CC BY
1UDC: 517.98 (YAK 517.98) MSC2010: 46B04, 46B42, 46E30, 46G10
LINEAR ISOMETRIES OF BANACH-KANTOROVICH LP-SPACES
© V. I. Chilin, G. B. Zakirova
Institute of Mathematics of the Academy of Sciences of the Republic of Uzbekistan 9, Universitet, Tashkent, 100174, Uzbekistan e-mail: [email protected]
Tashkent State Transport University 1, Odilxodjaev, Tashkent, 100167, Uzbekistan e-mail: [email protected]
Linear Isometries of Banaoh-Kantorovioh Lp-spaces. Chilin V. I., Zakirova G. B.
Abstract. Let B be a complete Boolean algebra, Q(B) be the Stone compact of B, and C^(Q(B)) be the commutative unital algebra of all continuous functions x : Q(B) ^ assuming possibly the values on nowhere-dense subsets of Q(B).
We consider the Banach-Kantorovich spaces Lp(B,m) C C^(Q(B)), associated with a measure m defined on B with the values in the algebra of measurable real functions. It is shown that in the case when the measure m has the Maharam property, for any linear isometry U : Lp(B,m) ^ Lp(B,m), 1 < p < x>, p = 2, there exist an injective normal homomorphisms T : CtXl(Q(B)) ^ CtXl(Q(B)) and an element y G Lp(B,m) such that U(x) = y • T(x) for all x G Lp(B, m).
Keywords: Banach-Kantorovich space, Maharam measure, vector integration, linear isometry.
Introduction
One of the important results in the theory of isometries of Banach function spaces is the theorem of J. Lamperti [5] , which describes the linear isometries of the spaces Lp(fi, 1 < p < <Xi,p = 2, for any measurable spaces (fi, with ^-finite measure
In the proof of this theorem, the property of preserving disjointness for such isometries was essentially used (see [2, Chapter 3]).
The development of the theory of integration for vector measures with values in vector lattices made it possible to construct Lp-spaces Lp(B,m), 1 < p < to, [3], [4], associated with the complete Boolean algebra B and a measure m defined on B with the values in the algebra L0 (fi, £, ^) of real measurable functions on the measure space (fi, £,
It is natural to expect that a similar description of linear isometries is valid for the linear isometries of the spaces Lp(B,m). In this paper, we establish the explicit form of
the linear isometries acting in the space Lp(B,m), 1 < p < to, p = 2, in the case when the measure m possesses the Maharam property.
We use the terminology and notation of the theory of Boolean algebras from [6], the theory of vector lattices from [7] and the theory of vector integration from [3].
1. Preliminaries
Let X be a real vector space, F be a complete vector lattice and let F+ = {f 6 F : f > 0}. A mapping || ■ || : X ^ F+ is called an F-valued norm if for any x,y 6 X, and real number A 6 r, the following holds: ||x|| =0 ^ x = 0; ||Ax|| = |A| ||x||; ||x + y|| < ||x|| + ||y||.
An F-valued norm || ■ || is said to be decomposable if for any fi, f2 6 F+ and x 6 X with ||x|| = fi + f2, there exist xi,x2 6 X such that x = xi + x2 and ||xj|| = fj, i = 1, 2.
A pair (X, || ■ ||) with F-valued norm is called a lattice normed space. If, in addition, the norm || ■ || is decomposable, then (X, || ■ ||) is called decomposable.
We say that a net {xa}aeA from a lattice normed space (X, || ■ ||) (bo)-converges to an element x 6 X if the net (||x — xa||)agA (o)-converges to zero in the lattice F, that is, there exists a net {fa}a^A C F+ such that fa ^ 0, and ||x — xa|| < fa for all a 6 A. A net (xa)agA C X is called (bo)-fundamental if the net (xa — x^)(a,^)gAxA (bo)-converges to zero.
A lattice normed space is called (bo)-complete if every (bo)-fundamental net in it (bo)-converges to an element of this space. A decomposable (bo)-complete lattice normed space is called a Banach-Kantorovich space.
The F-valued norm || ■ || on a vector lattice X is said to be monotonic if condition |x| < |y|, x, y 6 X, implies that ||x|| < ||y|. If a Banach-Kantorovich space (X, || ■ ||x) is a vector lattice and the norm || ■ ||x is monotonic, then it is called a Banach-Kantorovich lattice.
