CC BY
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Владикавказский математический журнал 2019, Том 21, Выпуск 4, С. 56-62
УДК 510.898, 517.98
DOI 10.23671 /VNC.2019.21.44624
UNBOUNDED ORDER CONVERGENCE AND THE GORDON THEOREM*
E. Y. Emelyanov12, S. G. Gorokhova3 and S. S. Kutateladze2
1 Middle East Technical University,
1 Dumlupinar Bulvari, Ankara 06800, Turkey;
2 Sobolev Institute of Mathematics,
4 Koptyug prospect, Novosibirsk 630090, Russia; 3 Southern Mathematical Institute VSC RAS, 22 Marcus St., Vladikavkaz 362027, Russia E-mail: eduard@metu.edu.tr, emelanov@math.nsc.ru, lanagor71@gmail. com, sskut@math.nsc.ru
Dedicated to Professor E. I. Gordon on occasion of his 70th birthday
Abstract. The celebrated Gordon's theorem is a natural tool for dealing with universal completions of Archimedean vector lattices. Gordon's theorem allows us to clarify some recent results on unbounded order convergence. Applying the Gordon theorem, we demonstrate several facts on order convergence of sequences in Archimedean vector lattices. We present an elementary Boolean-Valued proof of the Gao-Grobler-Troitsky-Xanthos theorem saying that a sequence xn in an Archimedean vector lattice X is uo-null (uo-Cauchy) in X if and only if xn is o-null (o-convergent) in Xu. We also give elementary proof of the theorem, which is a result of contributions of several authors, saying that an Archimedean vector lattice is sequentially uo-complete if and only if it is a-universally complete. Furthermore, we provide a comprehensive solution to Azouzi's problem on characterization of an Archimedean vector lattice in which every uo-Cauchy net is o-convergent in its universal completion.
Key words: unbounded order convergence, universally complete vector lattice, Boolean valued analysis. Mathematical Subject Classification (2010): 03H05, 46S20, 46A40.
For citation: Emelyanov, E. Y., Gorokhova, S. G., Kutateladze, S. S. Unbounded Order Convergence and the Gordon Theorem, Vladikavkaz Math. J., 2019, vol. 21, no. 4, pp. 56-62. DOI: 10.23671/VNC.2019.21.44624.
1. Introduction
Throughout the paper, we let X stand for a vector lattice, and all vector lattices are assumed to be real and Archimedean. We refer to [1, 2] for the unexplained terminology and facts on vector lattices and start with recalling some definitions and results. A vector lattice X is said to be Dedekind (a-Dedekind) complete if each nonempty order bounded (countable) subset of X has a supremum. A Dedekind complete (a-Dedekind complete) vector lattice X is said to be universally (a-universally) complete if each nonempty pairwise disjoint (countable)
#The research was partially supported by the Science Support Foundation Program of the Siberian Branch of the Russian Academy of Sciences; № I.1.2, Project № 0314-2019-0005. © 2019 Emelyanov, E. Y., Gorokhova, S. G. and Kutateladze, S. S.
subset of X+ has a supremum. Clearly, each universally complete vector lattice has a weak unit. It is well known that X possesses Dedekind and universal completions unique up to lattice isomorphism which are denoted by Xs and Xu. We will always suppose that X C Xs C Xu, whereas Xs is an ideal in Xu.
A sublattice Y of X is said to be regular if ya l 0 in Y implies ya l 0 in X; while Y is order dense in X if for every 0 = x € X+ there exists y € Y satisfying 0 < y ^ x. Obviously, the ideals and order dense sublattices are regular. In what follows, we will freely use the regularity of X in Xu. Note also that X is atomic iff X is lattice isomorphic to an order dense sublattice of RC (cf. [1, Theorem 1.78]).
A net (xa)aeA in X o-converges to x if there exists a net (zY)7er in X satisfying zY l 0 and, for each y € r, there is aY € A with |xa — x| ^ zY for all a ^ aY. In this case we write xa -— x. This definition is used for instance in [2, 3]. Sometimes (in particular, see [1, 4, 5]) the slightly different definition of o-convergence appears: (xa)aeA o-converges to x € X if there is a net (za)aeA such that za l 0 and |xa — x| ^ za for all a. These two definitions agree in the case of order bounded nets in Dedekind complete vector lattices (cf. [3, Remark 2.2]). The article [6] contains a more details discussion of the definitions of o-convergence. By [7, Theorem 1] (cf. also [8, Theorem 2]), o-convergence in X is topological iff X is finite dimensional.
