b-Property of sublattices in vector lattices
aa r X i v : . [ m a t h . F A ] F e b b -Property of sublattices in vector lattices March 1, 2021
S¸afak Alpay , Svetlana Gorokhova Abstract
We study b -property of a sublattice (or an order ideal) F of a vectorlattice E . In particular, b -property of E in E δ , the Dedekind completionof E , b -property of E in E u , the universal completion of E , and b -propertyof E in ˆ E (ˆ τ ), the completion of E . keywords: vector lattice, universal completion, Dedekind completion, b -property, local solid vector lattice MSC2020:
Vector lattices considered here are all real and Archime-dean. Vector topologies are assumed to be Hausdorff.
Definition 1.
A sublattice F of a vector lattice E is saidto have b -property in E , if x α is a net in F + and 0 ≤ x α ↑≤ e for some e ∈ E , then there exists f ∈ F with0 ≤ x α ↑≤ f .Recall that a subset F of E is said to be majorizing in E if, for each 0 < e ∈ E , there exists f ∈ F with 0 ≤ e ≤ f .A subset U of a vector lattice (VL) is called solid if | u | ≤ | v | , v ∈ U , imply u ∈ U . A linear topology τ on a L E is called locally solid if τ has a base of zero consistingof solid sets.A locally solid VL E (LSVL) satisfies the Lebesgue prop-erty if x α ↓ E implies x α τ → E ( τ ) satisfies the Fatou property if τ has a baseof zero consisting of solid and order closed sets.A sublattice F in a VL E is regular if inf A is the sameas in F and E whenever A ⊂ F whose infimum exists in F . Ideals are regular in E . E is called σ -laterally complete if the supremum of everydisjoint sequence exists in E + and laterally complete ifsupremum of every disjoint subset in E + exists in E . Example 1. [1, p.198] Let X be a topological space. Afunction f : X → R is called a step function if there ex-ists a collection of mutually disjoint subsets { V i } of X suchthat S i V i = X , f is constant on each V i , and f ∈ C ∞ ( X ).Let S ∞ ( X ) be the space of step functions on an extremallydisconnected topological space X . Then S ∞ ( X ) is a lat-erally complete VL.Lateral completion E λ of a VL E is defined to be the in-tersection of all laterally complete vector lattices between E and E u .Universal completion ( σ -universal completion) of a VL E is a laterally ( σ -laterally) and Dedekind complete (De-dekind σ -complete) vector lattice E u (resp., E s ) whichcontains E as an order dense sublattice. Every VL E hasa unique universal completion [1, Theorem 7.21] Example 2.
Let X be an extremally disconnected topo-logical space. C ∞ ( X ), the space of all extended contin-uous functions on X with the usual algebraic and latticeoperations is a universally complete VL. net ( x α ) α ∈ A in a VL E is order convergent to x ∈ E ifthere exists a net ( x β ) β ∈ B , possibly over a different indexset, such that x β ↓ β ∈ B , there exists α ∈ A with | x α − x | ≤ x β for all α ≥ α . In this case wewrite x α o → x .A net x α in E uo -converges to x ∈ E if | x α − x | ∧ u o → u ∈ E + . In this case we write x α uo → x .Let E ( τ ) be a LSVL. A net x α in E is uτ -convergent to x ∈ E if | x α − x | ∧ u τ → u ∈ E + . A net x α in E is called order Cauchy ( uo -Cauchy ) if the doubly indexednet ( x α , x α ′ ) ( α,α ′ ) is order convergent ( uo -convergent) tozero. E ( τ ) is called uo -complete if every uo -Cauchy net is uo -convergent in E .The b -property of a VL E was defined in [2] as: a VL E has b -property if every subset A in E which is orderbounded in ( E ∼ ) ∼ , remains to be order bounded in E .Equivalently, a VL E has b -property iff each net x α in E ,with 0 ≤ x α ↑ x for some x ∈ ( E ∼ ) ∼ , is order bounded in E ([2]). Example 3.
