Convergence structures and Hausdorff uo-Lebesgue topologies on vector lattice algebras of operators

A vector sublattice of the order bounded operators on a Dedekind complete vector lattice can be supplied with the convergence structures of order convergence, strong order convergence, unbounded order convergence, strong unbounded order convergence, and, when applicable, convergence with respect to a Hausdorff uo-Lebesgue topology and strong convergence with respect to such a topology. We determine the general validity of the implications between these six convergences on the order bounded operator and on the orthomorphisms. Furthermore, the continuity of left and right multiplications with respect to these convergence structures on the order bounded operators, on the order continuous operators, and on the orthomorphisms is investigated, as is their simultaneous continuity. A number of results are included on the equality of adherences of vector sublattices of the order bounded operators and of the orthomorphisms with respect to these convergence structures. These are consequences of more general results for vector sublattices of arbitrary Dedekind complete vector lattices. The special attention that is paid to vector sublattices of the orthomorphisms is motivated by explaining their relevance for representation theory on vector lattices.


INTRODUCTION AND OVERVIEW
In an earlier paper [15] the authors studied aspects of locally solid linear topologies on vector lattices of order bounded linear operators between vector lattices. Particular attention was paid to the possibility of introducing a Hausdorff uo-Lebesgue topology on such vector lattices.
Such vector lattices of operators carry two, and possibly three, natural convergence structures (order convergence, unbounded order convergence, and, when applicable, convergence with respect to the Hausdorff uo-Lebesgue topology), as they can be defined for arbitrary vector lattices. For vector lattices of operators, however, besides these 'uniform' convergence structures, there are also two, or possibly three, corresponding 'strong' counterparts that can be defined in the obvious way. Several relations between the resulting six convergence structures on vector lattices of operators were also investigated in [15]. In view of their relevance for representation theory in vector lattices, special emphasis was put on the orthomorphisms on a Dedekind complete vector lattice. In this case, implications between convergences hold that do not hold for more general vector lattices of operators. Furthermore, it was shown that the orthomorphisms are not only order continuous, but also continuous with respect to unbounded order convergence on the vector lattice and with respect to a possible Hausdorff uo-Lebesgue topology on it.
Apart from their intrinsic interest, the results in [15] can be viewed as a part of the groundwork that has to be done in order to facilitate further developments of aspects of the theory of vector lattices of operators. The questions that are asked are natural and basic, but even so the answers are often more easily formulated than proved.
In the present paper, we take this one step further and study these six convergence structures in the context of vector lattice algebras of order bounded linear operators on a Dedekind complete vector lattice. Also here there are many natural questions of a basic nature that need to be answered before one can expect to get much further with the theory of such vector lattice algebras and with representation theory on vector lattices. For example, is the left multiplication by a fixed element continuous on the order bounded linear operators with respect to unbounded order convergence? Is the multiplication on the order continuous linear operators simultaneously continuous with respect to a possible Hausdorff uo-Lebesgue topology on it? Given a vector lattice subalgebra of the order continuous linear operators, is the closure (we shall actually prefer to speak of the 'adherence') in the order bounded linear operators with respect to strong unbounded order convergence again a vector lattice subalgebra? Is there a condition, sufficiently lenient to be of practical relevance, under which the order adherence of a vector lattice subalgebra of the orthomorphisms coincides with its closure in a possible Hausdorff uo-Lebesgue topology? Building on [15], we shall answer these questions in the present paper, together with many more similar ones. As indicated, we hope and expect that, apart from their intrinsic interest, this may serve as a stockpile of basic, but non-elementary, results that will facilitate a further development of the theory of vector lattice algebras of operators and of representation theory in vector lattices. This paper is organised as follows.
Section 2 contains the necessary notation, definitions, and conventions, as well as a few preparatory results that are of interest in their own right. Proposition 2.6 shows that, in many cases of practical interest, a unital positive linear representation of a unital f-algebra on a vector lattice is always an action by orthomorphisms. Its consequence Corollary 2.7 specialises this to the case of left and right multiplications of order bounded operators by orthomorphisms.
In Section 3, we study the validity of each of the 36 possible implications between the 6 convergences that we consider on vector lattice algebras of order bounded linear operators on a Dedekind complete vector lattice. We do this for the order bounded linear operators as well as for the orthomorphisms.
The results that are already in [15] and a few additional ones are sufficient to complete the tables 3.1 and 3.2.
Section 4 contains our results on the continuity of the left and right multiplications by a fixed element with respect to each of the six convergence structures on the order bounded linear operators. For this, we distinguish between the multiplication by an arbitrary order bounded linear operator, by an order continuous one, and by an orthomorphism. By giving (counter) examples, we show that our results are sharp in the sense that, whenever we state that continuity holds for multiplication by, e.g., an orthomorphism, it is no longer generally true for an arbitrary order continuous linear operator, i.e., for an operator in the 'next best class'. We also consider these questions for the orthomorphisms. The results are contained in tables 4.14 to 4. 16.
In Section 5, we investigate the simultaneous continuity of the multiplication with respect to each of the six convergence structures. When there is simultaneous continuity, the adherence of a subalgebra is, of course, again a subalgebra. With only one exception (see Corollary 5.6 and Example 5.7), we give (counter) examples to show that our conditions for the adherence of an algebra to be a subalgebra again are 'sharp' in the sense as indicated above for Section 4. Section 6 is dedicated to the equality of various adherences of vector sublattices and vector lattice subalgebras. It is also indicated there how representation theory in vector lattices leads quite naturally to the study of vector lattice subalgebras of the orthomorphisms (see the Theorems 6.1 and 6.2), thus motivating in more detail the special attention that is paid to the orthomorphisms in [15] and in the present paper.