A net xa in X is said to be uo-convergent to x if |xa—x| Ay -— 0 for every y € X+. We write xa x. Following Nakano [9], uo-convergence is investigated as a generalization of almost everywhere convergence (see [3, 4, 10-18] and references therein). Note that o-convergence agrees with eventually order bounded uo-convergence. Furthermore, uo-convergence passes freely between X, Xs, and Xu [3, Theorem 3.2]. It was shown in [3, Corollary 3.5] that if e is a weak unit of X then xa —— x ^ |xa—x|Ae -— 0. By [3, Corollary 3.12] every uo-null sequence in X is o-null in Xu. This is untrue for arbitrary nets. By Theorem 4 below, or independently, by [18, Proposition 15.2], all uo-null nets in X are o-null in Xu if only if dim(X) < to. Although uo-convergence is not topological in many important cases (e.g., in Li[0,1] and in C[0,1]), it is topological in atomic vector lattices; see [7, Theorem 2].
A net xa is said to be o-Cauchy (uo-Cauchy) if the double net (xa — x^)(a,p) o-converges (uo-converges) to 0. Clearly, every o-Cauchy net is uo-Cauchy. In a Dedekind complete vector lattice with a weak unit e, a net xa is uo-Cauchy iff infa sup^ 7^a |x^ — xY| A e = 0 [13, Lemma 2.7]. It is well known that completeness with respect to o-convergence is equivalent to Dedekind completeness. By [3, Corollary 3.12], a sequence in X is uo-Cauchy in X iff it is o-convergent in Xu. As showed in Theorem 4, there is no net-version of the latter fact unless X is finite-dimensional. It was proved in [16, Theorem 3.9] (see also [15, Theorem 28]) that X is ^-universally complete iff X is sequentially uo-complete. In [15, Theorem 17], it was demonstrated that uo-completeness is equivalent to universal completeness. Thus, there is no need in any special investigation of (sequential) uo-completion.
The (always complete) Boolean algebra B(X) of all bands of X is called the base of X. If X has the projection property (e.g., if X is Dedekind complete), then B(X) can be identified with the Boolean algebra P(X) of all band projections in X and, if X has also a weak unit e, both B(X) and P(X) can be identified with the Boolean algebra C(e) of all fragments of e (cf. [2, Theorem 1.3.7(1)]).
2. Boolean-Valued Analysis and Unbounded Order Convergence
The classical Gordon's discovery [19, Theorem 2] (expressing the immanent connection between vector lattices and Boolean-valued analysis) reads shortly as follows: Each universally complete vector lattice is an interpretation of the reals R in an appropriate Boolean-valued
model Furthermore, each Archimedean vector lattice is an order dense ideal of the
descent of R within These facts are combined as follows (see [2, Theorems 8.1.2
and 8.1.6]):
Theorem 1 (Gordon's Theorem). Let X be an Archimedean vector lattice, while B = B(X) and R is the reals in the Boolean-valued model V(B). Then R 4 is a universally complete vector lattice including X as an order dense sublattice. Moreover,
bx ^ by
b < [x < yl (V b € B);(V x,y eRj).
By the Gordon Theorem, the universal completion X« of an Archimedean vector lattice X is the descent R 4 of the reals R in V(B(Xand the uniqueness of X« up to an order isomorphism follows from the uniqueness of R in V(B(X)) (see [2, 8.1.7]).
In [20] Kantorovich introduced Dedekind complete vector lattices and propounded his famous Heuristic Transfer Principle: The members of every Dedekind complete vector lattice are generalized reals (see [5] for further historical notes). This Kantorovich's motto was justified by the Gordon Theorem [19] published 42 years later in the same journal. The aim of the present paper, published another 42 years after [19], is to provide another illustration of the fruitfulness of the Gordon Theorem in exploring the theory of uo-convergence. To some extent, Archimedean vector lattices are commonly presented in the repertoire of the Boolean-valued orchestra, where the musicians are complete Boolean algebras and the orchestra director is the reals. To our knowledge, the present paper is a first attempt to apply Theorem 1 to uo-convergence. For the unexplained terminology and techniques of Boolean-valued analysis we refer the reader to [2, 5, 19, 21-25].
Let us turn to uo-convergence in X. Passing to X« = R 4, which has the weak unit 1, [1 is the multiplicative unit of R] = 1 we have, by [3, Corollary 3.5],
«O 0
xa ' 0
|xa|A 1 A 0 (xa € X).