Every perfect VL, and therefore every orderdual, have the b -property. Every reflexive BL and every KB -space have b -property [2, 3, 4, 5]. On the other hand,by considering the basis vectors e n in c , we see that c does not have the b -property in l ∞ .Let us note that Fremlin had considered subsets of aVL E that are order bounded in the universal completion E u of E . He proved that if E is a σ -Dedekind completeVL then E is σ -laterally complete iff E has the countable b -property in E u [1, Theorem 7.38]. Example 4.
Each projection band F in a vector lattice E has b -property in E . In particular, every band in a edekind complete vector lattice has b -property. An ele-ment u in a VL E is called an atom if whenever v ∧ w = 0,0 ≤ v ≤ u , and 0 ≤ w ≤ u imply either v = 0 or w = 0. If x is an atom in E , the principal band B x generated by x is a projection band and therefore has b -property in E . Example 5.
Every majorizing sublattice F has b -propertyin E . Let 0 ≤ x α ↑≤ e for some net x α ⊆ F , e ∈ E . As F is majorizing, there exists f ∈ F with e ≤ f . Then0 ≤ x α ≤ f . Since it is well-known that E is majoring in E δ , E has b -property in E δ . Example 6.
Every order ideal F in a vector lattice E with b -property in E is a band of E . Indeed, let x α be anet in F such that 0 ≤ x α ↑ e ∈ E , then x α is b -boundedin E and by the b -property of F , there exists f ∈ F with0 ≤ x α ≤ f . As x α ↑ e , we have 0 ≤ e ≤ f and as F is anideal, e ∈ F . Example 7.
Let F ⊆ E be a sublattice of E and I ( F )be the ideal generated by F in E . Then F has b -propertyin I ( F ). Having b -property is transitive: if E ⊆ F ⊆ G are sublattices of a VL X , then E has b -property in F ,and if F has b -property in G , then E has b -property in G .If E has b -property in G , then E has b -property in everysublattice of G containing E as a sublattice. Example 8.
Let ( E, k . k ) be a Banach lattice with ordercontinuous norm and F ⊆ E be a norm-closed sublattice.Let x n be a b -bounded sequence in F such that 0 ≤ x n ≤ e for some e ∈ E . Then x n is norm-Cauchy and is conver-gent to some x ∈ E . As F is norm-closed, x ∈ F andconsequently x n ↑ x . That is to say F has countable b -property in E . Order continuity of the ambient space is ssential in this example, if one takes E = l ∞ and F = c ,Then by considering the sequence e n in c , we see that c has no b -property in l ∞ . Example 9.
Generalizing Example 8, let E ( τ ) be LSVLwith Lebesgue property. Then every τ -closed order idealhas b -property in E ( τ ). This is because every τ -closedideal is a band and, as E ( τ ) is Dedekind complete, it is aprojection band. Example 10.
Given a LSVL E ( τ ), let us denote by E λ its lateral completion and E u its universal completion.Then the equality ( E λ ) δ = ( E δ ) λ = E u (see [1, Exer.10on p.213]) shows that each laterally complete E ( τ ) has b -property in its universal completion. Example 11. If E ( τ ) is a laterally complete LSVL, thenit has the projection property and every band on E has b -property. Furthermore, a subset A ⊂ E + of a later-ally complete VL E is order bounded in E u iff it is orderbounded in E by [1, Theorem 7.14]Let us observe that all Lebesgue topologies on a LSVL E ( τ ) induce the same topology on order bounded subsetsof E . Therefore, if F is a sublattice of E then on allsubsets of F with b -property in E all Lebesgue topologieson E induce the same topology. Example 12.
Let F be an order dense sublattice of avector lattice E . If F is laterally complete in its ownright, then F majorizes E and therefore has b -property in E . We refer to [1, 10] for all undefined terms. Main results
Lemma 1.