PRELIMINARIES
In this section, we give the notation, conventions, and definitions used in the sequel. We also include a few preliminary results.
All vector spaces are over the real numbers and all vector lattices are supposed to be Archimedean. We let E + denote the positive cone of a vector lattice E. The identity operator on a vector lattice E will be denoted by I, or by I E when the context requires this. The characteristic function of a set S is denoted by χ S .
Let E be a vector lattice, and let x ∈ E. We say that a net (x α ) α∈A in E is order convergent to x ∈ E (denoted by x α o − → x) when there exists a net (x β ) β∈B in E such that y β ↓ 0 and with the property that, for every β 0 ∈ B, there exists an α 0 ∈ A such that |x − x α | ≤ y β 0 whenever α in A is such that α ≥ α 0 . Note that, in this definition, the index sets A and B need not be equal.
A net (x α ) α∈A in a vector lattice E is said to be unbounded order convergent Order convergence implies unbounded order convergence to the same limit. For order bounded nets, the two notions coincide.
Let E and F be vector lattices. The order bounded linear operators from E into F will be denoted by ob (E, F); this is a Dedekind complete vector lattice when F is. We write E for ob (E, ). A linear operator T : E → F between two vector lattices E and F is order continuous when, for every net (x α ) α∈A in An order continuous linear operator between two vector lattices is automatically order bounded; see [4, Lemma 1.54], for example. The order continuous linear operators from E into F will be denoted by oc (E, F). We write E oc for oc (E, ).
Let F be a vector sublattice of a vector lattice E. Then F is a regular vector sublattice of E when the inclusion map from F into E is order continuous. Ideals are regular vector sublattices. For a net in a regular vector sublattice F of E, its unbounded order convergence in F and in E are equivalent; see [18,Theorem 3.2].
An orthomorphism on a vector lattice E is a band preserving order bounded linear operator. We let Orth(E) denote the orthomorphisms on E. Orthomorphisms are automatically order continuous; see [4,Theorem 2.44]. An overview of some basic properties of the orthomorphisms that we shall use can be found in the first part of [15,Section 6], with detailed references included.
A topology τ on a vector lattice E is a uo-Lebesgue topology when it is a (not necessarily Hausdorff) locally solid linear topology on E such that, for a net For the general theory of locally solid linear topologies on vector lattices we refer to [3]. A vector lattice need not admit a uo-Lebesgue topology, and it admits at most one Hausdorff uo-Lebesgue topology; see [8, Propositions 3.2, 3.4, and 6.2] or [26, Theorems 5.5 and 5.9]). In this case, this unique Hausdorff uo-Lebesgue topology is denoted by τ E .
Proof. The equivalence of the parts (1) and (2)  Let X be a non-empty set. As in [15], we define a convergence structure on X to be a non-empty collection of pairs ((x α ) α∈A , x), where (x α ) α∈A is a net in X and x ∈ X , such that: Replacing nets by sequences and subnets by subsequences gives the usual sequential convergence structure, as in [5, Definition 1.7.1].

Remark 2.2.
Our definition of a convergence structures is actually a possible definition of a so-called net convergence structure. The theory of their counterparts, the so-called filter convergence structures, has been canonised in [5]. It is only recently that a definition of a net convergence structure (more sophisticated than ours) has been given that can be shown to yield a natural bijection between the net convergence structures and the filter convergence structures on a set; see [24]. In this definition, not all index sets for the nets in the structure are admitted, the admissible index are allowed to be merely pre-ordered, and property (1) in the above definition is replaced with two others. We refer to [24] for further details, and content ourselves with our definition above that is sufficient for our merely descriptive purposes.
Suppose that is a convergence structure on a non-empty set X . For a nonempty subset S ⊆ X , we define the -adherence of S in X as We set a ( ) := . A subset S of X is said to be -closed when a (S) = S. It is evident how define the adherence of a subset in the case of a sequential convergence structure. The following result, which was mentioned in [15, Section 1] without proof (see also [16,Section 8] for special cases), was already established in a context of order convergence on partially ordered vector spaces as [27,Theorem 3.1]. Its sequential version is proved using a similar argument. Lemma 2.3. Let X be a non-empty set, and let be a convergence structure on X . Then the -closed subsets of X are the closed sets of a topology τ on X .
Proof. It is trivial that and X are -closed, and it is immediate that an arbitrary intersection of -closed subsets of X is -closed. We claim that a (S 1 ∪ S 2 ) = a (S 1 ) ∪ a (S 2 ) for arbitrary S 1 , S 2 ⊆ X , which implies that finite unions of -closed subsets are again -closed. Since it is obvious that a (S 1 ∪ S 2 ) ⊇ a (S 1 ) ∪ a (S 2 ), we need to show only the reverse inclusion. We may suppose that S 1 , S 2 = . Take an x ∈ a (S 1 ∪ S 2 ). Then there exists a net (x α ) α∈A in S 1 ∪ S 2 such that ((x α ) α∈A , x) ∈ . If there is a tail (x α ) α≥α 0 that is contained in S 1 , then it follows from property (1) of that x ∈ a (S 1 ). If no tail of (x α ) α∈A is contained in S 1 , then the set B := {β ∈ A : x β ∈ S 2 } is a co-final subset of A. Hence (x β ) β∈B can canonically be viewed as a subnet of (x α ) α∈A that is contained in S 2 , and property (1) implies that x ∈ a (S 2 ). In both cases, It is not generally true that a (S) is τ -closed. We have the following result, the final statement of which was already mentioned without proof for special cases in [16,Section 8]. Its sequential version is valid by essentially the same proof.
Lemma 2.4. Let X be a non-empty set, let be a convergence structure on X , and let S ⊆ X . Then Proof. Property (2) of implies that S ⊆ a (S), and the obvious monotony of the adherence set map implies that S τ = a (S τ ) ⊇ a (S). The remaining two statements follow easily from the chain of inclusions.
On a vector lattice E, the set of all pairs of order convergent nets and their order limits forms a convergence structure o on E. Likewise, there is a convergence structure uo on E and, when applicable, a topological convergence structure τ E . For a subset S of E, we shall write a o (S) for a o (S), a uo (S) for a uo (S), and, when applicable, S τ E for a τ E (S). The corresponding sequential convergence structures are denoted by σo , σuo , and, when applicable, σ τ E , respectively. There are self-explanatory notations a σo (S), a σuo (S), and, when applicable, a σ τ E (S). We shall also speak of the order adherence (or oadherence) of a subset, rather than of its o -adherence; etc. Note that the order adherence a o (S) of S is what is called the 'order closure' of S in other sources. Since this 'order closure' need not be closed in the τ o -topology on E, we shall not use this terminology that is prone to mistakes.
Let E and F be vector lattices, where F is Dedekind complete. Suppose that is a vector sublattice of ob (E, F). As for general vector lattices, we have the convergence structures o ( ), uo ( ) and, when applicable, a convergence structure τ on . In addition to these 'uniform' convergence structures, there are in this case also 'strong' ones that we shall now define. Let (T α ) α∈A be a net in , and let T ∈ . Then we shall say that (T α ) α∈A is strongly order convergent The set of all pairs of strongly order convergent nets in and their limits forms a convergence structure SO on . Likewise, the net is strongly unbounded order convergent to T (denoted by T α SUO −−→ T ) when it is pointwise unbounded order convergent to T , resulting in a convergence structure SUO on . When E admits a Hausdorff uo-Lebesgue topology τ E , then a net is strongly convergent with respect to τ E to T (denoted by T α S τ E − − → T ) when it is pointwise τ E convergent to T , yielding to a convergence structure S τ on . As for the three convergence structures on general vector lattices, we shall simply write a SUO ( ) for the SUO -adherence a SUO ( ) of a subset of ; etc. We shall use a similar simplified notation for adherences corresponding to the sequential strong convergence structures that are defined in the obvious way.
The adherence of a set in a convergence structure obviously depends on the superset, since this determines the available possible limits of nets. In an ordered context, there can be additional complications because, for example, the notion of order convergence of a net itself depends on the vector lattice that the net is considered to be a subset of. It is for this reason that, although we have not included the superset in the notation for adherences, we shall always indicate it in words. Let X be a convergence structure on a non-empty set X , and let Y be a convergence structure on a non-empty set Y . A map Φ : X → Y is said to be X -Y continuous when, for every pair ((x α ) α∈A , x) in X , the pair ((Φ(x α )) α∈A , Φ(x)) is an element of Y . We shall speak of S τ E -o continuity rather than of S τ Eo continuity; etc.
Let E be a vector lattice. For T ∈ ob (E), we define ρ T , λ T : ob (E) → ob (E) by setting ρ T (S) := ST and λ T (S) := T S for S ∈ ob (E). We shall use the same notations for the maps that result in other contexts when compositions with linear operators map one set of linear operators into another.
For later use in this paper, we establish a few preparatory results that are of some interest in their own right.

