznq in the positive cone of Ď such that }zn´py´xq}LppL,ϕq Ñ0 as
nÑ8. By the definition of the norms we have that }y}ě}x}.
To verify the saliency of C , assume that ˘z P C . Then 0 ď ˘z ď 0. Thus
` `
}z}LppL,ϕq “ 0 by the previous observation. Since we defined } ¨ }LppL,ϕq as a
completion of the norm of X (instead of working with X), we obtain that z “ 0
which shows the saliency of C . (cid:3)
`
4.1. Examples. Here finite sequence spaces are considered with coordinate-wise
order. Below we will write all order relations explicitly visible. If L“t0,A,B,1u
with 0 ď A,B ď 1 (thus A^B “ 0, A_B “ 1) and ϕpAq “ ϕpBq “ 1, then
2
LppL,ϕqisorder-isomorphicallyisometrictoℓpp2qfor1ďpă8(the2-dimensional
ℓp space).
If L “ t0,A,B,C,1u with 0 ď A,B,C ď 1 (i.e. smallest non-distributive but
modular lattice) and ϕpAq “ ϕpBq “ ϕpCq “ 1, then LppL,ϕq is 1-dimensional.
2
8 JARNOTALPONEN
Indeed,writepairwisedistincti,j,kPt1,2,3uandD1 “A, D2 “B,D3 “C,then
we have
1bD ´1bD “p1bD `1bD q´p1bD `1bD q
i j k i k j
“p1b1´1b0q´p1b1´1b0q
“1b1´1b1“0b1“0.
Thus 1bD “1bD “1bA. Hence
i j
1b1“1bD `1bD “1bA`1bA“2bA.
i j
This means that the space is spanned by 1bA.
If L “ t0,A,B,C,1u with 0 ď A ď 1, 0 ď B ď C ď 1 (i.e. smallest non-
modular lattice) and 0 ă ϕpAq, 0 ă ϕpBq ď ϕpCq, then LppL,ϕq is isometric to
ℓpp2q. The linear isometry is given by
1 1
1bAÞÑφpAqp e1, 1bB ÞÑφpBqp e2.
Indeed,
1bB´1bC “p1bA`1bBq´p1bA`1bCq“1b1´1b1“0,
so that the linear space reduces to the first example. In the definition of the norm
we clearly choose B instead of C, since ϕpBqďϕpCq.
If pΣ{ ,mq is the measure algebra on the unit interval, then LppL,ϕq with
„
L “ Σ{ , ϕ “ m, is order-isomorphically isometric to Lp. The isomorphism is
„
given by the extension of
a brA s ÞÝÑ a 1 .
i i „ i Ai
ÿi «ÿi ffa“.e.
4.2. Complementationandfunctionspaces. SupposethatLisadditionallyan
orthomodularlattice. ThenforeachM PLwemaydefinenorm-1linearprojections
P ,Q : LppL,ϕqÑLppL,ϕqasfollows. FirstwechooseaHamelbasist1bN u
M M α α
with N ďM or N ďMK of X (i.e. a maximal linearly independent family) and
α α
then we study operators X ÑX as follows :
P : a bN ÞÑ a bpM ^N q,
M i αi i αi
i i
ÿ ÿ
Q : a bN ÞÑ a bpMK^N q.
M i αi i αi
i i
ÿ ÿ
BytheorthomodularityofLwegetpM^N q^pMK^N q“0andpM^N q_
αi αi αi
pMK^N q“N above. Thus, by the construction of X we get
αi αi
P Q “Q P : X Ñt0u, P `Q “Id.
M M M M M M
Note that the preorder can be alternatively generated by replacing the sets A and
B with sets of the type A^M, B^M, A^MK and B^MK. Since P and Q
M M
are linear and preserve the conditions 1bA Ď 1bB and abA Ď bbA, we can
see that xĎy, x,y PX, if and only if P pxqĎP pyq and Q pxqĎQ pyq.
M M M M
Thus, applying P and Q to the condition ˘ a b A Ď |b | b B ,
M M i i i k k k
appearingin the definition of the semi-norm,we getthat both P and Q are bi-
M M
contractiveprojections. ThustheyextendasbicontracřtiveprojectionsřonLppL,ϕq.
GENERALIZED MEASURE ALGEBRAS 9
Moreover,if ϕpA^Mq`ϕpA^MKq“ϕpAq should hold for all APL, then
(4.1) }x}p “}P x}p `}Q x}p , 1ďpă8, xPLppL,ϕq.
LppL,ϕq M LppL,ϕq M LppL,ϕq
Theorem 4.3. Let 1 ď p ă 8, L be an orthomodular lattice with an order-
preserving map ϕ: L Ñ r0,1s, as above. Let pΩ,Σ,µq be a probability space and
Σ0 Ă Σ a Boolean algebra which σ-generates Σ. Let us assume that ϕpM _Nq “
ϕpMq ` ϕpNq whenever N ď MK. Suppose that there is an order-embedding
: Σ0 Ñ L such that µpMq “ ϕpMq for all M P Σ0. (We are not assuming
here that  respects the orthocomplementation operation.) Then
ai1Ai ÞÑ aibpAiq, Ai PΣ0
i i
ÿ ÿ
extends to a linear isometry LppΩ,Σ,µqÑLppL,ϕq.