Let F be a sublattice of a LSVL E ( τ ) . Theneach b -bounded subset B of F is τ -bounded subset withrespect to induced topology on F .Proof. To say that B is b -bounded is to say that B isorder bounded in E . So, if U is a neighborhood of 0 in τ then B ⊆ λU for some λ >
0. Then B ⊆ λU ∩ F = λ ( U ∩ F ). Lemma 2.
Let E be a vector lattice and F be an orderdense sublattice of E . Then TFAE:i ) F has b -property in E ;ii ) F is majorizing in E .Proof. i ) = ⇒ ii ): Let 0 ≤ x ∈ E be arbitrary, as F is order dense in E , there exists a net x α in F such that0 ≤ x α ↑ x . As x α is b -bounded by assumption, thereexists x ∈ F + with 0 ≤ x α ≤ x for all α , as x α ↑ x , wehave x ≤ x and F is majorizing. ii ) = ⇒ i ): Let x α be a net in F with 0 ≤ x α ↑≤ x for some x ∈ E . Since F is assumed to be majorizing E ,there exists y ∈ F with x ≤ y . Consequently, 0 ≤ x α ≤ y and F has b -property in E .This yields: E has b -property in E u iff E is majorizingin E u . We also have if E ( τ ) is a LSVL where E is anideal of ˆ E (ˆ τ ), where ˆ E is the completion. Then E has b -property in ˆ E (ˆ τ ).On the other hand, if E ( τ ) is a LSVL with Fatou prop-erty, then every increasing τ -bounded net of E + is orderbounded in E u i.e. every increasing τ -bounded net of E + is b -bounded in E u by [1, Theorem 7.51]The following property was introduced in [8] and [9]. efinition 2. A locally solid vector lattice E ( τ ) is called boundedly order bounded (BOB) if every τ -bounded net in E + is order bounded in E .We show BOB is equivalent to b -property if the LSVL E ( τ ) has Fatou property. Lemma 3.
Let E ( τ ) be a LSVL with Fatou property.Then E has b -property in E u iff E is BOB.Proof. Suppose E is BOB and x α be a net in E with0 ≤ x α ↑≤ x for some x ∈ E u . Then, by Lemma 1, x α is τ -bounded in E and, by assumption that E is BOB,0 ≤ x α ≤ x for some x ∈ E .Conversely, suppose x α is τ -bounded increasing net in E + , then by [1, Theorem 7.50], x α is order bounded in E u .Thus by b -property of E in E u , there exists x ∈ E with0 ≤ x α ≤ x and E ( τ ) is BOB.[1, Theorem 7.49] shows that, in a laterally σ -completeLSVL E ( τ ), every disjoint sequence in E + converges tozero with respect to any LS topology on E . We show asimilar result. The proof is similar. Proposition 1.
Let E ( τ ) be a LSVL which has countable b -property in its σ -lateral completion. Then every disjointsequence in E + converges to zero with respect to any lo-cally solid topology on E . In particular, every locally solidtopology on E has the pre-Lebesgue property.Proof. Let x n be a disjoint sequence in E + . Then nx n isalso a disjoint sequence in E + . Then x = W ∞ n =1 nx n existsin the σ -lateral completion, and we have 0 ≤ x n ≤ n x forall n . Countable b -property of E in its lateral completionyields a vector e ∈ E with 0 ≤ x n ≤ n e for all n . Thus x n onverges to zero with respect to any locally solid topologyon E . Corollary 1.
Let E ( τ ) be a LSVL with Lebesgue property.If E has countable b -property in its σ -lateral completionthen the topological completion ˆ E of E ( τ ) is E u .Proof. It follows from [1, Theorem 7.51].
Proposition 2.
A laterally complete vector lattice E has b -property in every vector lattice which contains E as anorder dense sublattice.Proof. In this case E majorizes the vector lattice thatcontains it. The result now follows from [1, Theorem 7.15].In [11, Proposition 2.22] it is proved that if E ( τ ) isa LSVL with Lebesgue topology, then a sublattice F of E is uτ -closed in E iff it is τ -closed. It was asked in[11, Question 2.24] whether Lebesgue assumption couldbe removed. The next result yields an answer utilizing b -property. Proposition 3.