Lemma 2.5.
Let be an f-algebra with an identity element e, and let E be a vector lattice with the principal projection property. Let a ∈ + , and suppose that is a positive linear map such that π(e) = I. Then π(a) ∈ Orth(E).
Proof. It is obvious that π(a) ∈ ob (E), so it remains to be shown that π(a) is band preserving on E. We know from [4, Theorem 2.57] that for n ≥ 1. Take x ∈ E + . Then we have for n ≥ 1. Let B x be the band generated by x in E, and let P x ∈ ob (E) be the order projection onto B x . Using that π(a)x ≥ 0 and equation (2.1), we have for all n ≥ 1. Hence (I − P x )[π(a)x] = 0, so that π(a)x ∈ B x . Since x was arbitrary, this shows that π(a) is band preserving.

Proposition 2.6.
Let be an f-algebra with an identity element e, and let E be a vector lattice with the principal projection property. Suppose that π : → ob (E) is a positive linear map such that π(e) = I. Then π( ) ⊆ Orth(E). If, in addition, π preserves the multiplication, then π is a unital vector lattice algebra homomorphism from into Orth(E).
Proof. It is clear from Lemma 2.5 that π maps into Orth(E). Suppose that π also preserves the multiplication. In this case, we note that Orth(E) is a unital Archimedean f-algebra (see [4,Theorem 2.59]), so that it is semiprime by [11,Corollary 10.4]. Since is likewise semiprime, it follows from [11, p. 96] (see also [11, part (i) of Theorem 3.7]) that π is a vector lattice homomorphism.
The following is immediate from Proposition 2.6 and-for its first part-the commutativity of Orth(E) (see [ show that the (then coinciding) maps ρ and λ even provide a vector lattice algebra isomorphisms between the Orth(E) and Orth(Orth(E). This is also true when E is not Dedekind complete.

IMPLICATIONS BETWEEN CONVERGENCES ON VECTOR LATTICES OF OPERATORS
In this section, we investigate the implications between the six convergences on the order bounded linear operators and on the orthomorphisms on a Dedekind complete vector lattice. Without further ado, let us simply state the answers and explain how they are obtained.
For a general net of order bounded linear operators (resp. orthomorphisms) on a general Dedekind complete vector lattice, the implications between order convergence, unbounded order convergence, convergence in a possible Hausdorff uo-Lebesgue topology, strong order convergence, strong unbounded order convergence, and strong convergence with respect to a possible Hausdorff uo-Lebesgue topology, are given in Table 3.1 (resp. Table 3.2).
In Orth(E), uo and SUO convergence of nets coincide, as do a possible τ Orth (E) and S τ E convergence.
In these tables, the value in a cell indicates whether the convergence of a net in the sense that labels the row of that cell does (value 1) or does not (value 0) in general imply its convergence (to the same limit) in the sense that labels the column of that cell. For example, the value 0 in the cell (uo, S τ E ) in Table 3.1 indicates that there exists a net of order bounded linear operators on a Dedekind complete vector lattice E that admits a Hausdorff uo-Lebesgue topology τ E , such that this net is unbounded order convergent to zero in ob (E), but not strongly convergent to zero with respect to τ E . The value 1 in the cell (uo, S τ E ) in Table 3.2, however, indicates that every net of orthomorphisms on an arbitrary Dedekind complete vector lattice E that admits a Hausdorff uo-Lebesgue topology τ E , such that this net is unbounded order convergent to zero, is strongly convergent to zero with respect to τ E .
We shall now explain how these tables can be obtained. Obviously, the order convergence of a net of operators implies its unbounded order convergence, which implies its convergence in a possible Hausdorff uo-Lebesgue topology. There are similar implications for the three associated strong convergences. Furthermore, an implication that fails for orthomorphisms also fails in the general case. Using these basic facts, it is a logical exercise to complete the tables from a few 'starting values' that we now validate.
For Table 3.1, we have the following 'starting values': It is easily checked that the above information suffices to complete both tables.

Remark 3.3.
(1) Every order bounded net of orthomorphisms on an arbitrary Dedekind complete vector lattice E that is strongly order convergent to zero, is order convergent to zero in Orth(E); see [15, Theorem 9.4]; (2) Every sequence of orthomorphisms on a Dedekind complete Banach lattice E that is strongly order convergent to zero, is order convergent to zero in Orth(E); see [15, Theorem 9.5]; (3) The validity of all zeroes in Table 3.1 (resp. Table 3.2) follows from the existence of a net of order bounded linear operators (resp. orthomorphisms) on a Dedekind complete Banach lattice for which the implication in question does not hold. With the cell (SO, o) in Table 3.2 as the only exception, such a net of operators on a Banach lattice can even be taken to be a sequence. This follows from an inspection of the (counter) examples referred to above when validating the 'starting' zeroes in the tables.