By [2, 8.1.4], for every net s = (xa)a€A in R 4, the standard name AA of A in V (B) (see [2, p. 401]) is also directed and (s t) : Aa a R is a net in R (within V(B)); moreover,
b < [lim(s t) = x]
o — lim x(b) o s = x(b)x
for every b € B = B(X) = p(R 4.) and every x 6R| [2, 8.1.4 (3)]. Thus,
xa —> x ^ o — lim(|xa — x| A 1) = 0 ^
^ lim(|xa — x| A 1) = 0
In the case of a sequence, A = N, AA = NA = N [25, p. 330]), and hence
«o, n • ^ , , «o „ . _ ,
xn —> 0 in X ^^ xn —> 0 in R 4 ^^ ^ [lim |xn| = 0] = 1 ^ xn
Similarly,
lim (|xn| A 1) = 0 A 0 in R 4= Xu.
= 1.
=
(1)
(2)
xn is uo-Cauchy in X
o — lim |xk — xm| A 1 = 0
xn is uo-Cauchy in R 4
lim (|xk — xm| A 1) = 0
U-N2 = ( NxN)A3(k,m)^w
=
lim |xk — xm| = 0
= 1 ^^ [xn is Cauchy in R] = 1 [(3 z € R) lim xn = z] = 1
lim xn = z] = 1, for some z € R 4 ;
z € R 4= Xu
n
The last equivalence in (3) is actually due to Gordon [19, Theorem 4] (see also [22]). Clearly, (3) implies that Xu is always sequentially uo-complete. The equivalences of (2) are exactly the first part of the following theorem (see [3, Corollary 3.12]), whereas (3) is its second part.
Theorem 2 (Gao-Grobler-Troitsky-Xanthos). A sequence xn in an Archimedean vector lattice X is uo-null in X iff xn is o-null in Xu; while xn is uo-Cauchy in X iff xn is o-convergent in Xu.
The presented proof of Theorem 2 is based on the fundamental fact that the standard name NA of the naturals is the naturals N in It seems to be the main obstacle in
obtaining the net versions of this theorem which are indeed impossible due to Theorem 4.
The following theorem, stated and proved in [16, Theorem 3.9] and [15, Theorem 28], is a result of contributions of several authors (cf. also [3, Theorem 3.10], [3, Proposition 5.7], and [13, Proposition 2.8]).
Theorem 3. X is sequentially uo-complete iff X a-universally complete.
< For the "if part" we remark firstly that the fact that every (sequentially) uo-complete vector lattice is (a-) Dedekind complete is already contained in the proof of [3, Proposition 5.7]. Now, the (a-) lateral completeness of a (sequentially) uo-complete vector lattice follows from the o-summability of every (countable) order bounded disjoint family in a (a-) Dedekind complete vector lattice (cf. [2, 1.3.4]).
The "only if part" is exactly [3, Theorem 3.10]. >
It could be illustrative to present some Boolean-valued proof of Theorem 3 as well as a Boolean-valued proof of Azouzi's Theorem [15, Theorem 17] which yields the equivalence of uo-completeness and universal completeness.
We conclude our paper with the following theorem which provides, among other things, an answer to Azouzi's question [15, Problem 23].
Theorem 4. Let X be an Archimedean vector lattice. Then the following are equivalent:
(1) dim(X) < to;
(2) every uo-Cauchy net in X is eventually order bounded in Xu;
(3) every uo-Cauchy net in X is o-convergent in Xu;
(4) every uo-null net in X is o-null in Xu;
(5) every uo-null net in X is eventually order bounded in Xu;
(6) every uo-convergent net in X is eventually order bounded in Xu;
(7) every uo-convergent net in X is eventually order bounded in X;
(8) every uo-convergent net in X o-converges in Xu to the same limit;
(9) every uo-convergent net in Xu o-converges in Xu to the same limit.
Before proving the theorem, we include the following modification of [13, Example 2.6]. Given a nonempty subset A c X, prA stands for the band projection in Xu onto the band in Xu generated by A.
Example 1. In any infinite-dimensional Archimedean vector lattice X there exists a uo-null net which is not eventually order bounded in Xu.
As dim(X) = to, there is a sequence en of pairwise disjoint positive nonzero elements of X. Let N2 be the coordinatewise directed set of pairs of naturals. A net in X is defined via x(n,m) = (n V m) ■ e„Am. Since {x^m) : (n, m) € N2} C B{efc:k&i} and
lim pr{ek} (x(n,m)) = lim (n V m)pr{efc}(enAm)=0 (V k € N),
then x(nm) —— 0 as (n,m) — to (e.g., it can be seen by use of [3, Corollary 3.5.] for a weak unit u in Xu s.t. u A ek = ek for all k). If x(n m) is eventually order bounded by some y € Xu,
then for some (no, m0) € N2 we have y ^ x(n,m) (V (n, m) ^ (no, mo)). Since n A m0 = m0 and (n, m0) ^ (n0, m0) for n ^ n0 V m0, then
y ^ x(n,m0) = (n V m-0) ■ enAmc = (n V m-0) ■ emo = n ■ emo > 0 (V n ^ n0 V m0)
which is impossible. Therefore, the net x(n m) is not eventually order bounded in X«. < Proof of Theorem 4. (1) ^ (2), (4) ^ (5) ^ (6), and (7) ^ (6) are trivial.