Let F be an order ideal of a LSVL E ( τ ) .If F has b -property in E , then F is uτ -closed iff it is τ -closed in E .Proof. As uτ is coarser than τ , the forward implication isclear.Now, suppose F is τ -closed and y α is a net in F with y α uτ → x for some x ∈ E . We will show x ∈ F . The latticeoperations are uτ -continuous, so that y ± α uτ → x . Therefore, LOG we may assume 0 ≤ y α for all α . Let z ∈ E + bearbitrary, then | y α ∧ z − x ∧ z | ≤ | y α − x | ∧ z τ → . Since 0 ≤ y α ∧ x ≤ y α for all α , and F is an order ideal,we have y α ∧ x ∈ F for all α and y α ∧ x τ → x ∧ x .Take y ∈ F , then y α ∧ y τ → x ∧ y , since F is τ -closed wehave x ∧ y ∈ F for each y ∈ F + . If z ∈ F d , then y α ∧ z = 0for all α and we have x ∧ z = 0. Thus x ∈ F dd . That is, x is in the band generated by F in E . Hence there existsa net z β in F + such that 0 ≤ z β ↑ | x | . Therefore z β is b -bounded in E , by b -property of F in E , 0 ≤ z β ≤ x for some x ∈ F and | x | ≤ x . Hence x ∈ F as F is anideal.It shown in [1, Theorem 7.39] that a Dedekind com-plete vector lattice is universally complete iff it is univer-sally σ -complete and has a weak unit. In the next result,we replace universally σ -completeness with countable b -property of E in E u . Proposition 4.
Let E be a Dedekind complete vector lat-tice with countable b -property in E u and a weak order unit.Then E = E u .Proof. If E = E u then E has b -property in E u and hasa weak unit. Now we prove the converse. Let 0 < e be aweak order unit for E . Then E is an order ideal in E u by[1, Theorem 1.40]. Let 0 < u ∈ E u be arbitrary. Since e isalso a weak unit for E u ( E is order dense in E u ), we have0 < u ∧ ne ↑ u . As u ∧ ne ∈ E for each n , we see thatthe sequence u ∧ ne is b -bounded in E u . Therefore thesequence u ∧ ne has an upper bound in E by assumption.Thus 0 ≤ u ∧ ne ≤ x for some x ∈ E . As E is an orderideal in E u , we have u ∈ E . t is well known that if E ( τ ) is a LSVL with Levi prop-erty and τ -complete order intervals, then E is complete. Inthe following we reach to the same conclusion by replac-ing Levi property with weaker condition that E having b -property in ˆ E (ˆ τ ). Proposition 5.
Let E ( τ ) be a LSVL with τ -complete or-der intervals. If E ( τ ) has b -property in ˆ E , then E is com-plete.Proof. The assumption on order intervals implies that E is an order ideal of ˆ E by [1, Theorem 2.42]. Let 0 < ˆ x ∈ ˆ E be arbitrary. Since E is order dense in ˆ E , there exists anet x α such that 0 ≤ x α ↑ ˆ x . By the b -property of E inˆ E , we can find x ∈ E with 0 ≤ x α ≤ x . But then since x α ↑ x , we have x ≤ x and x ∈ E . Proposition 6.
Let F be a sublattice of an order completevector lattice E . Suppose F is order dense and majorizingin E . Then each increasing b -bounded net in F is uo -Cauchy in F .Proof. Let x α be a b -bounded net in F so that 0 ≤ x α ↑≤ e for some e ∈ E + . Since E is order complete, x α ↑ x forsome x ∈ E + . Then x α is order convergent in E and hence o -Cauchy in E , thus x α is uo -Cauchy in F by [7, Theorem2.3]It was observed that in [7, Theorem 3.2] for a net x α in a regular sublattice F of a vector lattice E , x α uo → F iff x α uo → E . However this may fail for uτ -convergence. uτ -Convergence in a sublattice may not im-ply uτ -convergence in the entire space. For example, thestandard unit vectors e n in l ∞ is easily seen to be a null equence in the unbounded norm topology of c but notso in l ∞ . Proposition 7.