CONTINUITY OF LEFT AND RIGHT MULTIPLICATIONS
In this section, we study continuity properties of left and right multiplication operators. For example, take an arbitrary T ∈ ob (E), where E is an arbitrary Dedekind complete vector lattice that admits a Hausdorff uo-Lebesgue topology τ ob (E) . Is it then true that λ T : ob (E) → ob (E) maps unbounded order convergent nets in ob (E) to τ ob (E) convergent nets (with corresponding limits)? If not, is this then true when we suppose that T ∈ oc (E)? If not, is this true when we suppose that T ∈ Orth(E)? One can ask a similar combination of questions, specifying to classes of increasingly well-behaved operators, for each of the 6 · 6 = 36 combinations of convergences of nets in ob (E) under consideration in this paper. There are also 36 combinations to be considered for right multiplication operators. This section provides the answers in all 72 cases; the results are contained in the Tables 4.14 and 4.15. For the example that we gave, the answer is negative and remains so for arbitrary T ∈ oc (E), but it becomes positive for arbitrary T ∈ Orth(E). For Orth(E), there are similar questions to be asked for its left and right regular representation, but their number is smaller. Firstly, we see no obvious better-behaved subclass of Orth(E) that we should also consider. Secondly, since Orth(E) is commutative, there is only one type of multiplication involved. Thirdly, as in Table 3.2, there are two pairs of coinciding convergences. All in all, there are only 4 x 4 = 16 possible combinations that actually have to be considered for the regular representation of Orth(E). Also in this case, all answers can be given; the results are contained in Table 4. 16. As it turns out, Table 4.16 is identical to Table 3.2. There appears to be no a priori reason for this fact; it is simply the outcome.
We shall now set out to validate the Tables 4.14, 4.15, and 4.16. Fortunately, we do not need individual results for every cell. Upon considering the multiplications by the orthomorphism that is the identity operator, the zeroes in the Tables 3.1 and 3.2 already determine the values in many cells. For the remaining ones, the combination of the 'standard' implications that were already used for the Tables 3.1 and 3.2 and a limited number of results and (counter) examples already suffices. We shall now start to collect these.
We start with o-o and SO-SO continuity.
Proof. We prove the parts (1) and (2). Take T ∈ ob (E), and let (S α ) α∈A ⊆ ob (E) be a net such that S α o − → 0 in ob (E). By passing to a tail, we may assume that (|S α |) α∈A is order bounded in ob (E).
The parts (3) and (4) are immediate consequences of the definitions.
We now show that the condition in the parts (2) and (4) of Proposition 4.1 that T ∈ oc (E) cannot be relaxed to T ∈ ob (E).
n=1 be the sequence of standard unit vectors in E, and let c denote the sublattice of E consisting of the convergent sequences. We define a positive linear functional f c on c by setting . We define S n ∈ oc (E) for n ≥ 1 by setting The sequence (S n ) ∞ n=1 , being order convergent to S, is also strongly unbounded order convergent to S in ob (E).

Remark 4.3.
Examples 4.2 also shows that, already for a Banach lattice E, λ T need not even be sequentially o-τ ob (E) continuous, sequentially o-S τ E continuous, sequentially SO-τ ob (E) continuous, or sequentially SO-S τ E continuous for arbitrary T ∈ ob (E).

Remark 4.4.
The o-o continuity (appropriately defined) of left and right multiplications on ordered algebras is studied in [2]. It is established on [2, p. 542-543] that, for a Dedekind complete vector lattice E, the right and left multiplication by an element T of the ordered algebra L(E) of all (!) linear operators on E are both order continuous on L(E) in the sense of [2] if and only if the left multiplication is, which is the case if and only if T ∈ oc (E). The proof refers to [1, Example 2.9 (a)], which is concerned with multiplications by a positive operator T on the ordered Banach algebra L(E) of all (!) bounded linear operators on a Dedekind complete Banach lattice E. It is established in that example that the simultaneous order continuity of the right and left multiplication by T on L(E) in the sense of [1] is equivalent to T being order continuous. On [1, p. 151] it is mentioned that this criterion for the order continuity of an operator can also be presented for an arbitrary Dedekind complete vector lattice. Although it is not stated as such, and although a proof as such is not given, the author may have meant to state, and have known to be true, that, for a Dedekind complete vector lattice E and T ∈ ob (E), λ T and ρ T are both o-o continuous on ob (E) in the sense of the present paper if and only if λ T is, which is the case if and only if T ∈ oc (E). Using arguments as on [1, p. 151] and [2, p. 542-543], the authors of the present paper have verified that-this is the hard part-for T ∈ ob (E), the o-o continuity of λ T on ob (E) in the sense of the present paper does imply that T ∈ oc (E). Hence the three properties of T ∈ ob (E) mentioned above are, indeed, equivalent; a result that is to be attributed to the late Egor Alekhno.
We use the opportunity to establish the following side result, which follows easily from combining each of [25, Satz 3.1] and [7, Proposition 2.2] with the parts (1) and (2) of Proposition 4.1.

Proposition 4.5. Let E be a Dedekind complete vector lattice. Then:
(1) the map T → ρ T defines an order continuous lattice homomorphism ρ : (2) the map T → λ T defines an order continuous lattice homomorphism λ : . Remark 4.6. In [28, Problem 1], it was asked, among others, whether, for a Dedekind complete vector lattice E, the left regular representation of ob (E) is a lattice homomorphism from ob (E) into ob ( ob (E)). In [10,Theorem 11.19], it was observed that the affirmative answer is, in fact, provided by [25, Satz 3.1]. Part (2) of Proposition 4.5 gives still more precise information.
Part (1), which relies on [7, Proposition 2.2], implies that the right regular representation of oc (E) is an order continuous lattice homomorphism from oc (E) into ob ( oc (E)), with an image that is, in fact, contained in oc ( oc (E)). After this brief digression, we continue with the main line of this section, and consider uo-uo and SUO-SUO continuity of left and right multiplication operators.
We prove part (4). We now show that the condition in the parts (1), (2), and (4) of Proposition 4.7 that T ∈ Orth(E) cannot be relaxed to T ∈ oc (E).

Examples 4.9.
(1) We first give an example showing that λ T and ρ T need not be uo-uo continuous on ob (E) for T ∈ oc (E).
for f ∈ E. For n ≥ 1, we define the positive operator S n on E by setting We claim that (S n ) ∞ n=1 is a disjoint sequence in ob (E). Let m, n ≥ 1 with m > n. Take a k ≥ 1 such that 1/k < 1/n − 1/m. For every f ∈ E + , [4, Theorem 1.51] then implies that and S n ( f · χ [(i−1)/k,i/k] ) are disjoint for i = 1, . . . , k. Hence S m ∧ S n = 0, as claimed.
By [18,Corollary 3.6], the disjoint sequence (S n ) ∞ n=1 is unbounded order convergent to zero in ob (E). On the other hand, it is easy to see that ρ T (S n ) = λ T (S n ) = T = 0 for all n ≥ 1. Hence neither of (ρ T (S n )) ∞ n=1 and (λ T (S n )) ∞ n=1 is unbounded order convergent to zero in ob (E). This shows that neither ρ T nor λ T is uo-uo continuous on ob (E).
(2) We now give an example showing that λ T need not be SUO-SUO continuous on ob (E) for T ∈ oc (E).
For n ≥ 1, define the positive operator S n on E by setting S n f : n=1 is strongly unbounded order convergent to zero. On the other hand, it is easily seen that λ T (S n )χ [0,1] = χ [0,1] = 0 for n ≥ 1. This implies that (λ T (S n )) ∞ n=1 is not strongly unbounded order convergent to zero, so that λ T is not SUO-SUO continuous on ob (E).