Let B be a complete Boolean algebra with zero 0 and unit 1. The supremum and infimum of a set {e, q} C B are denoted by e V q and e A q, and by Ce is denoted the complement to an element e. A non-empty set E of nonzero elements from B is said to be disjoint if e A q = 0 for any e, q 6 E, e = q.
A Boolean subalgebra A in B is called regular if sup E 6 A and inf E 6 A for any subset E C A. Every regular Boolean subalgebra in B is a complete Boolean algebra.
Let Q(B) be the Stone compact of B, and let L0(B) := C^(Q(B)) be the commutative unital algebra over the field r of all continuous functions x : Q(B) ^ [—to, assuming possibly the values ±to on nowhere-dense subsets of Q(B) (see, for example,
, w , f -777, ifx(t) = 0,
i(x)(t) = ^ -(t) v A ; I 0, ifx(t) =0.
[3, Chapter 1, Section 1.4.2], [7, Chapter V]). With respect to the partial order
f < g & g(t) - f (t) > 0 for all t e Q(B) \ (f-1(±to) u g-1(±^)),
the algebra L0(B) is a complete order vector lattice, and the set V of all idempotents
in L0(B) is a complete Boolean algebra with respect to the partial order induced from
L0(B). In addition, V is isomorphic to the Boolean algebra B (see, for example, [7,
Chapter V]). It is known that the set C(Q(B)) of all continuous real functions on Q(B)
is a subalgebra in L0(B) and C(Q(B)) is a Banach space with respect to the uniform
norm ||x||^ = sup |x(t)|.
teQ(B)
We denote by s(x) := sup{|x| > n-1} the support of an element x e L0(B), where
n>1
{|x| > A} is the characteristic function \ex of the set E\ which is the closure of the set {t e Q(B) : |x(t)| > A}, A e R.
For any nonzero x e L0(B) define i(x) to be the inverse element to x on its support,
i.e.
0, ifx(t) = 0. It is clear that i(x) e L0(B) and i(x) ■ x = s(x).
Let (fi, E, be a measurable space with ^-finite measure
and let L0(fi) be the
algebra of all classes of almost everywhere equal real-valued measurable functions on (fi, E,^). With respect to the partial order f < g & g — f > 0 almost everywhere, the algebra L0(fi) is a complete order vector lattice, and the set B (fi) of all idempotents from L0(fi) is a complete Boolean algebra with respect to the partial order induced from L0(fi).
Since ^ is a ^-finite measure, it follows that B(fi) is a Boolean algebra of countable type, that is, any subset E C B(fi) of nonzero pairwise disjoint elements is at most countable. Thus for any increasing net xa ^ x e L0(fi), {xa}aeA C L0(fi) there exists a sequence a7 < a2 < ■ ■ ■ < an < ... such that xan t x (see, for example, [7, Chapter VI, §2]).
A mapping m : B ^ L0 (fi) is called a L0(fi)-valued measure if it satisfies the following conditions:
1) m(e) > 0 for all e e B;
2) m(e V g) = m(e) + m(g), for any e, g e B, e A g = 0;
3) m(ea) I 0 for any net ea | 0, {ea} C B.
A measure m is said to be strictly positive if m(e) = 0 implies e = 0. In this case, B is a Boolean algebra of countable type.
A strictly positive L°(fi)-valued measure m is said to be decomposable if for any e G B and a decomposition m(e) = / + /2, /i,/2 G L°(fi)+, there exist e1,e2 G B, such that e = ex V e2, and m(e«) = /¿, i = 1,2. A measure m is decomposable if and only if it is a Maharam measure, that is, for any e G B, 0 < / < m(e), / G L°(fi), there exists q G B, q < e, such that m(q) = / (see [8]).
We use the following important property of the Maharam measure.
Proposition 1. [8, Proposition 3.2]. For each L°(fi)-valued Maharam measure m : B ^ L°(fi) there exists a unique injective completely additive homomorphism p : B (fi) ^ B such that p(B (fi)) is a regular Boolean subalgebra of B, and
m(p(q)e) = qm(e) for all q G B(fi), e G B.