Let F be a sublattice of a LSVL E ( τ ) .Suppose F has b -property in E . For a net x α in F forwhich x α uτ → in F , we have x α uτ → in E ( τ ) .Proof. Suppose x α uτ → F . WLOG we may suppose0 ≤ x α for all α . Then 0 ≤ x α ∧ y τ → y ∈ F + .On the other hand, for each x ∈ E + , 0 ≤ x α ∧ x ≤ x andthe net 0 ≤ ( x α ∧ x ) is b -bounded in F , by the hypothesis,there exists y ∈ F + such that 0 ≤ x α ∧ x ≤ y for all α .Then 0 ≤ x α ∧ x ≤ x α ∧ y τ → x α ∧ x τ →
0. As x is arbitrary x α uτ → E ( τ ). Proposition 8.
Let E ( τ ) be a laterally complete vectorlattice, then E has b -property in ( E ∼ ) ∼ n .Proof. Recall that E is order dense in ( E ∼ ) ∼ n . Then E ismajorizing in ( E ∼ ) ∼ n by [1, Theorem 7.15]. Therefore E has b -property in ( E ∼ ) ∼ n Proposition 9.
Let E ( τ ) be a LSVL with Lebesgue prop-erty. Then every order closed sublattice F of E ( τ ) hascountable b -property in ˆ E (ˆ τ ) .Proof. Let x n be a b -bounded sequence in E . Then thereexists ˆ x ∈ ˆ E with 0 ≤ x n ↑ ˆ x . Since E ( τ ) is assumed tohave Lebesgue property, it has the σ -Lebesgue propertyas well as the Fatou property by [1, Theorem 4.8]. Sincethe topology ˆ τ of ˆ E is also Lebesgue, the sequence x n isˆ τ -Cauchy in ˆ E . Then x n ˆ τ → x for some x ∈ ˆ E . Since τ isFatou and F being order closed is τ -closed by [1, Theorem .20]. Thus x ∈ F . As x n ↑ , x n τ → x , hence x = sup x n ,and F has b -property in ˆ E . Proposition 10.
Let F be a uo -closed sublattice of aDedekind complete vector lattice E . Then F has b -propertyin E .Proof. Let x α be a net in F with 0 ≤ x α ↑ x for some x ∈ E . As E is Dedekind complete, x α ↑ ˆ x for someˆ x ∈ E . Then x α o → ˆ x , consequently x α uo → ˆ x in E as F is uo -complete, ˆ x ∈ F . Proposition 11.
Let E be a vector lattice admitting aminimal topology τ . Let x n be a b -bounded sequence in E u . Then x n is τ -Cauchy in E .Proof. Let x n be such that 0 ≤ x n ↑ x u for some x u ∈ E u .Since E u is Dedekind complete, x n being order bounded in E u , has a supremum in E u , let it be x . Therefore x n o → x ,it follows that x n is uo -Cauchy in E u . Since E is orderdense in E u , and order dense sublattices are regular, E isregular in E u and by [7, Theorem 3.2], x n is uo -Cauchy in E . As every minimal topology is Lebesgue, τ is Lebesgueand x n is uτ -Cauchy. As τ is unbounded, it follows that x n is τ -Cauchy on E . Definition 3.
A locally solid vector lattice E ( τ ) is called boundedly uo -complete if every τ -bounded uo -Cauchy netin E is uo -convergent. Proposition 12.
A boundedly uo -complete LSVL E ( τ ) has b -property in E u . roof. Let 0 ≤ x α ↑ x u , where x u ∈ E u , be a net in E . As x α is a b -bounded subset of E , it is τ -bounded byLemma 1. We show x α has an upper bound in E . As E u is Dedekind complete, sup x α exists in E u . Let thissupremum be x . Then 0 ≤ x α ↑ x in E u . Thus x α o → x .It follows that x α is uo -Cauchy in E as E is order denseand a regular sublattice of E u . Thus x α being uo -Cauchyand τ -bounded converges to some x ′ ∈ E . But as x α o → x we must have x = x ′ Definition 4.
A Banach lattice is monotonically complete(the Levy property) if every norm bounded increasing netin E + has supremum.We now show that every boundedly uo -complete Ba-nach lattice E has b -property in ( E ∼ n ) ∼ n . The proof usesan idea of [6] in that ( E ∼ n ) ∼ n is monotonically completeand the canonical map J : E → ( E ∼ n ) ∼ n maps a boundedincreasing net in E + to a net in ( E ∼ n ) ∼ n with similar prop-erties. Proposition 13.