Remark 4.10.
Examples 4.9 also shows that ρ T and λ T need not be uo-τ ob (E) continuous and that λ T need not be SUO-S τ E continuous on ob (E) for arbitrary T ∈ oc (E) = ob (E). In fact, the sequential versions of these continuity properties can already fail to hold, even in cases where E is a Banach lattice with an order continuous norm.
We now turn to the Hausdorff uo-Lebesgue topologies, where we shall make use of Theorem 2.1. Proposition 4.11. Let E be a Dedekind complete vector lattice that admits a (necessarily unique) Hausdorff uo-Lebesgue topology τ E , so that ob (E) also admits a (necessarily unique) Hausdorff uo-Lebesgue topology τ ob (E) . Then: Proof. We know from Corollary 2.7 that ρ T , λ T ∈ Orth( ob (E)) when T ∈ Orth(E), and then the parts (1) and (2)  We now show that the condition in the parts (1), (2), and (4) of Proposition 4.11 that T ∈ Orth(E) cannot be relaxed to T ∈ oc (E).

Examples 4.12.
(1) We first give an example showing that λ T and ρ T need not be τ ob (E)τ ob (E) continuous on ob (E) for T ∈ oc (E). For this, we resort to the context and notation of part (1) of Examples 4.9. In that example, we know that S n uo − → 0 in ob (E), and then certainly S n τ ob (E) − −−− → 0. Since ρ T (S n ) = λ T (S n ) = T = 0 for all n ≥ 1, we see that neither ρ S nor λ S is τ ob (E) -τ ob (E) continuous on ob (E).
(2) We give an example showing that λ T need not be S τ E -S τ E continuous on ob (E) for T ∈ oc (E). For this, we resort to the context and notation of part (2) of Examples 4.9. In that example, we know that Remark 4.13. Examples 4.12 also shows that ρ T and λ T need not even be τ ob (E) -τ ob (E) continuous and that λ T need not even be S τ ob (E) -S τ ob (E) continuous on ob (E) for arbitrary T ∈ oc (E) = ob (E). In fact, the sequential versions of these continuity properties can already fail to hold, even in cases where E is a Banach lattice with an order continuous norm.
We now have sufficient material at our disposal to determine the tables mentioned at the beginning of this section.
For right multiplications on ob (E), the results are in Table 4.14. The value in a cell with a row label indicating a convergence structure 1 and a column label indicating a convergence structure 2 is to be interpreted as follows: (1) A value {0} (resp. Orth(E), resp. oc (E)) means that ρ T is 1 -2 continuous on ob (E) for every Dedekind complete vector lattice E and for every T ∈ {0} (resp. T ∈ Orth(E), resp. T ∈ oc (E)), but there exist a Dedekind complete vector lattice E and a T ∈ Orth(E) (resp. T ∈ oc (E), resp. T ∈ ob (E)) for which this is not the case; (2) A value ob (E) means that ρ T is 1 -2 continuous on ob (E) for every Dedekind complete vector lattice E and for every T ∈ ob (E).
As mentioned in the beginning of this section, a zero in Table 3.1 gives {0} in Table 4.14. It is easily verified that the remaining values can be determined using that order convergence implies unbounded order convergence, which implies τ E convergence when applicable; that analogous implications hold for their strong versions; that order convergence implies strong order convergence; combined with Proposition 4.1, Proposition 4.7, Remark 4.10, Proposition 4.11, and Remark 4.13.
For left multiplications on ob (E), the results are in Table 4.15, with a similar interpretation of the values in the cells as for Table 4.14.
For Table 4.15, the values of the cells can be determined using the zeroes in Table 3.1, the 'standard implications' as listed for Table 4.14, combined with For multiplications on Orth(E), the continuity properties are given by Table 4.16. In that table, a value 1 in a cell with a row label indicating a convergence structure 1 and a column label indicating a convergence structure 2 means that the maps ρ T = λ T : Orth(E) → Orth(E) is 1 -2 continuous for all T ∈ Orth(E). A value 0 means that there exists a Dedekind complete vector lattice E and a T ∈ Orth(E) for which this is not the case.
In Orth(E), uo and SUO convergence of nets coincide, as do a possible τ Orth (E) and S τ E convergence.
The values in the cells of Table 4.16 can be determined using the zeroes in Table 3.2, the 'standard implications' as listed for Table 4.14; the fact that Orth(E) is a regular vector sublattice of ob (E); the facts that unbounded order convergence and strong unbounded order convergence coincide on Orth(E), as do a possible τ Orth(E) and S τ E convergence; combined with Proposition 4.1, Proposition 4.7, and Proposition 4.11.

SIMULTANEOUS CONTINUITY OF MULTIPLICATIONS AND ADHERENCES OF SUBALGEBRAS OF ob (E)
In this section, we study the simultaneous continuity of the multiplications in subalgebras of ob (E) (where E is a Dedekind complete vector lattice) with respect to the six convergence structures under consideration in this paper. This is motivated by questions of the following type. Suppose that E admits a Hausdorff uo-Lebesgue topology. Take a subalgebra (not necessarily a vector lattice subalgebra) of ob (E). Is its adherence a S τ E ( ) in ob (E) with respect to strong τ E convergence again a subalgebra of ob (E)? This is not always the case, not even when ⊆ oc (E); see Example 5.13. When ⊆ Orth(E), however, the answer is affirmative; see Corollary 5.12.
As is easily verified, it follows already from the continuity of the left and right multiplications with respect to strong τ E convergence (see Proposition 4.11) that a S τ E ( ) · a S τ E ( ) ⊆ a S τ E (a S τ E ( )) when ⊆ Orth(E), but that is not sufficient to show that a S τ E ( ) is a subalgebra. The simultaneous continuity of the multiplication in Orth(E) with respect to strong τ E convergence in Orth(E) would be sufficient to conclude this, and this can indeed be established; see Proposition 5.11.
For each of the remaining five convergence structures, we follow the same pattern. We establish (this also relies on the single variable results in Section 4) the simultaneous continuity of the multiplication with respect to the convergence structure under consideration, and then conclude that the pertinent adherence of a subalgebra is again a subalgebra. For the latter result it is-as the above example already indicates-essential to impose an extra condition on the subalgebra . This condition depends on the convergence structure under consideration. Natural extra conditions are that be a subalgebra of Orth(E) or of oc (E), and we do indeed obtain positive results under such conditions. We also have fairly complete results showing that the relaxation of the pertinent condition to the 'natural' next lenient one does, in fact, render the statement that the adherence is a subalgebra again invalid. This also implies that multiplication is then not simultaneously continuous.
In the cases where the lattice operations are known to be simultaneously continuous with respect to the convergence structure under consideration, it obviously also follows that the pertinent adherence of a vector lattice subalgebra is a vector lattice subalgebra again.
We shall now embark on this programme. We start with order convergence, which is the easiest case. For this, we have the following result on the simultaneous continuity of multiplication.