Let m : B ^ L°(fi) be a Maharam measure. We identify B with the Boolean algebra of idempotents in L°(B), i.e. we assume that B C L°(B). By Proposition 1, there exists a regular Boolean subalgebra V(m) in B and an isomorphism p from B (fi) onto V(m) such that m(p(q)e) = qm(e) for all q G B(fi), e G B. In this case, the algebra L°(fi) is identified with the algebra L°(V(m)) = C^(Q(V(m))) (the corresponding isomorphism will also be denoted by p). Thus, the algebra C^(Q(V(m))) can be considered as a subalgebra and as a regular vector sublattice in L°(B) = C^(Q(B)) (this means that the exact upper and lower bounds for bounded subsets of L°(V(m)) are the same in L°(B) and in L°(V(m))). In particular, L°(B) is an L°(V(m)) -module.
n
Denote by S(B) the vector sublattice in L°(B) of all step elements x = ^2, aie«,
i=1
where a« G R, e« G B, e« ■ ej = 0, i, j = 1,..., n. The equality
/n
xdm := a«m(e«) (x G S(B)) i=1
uniquely defines a linear operator Im : S(B) ^ L°(fi).
A positive element x G L°(B)+ is called m-integrable if there exists a sequence (xn}^=1 C S(B), 0 < xn t x, such that there is a supremum supIm(xn) in the lattice
n>1
L°(fi). In this case, the integral of the element x with respect to the measure m is defined by
Im(x) := / xdm := sup/m(xn).
n>1
It is known that the definition of the integral Im(x) does not depend on the choice of
a sequence (xn}^=1 C S(B), 0 < xn t x, for which there exists supIm(xn) (see, for n=1 n>1
example, [3, 6.1.3]).
An element x G L°(B) is called m-integrable if its positive x+ and negative x-parts are m-integrable. The set of all m-integrable elements is denoted by L1(B,m), and for every x G L1(B,m) we have
J xdm := J x+dm — J x-dm.
If ||x||1im := J |x|dm,x G L1(B,m), then the pair (L1(B,m), ||.||1im) is a lattice normed space over L°(fi) [3, 6.1.3]. Moreover, in the case when m : B 4 L°(fi) is a Maharam measure, the pair (L1(B,m), ||x||1im) is a Banach-Kantorovich space. In addition,
L°(V(m)) ■ L1 (B,m) C L1(B,m), J(p(a)x)dm = a J xdm, | J xdm| < J |x|dm,
for all x G L1(B,m), a G L°(fi) [3, Theorem 6.1.10]. Let p G [1, to), and let
Lp(B,m) = {x G L°(B) : |x|p G L1(B,m)},
||x|p,m := [J |x|pdm] p, x G Lp(B,m).
It is known that for a Maharam measure m the pair (L^(B,m), ||x||Pim) is a Banach-Kantorovich space [4, 4.2.2]. In addition,
p(a)x G Lp(B,m) for all x G Lp(B,m), a G L°(fi), 1 < p < to,
and ||p(a)x|p,m = |a||x|p,m.
We use the following important property of order continuity of the norm || ■ ||p,m.
Proposition 2. If {xn} C LP(B,m) and xn I 0, then ||xn||p,m I 0.
Proof. Since xp ^ 0, it follows that /m(xp) ^ 0 (see the Dominated Convergence Theorem 6.1.5 [3]). Thus ||xn||p,m ^ 0. □
Using Proposition 2 for any sequence
{xn} C Lp(B, m), xn t x G Lp(B,m),
we get the convergence ||x — xn||p,m ^ 0.
Recall that writing yn y for elements yn, y, n =1, 2,..., from a vector lattice F means that the sequence {yn} (o)-converges to the element y, i.e. there exist sequences an, bn G F such that an t y, bn i y and an < yn < bn for all n = 1, 2,... It is known that if yn -—4 y then Ayn Ay for any A G r and |yn — y| 0 (see, for example, [7, Chapter III, §7]).
Proposition 3. If x, xn G L°(B), n = 1, 2,..., and xn | 0, then |x| ■ xn | 0.