Let E be a boundedly uo -complete Ba-nach lattice with E ∼ n separating points of E . If ≤ x α ↑ isan increasing net in E + which is order bounded in ( E ∼ n ) ∼ n ,then x α has an upper bound in E .Proof. Since the net x α is order bounded in ( E ∼ n ) ∼ n , it isnorm bounded in ( E ∼ n ) ∼ n and hence norm bounded in E by Lemma 1.Let J : E → ( E ∼ n ) ∼ n be the natural embedding, where J ( x )( f ) = f ( x ) for each x ∈ E and f ∈ E ∼ n . The map J isa vector lattice isomorphism and the range J ( E ) in ( E ∼ n ) ∼ n is order dense in ( E ∼ n ) ∼ n by [1, Theorem 1.43]. Therefore, J ( E ) is a regular sublattice of ( E ∼ n ) ∼ n . y [10, 2.4.19], ( E ∼ n ) ∼ n is a monotonically complete Ba-nach lattice. Thus, the increasing net J ( x α ) has a supre-mum in ( E ∼ n ) ∼ n say x .So J ( x α ) ↑ x and J ( x α ) is order Cauchy in ( E ∼ n ) ∼ n .It follows that J ( x α ) is uo -Cauchy in ( E ∼ n ) ∼ n and in theregular sublattice J ( E ). As J is 1-1 and onto J ( E ) islattice isomorphism, x α is uo -Cauchy in E . E is boundedly uo -complete, x α uo → x for some x ∈ E . On the other hand0 ≤ x α ↑ , thus x α ↑ x and x α is order bounded in E . References [1] Aliprantis, C.D., Burkinshaw, O.: Locally Solid Riesz Spaces withApplications to Economics, 2nd edition. American Mathematical So-ciety, Providence, RI (2003)[2] Alpay, S¸., Altın, B., Tonyali, C.: On property b of vector lattices,Positivity, 7, 135-139 (2003),[3] Alpay, S¸., Altın, B., Tonyali, C.: A note on Riesz spaces with prop-erty b , Czechoslovak Math. J. 56 (131), 765-772 (2006)[4] Alpay, S¸., Emelyanov, E., Gorokhova, S.: Bibounded uo -convergence and bbuo -duals of vector lattices,https://arxiv.org/abs/2009.07401v1 (2020)[5] Alpay, S¸., Ercan, Z.: Characterizations of Riesz spaces with b -property, Positivity, 13 no. 1, 21-30 (2009)[6] Gao, N., Leung, D., Xanthos, F.: Dual representation of risk mea-sures on Orlicz spaces. Preprint: arXiv (2018)[7] Gao, N., Troitsky, V., Xanthos, F.: Uo -convergence and its applica-tions to Ces´aro means in Banach lattices. Isr. J. Math. 220, 649-689(2017)[8] Labuda, I.: Completeness type properties of locally solid Rieszspaces, Studia Math. 77, 349-372 (1984)[9] Labuda, I.: On boundedly order-complete locally solid Riesz spaces,Studia Math. 81, 245-258 (1985)[10] Meyer-Nieberg, P.: Banach Lattices. Universitext, Springer-Verlag,Berlin (1991)[11] Taylor, M.A.: Unbounded Convergences in Vector Lattices, Master’sthesis, University of Alberta, (2018)-convergence and its applica-tions to Ces´aro means in Banach lattices. Isr. J. Math. 220, 649-689(2017)[8] Labuda, I.: Completeness type properties of locally solid Rieszspaces, Studia Math. 77, 349-372 (1984)[9] Labuda, I.: On boundedly order-complete locally solid Riesz spaces,Studia Math. 81, 245-258 (1985)[10] Meyer-Nieberg, P.: Banach Lattices. Universitext, Springer-Verlag,Berlin (1991)[11] Taylor, M.A.: Unbounded Convergences in Vector Lattices, Master’sthesis, University of Alberta, (2018)