Proposition 5.1. Let E be a Dedekind complete vector lattice. Suppose that
Proof. It is clear that S ∈ oc (E). By passing to a tail, we may suppose that (|T β |) β∈B is bounded above by some R ∈ ob (E) + . Using the parts (1) and (2) of Proposition 4.1 for the final order convergence, we have that The following is now clear from Proposition 5.1 and the simultaneous order continuity of the lattice operations.

Corollary 5.2. Let E be a Dedekind complete vector lattice. Suppose that is a subalgebra of oc (E). Then the adherence a o ( ) in ob (E) is also a subalgebra of oc (E). When is a vector lattice subalgebra of oc (E), then so is a o ( ).
We now show that the condition in Corollary 5.2 that ⊆ oc (E) cannot be relaxed to ⊆ ob (E).

Example 5.3.
Take E = ℓ ∞ and let (e n ) ∞ n=1 be the standard sequence of unit vectors in E. We define T ∈ ob (E) as in Examples 4.2. For n ≥ 1, we now define S ′ n ∈ oc (E) by setting and S ′ ∈ oc (E) by setting It is easily verified that T 2 = 0, that S ′ n S ′ m = 0 for m, n ≥ 1, and that S ′ n T = T S ′ n = 0 for n ≥ 1. Hence := Span{T, S ′ n : n ≥ 1} is a subalgebra of ob (E). As S ′ n ↑ S ′ in ob (E), both S ′ and T are elements of a o ( ).
However, T S ′ / ∈ a o ( ). In fact, T S ′ is not even an element of a SO ( ) ⊇ a o ( ). To see this, we observe that T S ′ e 2 = e 1 = 0, and that, as is easily verified, T S ′ e 2 ⊥ Re 2 for all R ∈ . Hence there cannot exist a net Now we turn to the strong order adherences of subalgebras of ob (E). We start by showing that Orth(E) is closed in ob (E) under the convergences under consideration in this paper. We recall from Theorem 2.1 that either all of E, Orth(E), and ob (E) admit a Hausdorff uo-Lebesgue topology, or none does.

Lemma 5.4. Let E be a Dedekind complete vector lattice. Then Orth(E) is closed in ob (E) under order convergence, unbounded order convergence, strong order convergence, and strong unbounded order convergence. Suppose that E admits a (necessarily unique) Hausdorff uo-Lebesgue topology. Then Orth(E) is closed in
Proof. A band in a (not necessarily Dedekind complete) vector lattice is not only closed under order convergence, but also closed under unbounded order convergence (see [18,Proposition 3.15]) and under convergence in a Hausdorff locally solid linear topology on the lattice (see [3,Theorem 2.21(d)]. This implies that Orth(E) is closed in ob (E) under order convergence, unbounded order convergence, and convergence in a possible Hausdorff uo-Lebesgue topology on ob (E). It also implies that, for each of the three strong convergences in ob (E) under consideration, a limit in ob (E) of a net of orthomorphisms, i.e., of order bounded band preserving operators, is again an order bounded band preserving operator, i.e., an orthomorphism.

Proposition 5.5. Let E be a Dedekind complete vector lattice. Suppose that (S α ) α∈A
is a net in Orth(E) such that S α SO − → S in ob (E) for some S ∈ ob (E) and that Proof. Lemma 5.4 shows that S ∈ Orth(E). Take x ∈ E. By passing to a tail, we may suppose that (|T β x|) β∈B is bounded above by some y ∈ E + . By applying [4, Theorem 2.43] and the order continuity of |S| for the final convergence, we see that The following is now clear from Proposition 5.5. Corollary 5.6. Let E be a Dedekind complete vector lattice. Suppose that is a subalgebra of Orth(E). Then the adherence a SO ( ) in ob (E) is also a subalgebra of Orth(E).
We now show that the condition in Corollary 5.6 that ⊆ Orth(E) cannot be relaxed to ⊆ ob (E). At the time of writing, the authors do not know whether it might be relaxed to ⊆ oc (E).
Example 5.7. We resort to the context and notation of Example 5.3. In that example, we had operators T, S ′ ∈ a o ( ) such that T S ′ / ∈ a SO ( ). Since a o ( ) ⊆ a SO ( ), this example also provides an example as currently needed.
We turn to unbounded order adherences and strong unbounded order adherences.

Proposition 5.8. Let E be a Dedekind complete vector lattice. Suppose that
Then S, T ∈ Orth(E), and S α T β uo − → ST in ob (E). Seven similar statements hold that are obtained by, for each of the three occurrences of unbounded order convergence, either keeping it or replacing it with strong unbounded order convergence.
Proof. We start with the statement for three occurrences of unbounded order convergence. For this, we first suppose that S = T = 0.
For α ∈ A, let P α be the order projection in Orth(E) onto the band B α in Orth(E) that is generated by (|S α | − I) + . Then 0 ≤ P α I ≤ P α |S α | ≤ |S α | by [15,Lemma 6.9]. Hence P α I uo − → 0 in ob (E), so that also P α I uo − → 0 in the regular vector sublattice Orth(E) of ob (E) by [18,Theorem 3.2]. Since the net (P α I) α∈A is order bounded in Orth(E), we see that Combining the fact that |S α | ≤ I + P α |S α | by [15, Proposition 6.10(2)] with equation (5.2), we have, for α ∈ A, β ∈ B, For the case of general S and T , we first note that S, T ∈ Orth(E) as a consequence of Lemma 5.4. On writing  We now show that, neither for a uo ( ) to be a subalgebra of ob (E), nor for a SUO ( ) to be a subalgebra of ob (E), the condition in Corollary 5.9 that ⊆ Orth(E) can be relaxed to ⊆ oc (E). for n ≥ 1, and we define T ∈ oc (E) by setting x i e i ∈ E. Set S n := S 1,2 − S 1,n+3 for n ≥ 1. It is not hard to check that T 2 = T , that S n T = T S n = 0 for n ≥ 1, and that S m S n = 0 for m, n ≥ 1. Hence := Span{T, S n : n ≥ 1} is a subalgebra of oc (E). Using [4, Theorem 1.51], it is easy to see that (S 1,n+3 ) ∞ n=1 is a disjoint sequence in ob (E), so that S 1,n+3 uo − → 0 in ob (E) by [18, Corollary 3.6]. Hence S n uo − → S 1,2 in ob (E), showing that S 1,2 ∈ a uo ( ). Obviously, T ∈ a uo ( ). We claim that, however, T S 1,2 is not even an element of τ ob (E) ⊇ a uo ( ). In order to see this, we observe that T S 1,2 = S 1,3 and, using [4, Theorem 1.51], that S 1,3 ⊥ T and S 1,3 ⊥ S n for n ≥ 1. Hence T S 1,2 ⊥ , which implies that x i e i ∈ ℓ 1 , we have S 1,n+3 x = x 1 e n+3 for n ≥ 1. This implies that S 1,n+3 SUO −−→ 0 in ob (E), showing that S n SUO −−→ S 1,2 in ob (E). Hence S 1,2 ∈ a SUO ( ). Obviously, T ∈ a SUO ( ). We claim that, however, T S 1,2 is not even an element of a S τ E ( ) ⊇ a SUO ( ). In order to see this, it is sufficient to observe that T S 1,2 e 1 = e 3 = 0 and that T S 1,2 e 1 ⊥ Re 1 for all R ∈ . This implies that We turn to closures in a Hausdorff uo-Lebesgue topology and strong closures with respect to a Hausdorff uo-Lebesgue topology. We recall once more from Theorem 2.1 that either all of E, Orth(E), and ob (E) admit a Hausdorff uo-Lebesgue topology, or none does. If they do, then, by general principles (see [26,Proposition 5.12]), τ Orth(E) is the restriction of τ ob (E) to Orth(E).