Proof. It is clear that |x| ■ xn ^ y e L0(B), y > 0. If y = 0 then there exists e > 0 such that e = {y > e} = 0. Since
e ■ |x| ■ xn > e ■ y > ee,
it follows that e ■ |x| = 0. Thus there exists 5 > 0 such that p = {e ■ |x| > 5} = 0 and p < e. Therefore, i(p|x|) = 0 and
s(i(p|x|)) ■ xn = i(p|x|) ■ p|x| ■ xn > i(p|x|) ■ ep = 0 for all n =1, 2,...,
which is impossible, because
0 < s(i(p|x|)) ■ xn < xn I 0. Thus y = 0, that is, |x| ■ xn ^ 0. □
To describe the isometries of Banach-Kantorovich Lp-spaces, we need the notion of convergence in measure generated by a Maharam measure m.
Definition 1. We say that the sequence {xn}^=1 C L0(B) converges in measure m to an element x e L0(B) (notation xn —^ x) if
m({|xn — x| > e}) 0 for all e > 0. We need the following properties of measure convergence. Proposition 4. (i). If xn,x,y e L0(B), n = 1, 2..., and xn —^ x, xn —^ y, then
x = y.
(ii). If xn,x e L0(B), n = 1, 2 ..., and xn x, then xn —^ x.
(o)
(iii). If xn, x e L^B, m), n =1, 2 ..., 1 < p< to, and ||xn — x|p,m —^ 0, then
m
xn ^ x.
Proof. (i). Let us first show that for any a, b e L0(B), e > 0, the inequality
{|a + b| > e} < {|a| > 2} V {|b| > 2} (1)
is valid.
Let e = {|a| > f}, q = {|b| > |}. Since Ce = {|a| < |}, Cq = {|b| < |}, it follows that
(Ce A Cq) ■ |a + b| < (Ce A Cq) ■ |a| + (Ce A Cq) ■ |b| < 2 ■ | ■ (Ce A Cq) = e ■ (Ce A Cq). Thus {|a + b| < e} > Ce A Cq, and therefore
{|a + b| > e} = C({|a + b| < e}) < C(Ce A Cq) = e V q.
Let now xn, x, y e L0(B), n = 1, 2 ..., and xn —^ x, xn —^ y. Let
ek = {|x — y| > —}, k = 1, 2,..., and e = sup ek.
k fc>i
If pn,fc = {|xn — x| > fk}, qn,fc = {|xn — y| > fk} then using the equality x — y = (xn — y) + (x — xn) and the inequality (1), we obtain that ek < pn,k V qn,k. Thus
0 < m(efc) < m(pn,fc) + m(qn,fc) 0 as n ^ to.
Consequently, m(ek) = 0, that is, ek = 0 for any k = 1, 2,.... Therefore, e = sup ek = 0,
k>1
which implies that x = y.
(ii). Since xn x, it follows that |xn — x| 0. Thus there exists 0 < zn e L0(B) such that
|xn x| < zn 4" 0. Let e > 0 and en = {|xn — x| > e}. Using inequalities
0 < een < en|xn x| < enzn < zn 0,
we obtain that en —°4 0. Thus
qn = sup ek | 0 and m(en) < m(qn) | 0.
k>n
(o)
Consequently, m({|xn — x| > e}) = m(en) —> 0 for any e > 0, that is, x* n ^ .
(iii). Since
(o)
|xn — x|pdm = ||xn — x|pp m-► 0,
p,m
it follows that there exists {an}^=7 C L0(fi) such that an | 0 and
J |xn — x|pdm < an for all n =1, 2 ... Let e > 0 and en = {|xn — x| > e}. Using the inequalities
epm(en) = ep J endm < J en|xn — x|pdm < J |xn — x|pdm < an, we obtain that
0 < m(en) < —«n I 0, that is, m(en) 4 0. Thus xn —^ x. □
2. Description of isometries of Banach-Kantorovich Lp-spaces
Let B be a complete Boolean algebra and let m : B ^ be a Maharam
measure for which m(1B) = The algebra is identified with the algebra
L°(V(m)) = Cœ(Q(V(m))), which is a subalgebra and a regular vector sublattice in L°(B) = CM(Q(B)).
Repeating the proof of Theorem 2.1 and Corollary 2.1 from [5] we have the following version of these statements for the Magaram measure m.