Proof. First, by considering the order-embedding and the definition of X com-
patible with operations of simple functions on the Boolean algebra we can show
recursively that the values on the right side remain invariant in passing to a form
with A :s pairwise µ-almost disjoint. Thus it is easy to see from the construction
i
of X that the above linear map is in fact well defined.
Notethatthespecialsimplefunctionsoftheaboveform, a 1 PLppΩ,Σ,µq,
i i Ai
are dense in the set of all simple functions, since Σ0 σ-generates Σ. Consequently,
these special simple functions are dense in LppΩ,Σ,µq. Onřthe other hand, since
LppL,ϕq is complete by the construction, we are only required to verify that the
linear mapping appearing in the statement is norm-preserving.
Thus, pick an element a bpA q in X. Let εą0. Let |b |bB PX such
i i i j j j
that
ř ř
(4.2) ˘ a bpA qĎ |b |bB
i i j j
i j
ÿ ÿ
and
p
a bpA q ě |b |pϕpB q´ε.
i i j j
› ›
›ÿi › ÿj
› ›
Bythe beginningrem›arksofthe p›roofwemayassumewithoutlossofgenerality
› ›
thatA ^A “0abovefori‰j. Actually,wemayrefinetheaboverepresentations
i j
of elements of X. Namely, by using the orthomodular law recursively we obtain
that there are M1,...,Mk P L with Mj ď MiK for i ‰ j such that for each
K PtpAiq,Bj: i,ju there are ℓ1,ℓ2...,ℓl Pt1,...,ku such that K “ li“1Mℓi.
LetY ĂX bethek-dimensionalsubspaceofX spannedbytheelements1bM .
i
Ž
We may write (4.2) in the form
˘ c bM Ď d bM .
ℓ ℓ ℓ ℓ
ℓ ℓ
ÿ ÿ
Let P : Y Ñ r1bM s be the projection defined above. Since P is linear and
Mℓ ℓ Mℓ
order-preserving, it follows that |c | ď d . Because of the choice of M :s and the
ℓ ℓ ℓ
assumption on ϕ we observe that
1
p
d bM “ dpϕpM q .
ℓ ℓ ℓ ℓ
› › ˜ ¸
›ÿℓ › ÿℓ
› ›
› ›
› ›
10 JARNOTALPONEN
Clearly, putting d “ |c | is the optimal choice of the coefficients for this choice of
ℓ ℓ
pM q. We obtain
ℓ
p
a bpA q ď |c |pϕpM q“ |a |pϕpA q“ |a |pµpA q
i i ℓ ℓ i i i i
› ›
›ÿi › ÿℓ ÿi ÿi
› ›
where › ›
› › |c |pϕpM qď |b |pϕpB q.
ℓ ℓ j j
ℓ j
ÿ ÿ
Since εą0 was arbitrary it follows that
1
p
a bpA q “ |a |pµpA q .
i i i i
› › ˜ ¸
›ÿi › ÿi
Thus the investigated l››inear map is n››orm-preserving. (cid:3)
› ›
As pL,ϕq may contain such measure algebras in different dispositions, we con-
clude that the corresponding LppL,ϕq space is rich in a sense.
4.3. Disposition of lattices and submeasures. We note that
ϕ˚pAq:“}1bA}L1pL,ϕq, APL,
defines an order-preserving mapping ϕ˚: L Ñ r0,1s with ϕ˚ ď ϕ, ϕ˚p0q “ 0,
ϕ˚p1q“1 and
ϕ˚pA_Bqďϕ˚pAq`ϕ˚pBq
evenif ϕfails to satisfy the correspondingproperty,orfails to be order-preserving.
Also, if L is an orthomodular lattice then
ϕ˚pM _Nq“ϕ˚pMq`ϕ˚pNq
if N ď MK. Moreover, it holds that }x}LppL,ϕ˚q “ }x}LppL,ϕq for any mapping
ϕ: LÑr0,1sandany 1ďpă8. Therefore,is seems reasonableto assume that ϕ
satisfies the above properties of ϕ˚ in the first place.
Thefiniteexamplesoflatticescorrespondedtoℓppnqspacesanditseemsnatural
to ask whether Banach spaces of the type LppL,ϕq can be represented in terms of
classicalLppµq spaces (e.g. as subspaces or quotients). This appears aninteresting
problem for future research.
This calls for the following definition. Suppose that pA,µq is a probability mea-
sure algebra and pL,ϕq is a lattice endowed with a submeasure. Then we say that
A is an algebrification of L if one can form the following commuting diagram:
L ÝÝÝhÝÑ A
b b
Lpp§§L,ϕq ÝÝÝTÝÑ LppA§§,µq.
đ đ
whereh is a lattice homomorphism(A being understoodas a lattice inthe obvious
way),b denotesthe mappingAÞÑ1bAappearinginthe constructionofLppL,ϕq
spaces and T is an isometric isomorphism between Banach spaces. The definition
is not purely algebraic since it is affected by the (sub)measures of sets (e.g. h
necessarilymapsϕ-nullelementstotheminimalelementofA). Thisdefinitionalso
relies on the facts that the simple functions are dense in the spaces and that T is
an isometry, so that the diagram becomes rather rigid.