Then S, T ∈ Orth(E), and S
. Seven similar statements hold that are obtained by, for each of the three occurrences of τ ob (E) convergence, either keeping it or replacing it with strong τ E convergence.
Proof. We start with the statement for three occurrences of τ ob (E) convergence. For this, we first suppose that S = T = 0.
We can use parts of the proof of Proposition 5.8 here. In that proof, it was established that, for α ∈ A, there exists a band projection P α in Orth(E) such that . For the case of general S and T , we first note that S, T ∈ Orth(E) as a consequence of Lemma 5.4. On writing The following is now clear from Proposition 5.11 and the simultaneous continuity of the lattice operations with respect to the τ ob (E) topology.

Corollary 5.12. Let E be a Dedekind complete vector lattice that admits a (necessarily unique) Hausdorff uo-Lebesgue topology τ E . Suppose that is a subalgebra
of Orth(E). Then the closure τ ob (E) in ob (E) and the adherence a S τ E ( ) in ob (E) are equal, and are subalgebras of Orth(E). When is a vector lattice subalgebra of Orth(E), then so is We now show that, neither for τ ob (E) to be a subalgebra of ob (E), nor for a S τ E ( ) to be a subalgebra of ob (E), the condition in Corollary 5.12 that ⊆ Orth(E) can be relaxed to ⊆ oc (E).
Example 5.13. We return to the context and notation of Example 5.10. In that example, we saw that S n uo − → S 1,2 in ob (E). Then certainly S n τ ob −−→ S 1,2 in ob (E), so that both T and S 1,2 are elements of τ ob (E) . We saw in Example 5.10, however, that T S 1,2 / ∈ τ ob (E) .
It was also observed that S n SUO −−→ S 1,2 in ob (E). Since E is atomic, the unbounded order convergence of a net in E and its convergence in the Hausdorff uo-Lebesgue topology on E are known to coincide (see [9, Lemma 3.1] and [26,Lemma 7.4]). Thus also S n S τ E − − → S 1,2 , so that both T and S 1,2 are elements of a S τ E ( ). We saw in Example 5.10, however, that T S 1,2 / ∈ a S τ E ( ).