Theorem 1. Let x,y G Lp(B,m), 1 < p < to, p = 2. Then the equality
I|X + y||P,m + ||X - y||P,m = 2(||x||P,m + ||yyP,m)
is valid if and only if x ■ y = 0.
Let U : L^(B,m) ^ Lp(B,m), 1 < p < to, p = 2, be a linear isometry, that is, U is a linear map and ||U(x)||p,m = ||x||p,m for all x G L^(B,m). If x,y G LP(B,m) and x ■ y = 0, then by Theorem 1 we have
Il U (x) + U (y)||p,m + || U (x) - U (y)||p,m = || U (x + y)||p,m + || U (x - y)||p,m =
= llx + y||p,m + ||x - y|p,m = 2(||x||Pm + ||y|p,m) = 2(||U(x)|p,m + ||U(y)||prJ. Again using Theorem 1, we get U(x) ■ U(y) = 0. Thus, we have the following
Proposition 5. If U : Lp(B,m) ^ LP(B,m), 1 < p < to, p = 2, is a linear isometry and x, y G L^B, m), x ■ y = 0, then U (x) ■ U (y) = 0.
A linear operator T : L°(B) ^ L°(B) is called a homomorphism if T (x ■ y) = T (x) ■ T (y) for all x,y G L°(B ). If 0 < x G L° (B), then T (x) = T (y7^) ■ T (y7^) > 0, that is, the homomorphism T : L°(B ) ^ L°(B ) is a positive operator.
A positive linear operator T : L°(B) ^ L° (B) is called normal(resp., completely additive), if T(supxa) = supT(xa) for any increasing net
a a
0 < xa t x G L°(B), {xa} C L°(B) (resp., T(J^ ej) = sup T(e^), for every family of idempotents
igl agA jga
{ej}ig/ C B, ejej = 0, i = j, i, j G I,
where A = {a} is the net of all finite subsets of I, ordered by inclusion).
It is clear that the normality property for a positive linear operator implies that this operator is a completely additive one. In the case when T : L° (B) ^ L°(B) is a homomorphism, the converse implication is also valid.
Theorem 2. ([1, Theorem 4]). The homomorphism T : L0(B) ^ L0(B) is a normal operator if and only if T is completely additive operator.
Since m(1B) = and ^ is a ^-finite measure, it follows that there exists a sequence
{en} C B(fi) such that
men) < to, enek = 0, n = k, n, k =1, 2,..., sup en = 1,
n>1
and {en ■ m(q) : q e B} C L7(fi, E,^). Consequently, the function
oo „
v(q) = ^ / enm(q)d^ n=7 n
is a ^-finite numerical measure on the Boolean algebra B, in particular, B is a Boolean algebra of countable type. In this case, in the definition of normality (resp., completely additivity) of a positive linear operator T : L0(B) ^ L0(B), instead of an increasing net {xa} C L0(B) (resp., family of idempotents {ei}ig1 C B^e^ = 0, i = j, i,j e I), one should take a sequences {xn} C L0(B) (resp., countable family of idempotents {ei}i>i C B, e^ej = 0, i = j, i, j = 1, 2....)
The following Theorem is a vector-valued version of Lamperti's theorem (Theorem 3.1 [5]) for the Banach-Kantorovich space (Lp(B, m), || ■ ||p,m).
Theorem 3. Let U : Lp(B, m) ^ Lp(B,m), 1 < p < to, p = 2, be a linear isometry. Then there exist a normal homomorphism T : L0(B) ^ L0(B) and y e Lp(B,m) such that U(x) = y ■ T(x) for all x e Lp(B, m).
Proof. Define the mapping <£> : B ^ B, by setting <£>(e) = s(U(e)), e e B, where s(U(e)) is the support of the element U(e) e Lp(B,m). It is clear that <£>(e) = 0 if and only if e = 0. If e,g e B and e ■ g = 0, then U(e) ■ U(g) = 0 (see Proposition 5). Consequently, <£>(e) ■ <£>(g) = 0 and
^(e V g) = s(U(e + g)) = s(U(e) + U(g)) = s(U(e)) + s(U(g)) = = ^(e) + ^(g) = ^(e) V ^(g).