(2) ^ (3): Suppose xa is uo-Cauchy in X. Then xa is uo-Cauchy in X« by [3, Theorem 3.2], because X is regular in Xu. It follows from [15, Theorem 17] that xa —A y for some y € Xu. Since xa is eventually order bounded in Xu by the assumption, then xa A y.
(3) ^ (4) follows since every uo-null net is uo-Cauchy, o-convergent implies uo-convergent, and the uo-limit of any uo-convergent net is unique.
(5) ^ (1) is Example 1.
(6) ^ (7) follows from the equivalence (6) ^ (1) because (1) (7) is obvious.
(1) ^ (8) follows from the equivalence (1) ^ (4), since (8) is equivalent to the fact that every uo-null net in X is o-null in X«.
(1) ^ (9) follows from (1) ^ (8) since (X«)« = X« and dim(X) < to iff dim(X«) < to. > While preparing this paper, we became aware of the still unpublished work [18] by Taylor which provides the construction [18, Proposition 15.2] similar to Example 1. The equivalence (1) ^ (8) of Theorem 4 is also contained in [18, Corollary 15.3].
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Received 4 Jule, 2019
Eduard Y. Emelyanov Middle East Technical University, 1 Dumlupinar Bulvari, Ankara 06800, Turkey, Professor;
Sobolev Institute of Mathematics, 4 Acad. Koptyug Pr., Novosibirsk 630090, Russia, Leading Scientific Researcher
E-mail: eduard@metu.edu.tr, emelanov@math.nsc.ru
Svetlana G. Gorokhova
Southern Mathematical Institute VSC RAS,
22 Marcus St., Vladikavkaz 362027, Russia,
Scientific Researcher
E-mail: lanagor71@gmail. com
Semen S. Kutateladze
Sobolev Institute of Mathematics,
4 Acad. Koptyug Pr., Novosibirsk 630090, Russia,
Professor, Senior Principal Scientific Officer
E-mail: sskut@math.nsc.ru
https://orcid.org/0000-0002-5306-2788
Vladikavkaz Mathematical Journal 2019, Volume 21, Issue 4, P. 56-62
НЕОГРАНИЧЕННАЯ ПОРЯДКОВАЯ СХОДИМОСТЬ И ТЕОРЕМА ГОРДОНА Э. Ю. Емельянов1'2, С. Г. Горохова3, С. С. Кутателадзе2
1 Ближневосточный технический университет, Турция, 06800, Анкара, Думлупинар Булвари, 1; 2 Институт математики им. С. Л. Соболева СО РАН, Россия, 630090, пр. Академика Коптюга, 4; 3 Южный математический институт — филиал ВНЦ РАН, Россия, 362027, Владикавказ, ул. Маркуса, 22 E-mail: eduard@metu.edu.tr, emelanov@math.nsc.ru, lanagor71@gmail.com, sskut@math.nsc.ru
Аннотация. Знаменитая теорема Гордона является естественным инструментом для построения универсального пополнения архимедовой векторной решетки. Она позволяет нам уточнить некоторые недавние результаты о неограниченной порядковой сходимости. Применяя теорему Гордона, мы демон-
стрируем несколько фактов о сходимость последовательностей. В частности, приводится элементарное доказательство теоремы Гао — Гроблера — Троицкого — Хантоса о том, что последовательность в архимедовой векторной решетке ио-сходится к нулю (соответственно, является ио-фундаментальной) тогда и только тогда когда она порядково сходится к нулю (соответственно, является порядково сходящейся) в универсальном пополнении этой решетки. В статье дается простое доказательство известной теоремы о том, что архимедова векторная решетка секвенциально ио-полна тогда и только тогда когда она ст-универсально полна. Кроме того в статье дается полное решение недавней проблемы Азози о конечномерности всякой архимедовой векторной решетки в которой любая uo-фундаментальная последовательность порядково сходится в универсальном пополнении этой решетки.
Ключевые слова: неограниченная порядковая сходимость, расширенное пространство Канторовича, булевозначный анализ.
Mathematical Subject Classification (2010): 03H05, 46S20, 46A40.
Образец цитирования: Emelyanov E. Y., Gorokhova S. G. and Kutateladze S. S. Unbounded Order Convergence and the Gordon Theorem // Владикавк. мат. журн.—2019.—Т. 21, № 4.—С. 56-62 (in English). DOI: 10.23671/VNC.2019.21.44624.