Using mathematical induction, we obtain that
<£>( sup ei) = sup ^(ej)
1<i<n 1<i<n
for any finite set of pairwise disjoint idempotents {ej}n=1 C B.
Let {ej}°= 1 C B be a countable set of pairwise disjoint idempotents and let gn = sup ej, n = 1,2,... Then gn t supgn = sup ej := e, and by Proposition 2 we get
1<j<n n>1 j>1
n
|| y^ U(ej) — U(e)||p,m = IIU(gn) — u(e)||p,m = ||U(gn — e)||p,m = ||gn — e||p,m i 0, j=1
that is, U(ej) = U(e) (the series converges with respect to the || ■ ||p,m-norm). Since
j=1
U(ej) ■ U(e3-) = 0, i = j, i, j = 1, 2 ..., it follows that
sup p(ej)) = sup s(U(ej)) = s(U(e)) = p(e).
j>1 j>1
Moreover,
p(1) = p(e + Ce) = s(U (e + Ce)) = s(U (e) + U (Ce)) =
= s(U (e)) + s(U (Ce)) = p(e) + p(Ce),
that is, p(Ce) = p(1) — p(e).
Thus, the mapping p : B 4 B satisfies all the properties of a regular isomorphism from Definition 3.2.3 [2, Chapter III, §3.2], so p is an injective completely additive Boolean homomorphism [2, Chapter III, §3.2, Remarks 3.2.4].
Using now Theorems 3 and 4 from [1], we get that there exists a unique injective normal homomorphism T : L°(B) 4 L°(B) such that T(e) = p(e) for all e G B. In addition, the restriction A = T|c(q(b)) is || ■ ¡^-continuous injective homomorphism from C(Q(B)) into C(Q(B)). For any e G B we have that
U (e) = U (1 — (1 — e))s(U (e)) = U (1)p(e) — U (1 — e)p(1 — e)p(e) =
= U (1)p(e) — U (1 — e)p((1 — e)e) = U (1)p(e), that is, U(e) = U(1)p(e). If
n
x = ajej G S(B) C C(Q(B)), ej G B, ejej = 0, i = j, i, j = 1,... ,n, j=1
is a step element, then
nn
U(x) = ^ ajU(ej) = U(1) ^ atT(ej) = U(1) ■ T(x). j=1 j=1
Let x G C(Q(B)), and let {xn}^=1 C S(B) be a sequence of step elements such that ||xn — x||^ 4 0. Then
||T(xn) — T(x)|U = ||A(xn) — A(x)|U 4 0.
Therefore
||U(xn) - U(1)T(x)||p,m = ||U(1) ■ T(xn) - U(1)T(x)||p,m = ||U(1) ■ T(xn - x)||p,m <
< ||U (1) ||p,m ■ ||T (xn - x)|U^ 0.
Since
IIU(xn) - U(x)|p,m = IIU(xn - x)|p,m = ||xn - x|p,m < ||xn - ^ 0,
it follows that U(x) = U(1)T(x).
Let us show that Ux = U(1) ■ T(x) holds for all x G Lp(B,m). It suffices to check this equality for all 0 < x G Lp(B,m). Let 0 < x G Lp(B,m), and let 0 < xn = x ■ X{o<x<ni> G C(Q(B)). Since xn t x and the norm || ■ ||p,m is order continuous norm (Proposition 2), it follows that
IIU(1) ■ T(xn) - U(x)|p,m = ||U(xn) - U(x)||p,m = ||U(xn - x)|p,m = ||xn - x|p,m | 0.
Thus, U(1) ■ T(xn) - U(x) 0 (Proposition 4 (iii)), that is, U(1) ■ T(xn) U(x).
Using the normality of the homomorphism T we obtain that T(xn) t T(x). Consequently (Proposition 3),
|U(1) ■ T(xn) - U(1) ■ T(x)| = |U(1)| ■ |T(xn) - T(x)| I 0,
that is, U(1) ■ T(xn) U(1) ■ T(x). Using Proposition 4 (ii), we obtain that U(1) ■ T(xn) U(1) ■ T(x). Thus, U(x) = U(1) ■ T(x) (Proposition 4 (i)). Therefore, Ux = U(1) ■ T(x) for all x G LP(B,m). □
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