EQUALITY OF ADHERENCES OF VECTOR SUBLATTICES
In this section, we establish the equality of various adherences of vector sublattices with respect to convergence structures under consideration in this paper. We pay special attention to vector sublattices of the orthomorphisms on a Dedekind complete vector lattice. Apart from the intrinsic interest of the results, our research in this direction is also motivated by representation theory. We shall now explain this.
Suppose that E is a vector lattice, and that is a non-empty set of order bounded linear operators on E. Typical examples to keep in mind are those where is a group of order automorphisms of E (this arises naturally when considering positive representations of groups on vector lattices), or where is a (vector lattice) algebra of order bounded linear operators (this arises naturally when considering positive representations of (vector lattice) algebras on vector lattices). One of the main issues in general representation theory is to investigate the possible decompositions of a module into submodules. In our case, this is asking for decompositions E = F 1 ⊕ F 2 as an order direct sum of vector sublattices F 1 and F 2 that are both invariant under . It is well known (see [29,Theorem 11.3] for an even stronger result) that F 1 and F 2 are then projection bands that are each other's disjoint complements. Their respective order projections then commute with all elements of . Conversely, when an order projection has this property, then E is the order direct sum of its range and its kernel, and both are invariant under . All in all, the decomposition question for the action of on E is the same as asking for the order projections on E that commute with . This makes it natural to ask for the commutant of in Orth(E), where these order projections reside. This commutant is obviously an associative subalgebra of Orth(E). Somewhat surprisingly, it is quite often also a vector sublattice of Orth(E). For example, this is always true for Banach lattices, in which case the operators in need not even be regular. Being bounded is enough, as is shown by the following result, for which the Banach lattice need not even be Dedekind complete. Proof. It is obvious that ′ Orth is an associative subalgebra of Orth(E) that contains I and that is closed with respect to the coinciding operator norm and order unit norm · I . An appeal to [15, Theorem 6.1] then finishes the proof.
For Dedekind complete vector lattices, we have the following. Proof. For T ∈ Orth(E), we let λ T (resp. ρ T ) denote the corresponding left (resp. right) multiplication operator on ob (E). Since Corollary 2.7 implies that λ T − ρ T is an orthomorphism on ob (E), its kernel is a band in ob (E) by [4,Theorem 2.48]. It follows from this that the commutants of and B in Orth(E) are equal.
We shall now show that this common commutant in Orth(E) is a vector sublattice of Orth(E). In view of what we have already established, we may suppose that consists of one positive operator S on E. We shall show that for T 1 , T 2 ∈ Orth(E), T 1 ∨ T 2 commutes with S whenever T 1 and T 2 do. Obviously, (T 1 ∨ T 2 )S = λ T 1 ∨T 2 (S) which, by part (2) of Corollary 2.7 equals (λ T 1 ∨λ T 2 )(S). Using once more from part (2) of Corollary 2.7 that left multiplications by elements of Orth(E) are orthomorphisms on ob (E), [4,Theorem 2.43 As a consequence of the assumption, this equals (ST 1 ) ∨ (ST 2 ). By a reasoning similar to the one just given, but now using part (1) of Corollary 2.7, this equals S(T 1 ∨ T 2 ). Hence ′ Orth is a vector sublattice of Orth(E).
It is clear that ′ Orth is an associative subalgebra of Orth(E) containing I and that I, which is a weak order unit of Orth(E), is also one of ′ Orth . We turn to the remaining statements when S ⊆ oc (E). Suppose that (T α ) α∈A is a net in ′ Orth , that T ∈ ob (E), and that T α o − → T in ob (E). Then certainly T ∈ Orth(E). Using that ⊆ oc (E), it follows from Proposition 4.1 that T commutes with all elements of . Hence T ∈ ′ Orth , and we conclude that ′ Orth is an order closed vector sublattice of ob (E). Obviously, it is then also order closed in every regular vector sublattice of ob (E) containing it. We have thus established part (1). In Theorem 6.2, when ⊆ oc (E), then the vector lattice ′ Orth is a Dedekind complete vector lattice with the identity operator I as a weak order unit. The unbounded version of Freudenthal's spectral theorem (see [23,Theorem 40.3], for example) then shows that an arbitrary element T ∈ ′ Orth is an order limit of a sequence of linear combinations of the components of I in ′ Orth . Since the latter are precisely the order projections that commute with we see that, in this case, ′ Orth does not only contain all information about the collection of bands in E that reduce , but that it is also completely determined by this collection.
On a later occasion, we shall report more elaborately on the procedures of taking commutants and also of taking bicommutants in the context of operators on vector lattices and Banach lattices, as well as on their relations with reducing projection bands for sets of operators; see [13,14]. For the moment, we content ourselves with the general observation that the study of vector lattice subalgebras of the orthomorphisms is relevant for representation theory on vector lattices.
We shall now set out to study one particular aspect of this, namely, the equality of the adherences of vector sublattices of the orthomorphisms with respect to several of the convergence structures under consideration in this paper. Although from a representation theoretical point of view it would be natural to require that they also be associative subalgebras, this does, so far, not appear to be relevant for these issues. Such results on equal adherences can then also be obtained for associative subalgebras of the orthomorphisms on a Banach lattice, as a consequence of the fact that their norm closures in the orthomorphisms are. in fact, vector sublattices to which the previous results can be applied.
Regarding the results below that are given for vector sublattices of the orthomorphisms, we recall that, for a Dedekind complete vector lattice, several adherences coincide for subsets of the orthomorphisms. Indeed, since, for nets of orthomorphisms, unbounded order convergence coincides with strong unbounded order convergence, and since the convergence in a possible Hausdorff uo-Lebesgue topology coincides with the corresponding strong convergence, the corresponding adherences of subsets of the orthomorphisms are also equal. The same holds for sequential adherences. For reasons of brevity, we have refrained from including these 'obviously also equal' adherences in the statements.
Although our motivation leads us to study vector sublattice of the orthomorphisms, the results as we shall derive them for these are actually consequences of more general statements for vector lattices that need not even consist of operators. These are of interest in their own right. Other such results are [ in .
Before giving the proof, we mention the following fact that is easily verified. Suppose that X is a topological space that is supplied with two topologies τ 1 and τ 2 , where τ 2 is weaker than τ 1 . Then S Proof. It follows from [15, Theorem 6.1] that · e is a Banach f-subalgebra of . Being a vector sublattice of , Proposition 6.4 shows that the sets in the second line of equation (6.1) are equal. Since the convergence of a net in the order unit norm · e implies its order convergence to the same limit (and then also its convergence in τ to the same limit), we are done by an appeal to the remark preceding the proof.
The following is now clear from Theorem 6.6 and the argument in the proof of Corollary 6.5. in Orth(E); here · denotes the coinciding operator norm, order unit norm with respect to the identity operator, and regular norm on Orth(E).
We shall now continue by establishing results showing that the closures of vector sublattices (or associative subalgebras) in a possible Hausdorff uo-Lebesgue topology coincide with their adherences with respect to various convergence structures on the enveloping vector lattices (or vector lattice algebras) under consideration in this paper.
Needless to say, under appropriate conditions, 'topological' results as obtained above may apply at the same time as 'adherence' results to be obtained below. For reasons of brevity, we have refrained from formulating such 'combined' results.
Let us also notice at this point that the results below imply that the adherences of vector sublattices that occur in the statements are closed with respect to the pertinent convergence structures. Indeed, these adherences are set maps that map vector sublattices to vector sublattices. When they agree on vector sublattices with the topological closure operator that is supplied by the Hausdorff uo-Lebesgue topology, then they, too, are idempotent. For example, the unbounded order adherence of the vector sublattice F in Proposition 6.8, is unbounded order closed. For reasons of brevity, we have refrained from including such consequences in the results.
We start by considering two cases where the enveloping vector lattices have weak order units.
On combining Theorem 2.1, Proposition 6.8, and [15, Proposition 6.5], the following is easily obtained. We recall that a subset of a vector lattice is said to be an order basis when the band that it generates is the whole vector lattice. We continue by considering cases where the enveloping vector lattice (or vector lattice algebra) has a strong order unit.
It is known that the o-adherence of a vector sublattice of a Dedekind complete Banach lattice E with a strong order unit can be a proper sublattice of its uo-adherence; see [17, Lemma 2.6] for details. When the vector sublattice contains a strong order unit of E, however, then this cannot occur, not even in general vector lattices. This is shown by the following preparatory result. Lemma 6.10. Let E be a vector lattice with a strong order unit. Suppose that F is a vector sublattice of E that contains a strong order unit of E. Then a o (F) = a uo (F) and a σo (F) = a σuo (F) in E.
Proof. We prove that a o (F) = a uo (F). It is clear that a o (F) ⊆ a uo (F). For the reverse inclusion, we choose a positive strong order unit e of E such that e ∈ F. Take x ∈ a uo (F), and let (x α ) α∈A be a net in F such that x α uo − → x in E. There exists a λ ∈ ≥0 such that |x| ≤ λe. For α ∈ A, set y α := (−λe ∨ x α ) ∧ λe. Clearly, ( y α ) α ⊆ F and y α uo − → (−λe ∨ x) ∧ λe = x. Since the net ( y α ) α∈A is order bounded in E, we have that y α o − → x in E. Hence x ∈ a o (F). We conclude that a uo (F) ⊆ a o (F).
The proof for the sequential adherences is a special case of the above one.
Remark 6.11. For comparison, we recall that, for a regular vector sublattice F of a vector lattice E, it is always the case that a o (F) = a uo (F) in E, and that these coinciding subsets are order closed subsets of E; see [17,Theorem 2.13]. For this to hold, no assumptions on E are necessary.
The following is immediate from Proposition 6.8 and Lemma 6.10.  The following result follows from the combination of Theorem 2.1, Theorem 6.12, and [15,Proposition 6.5]. In view of [15, Proposition 6.5], the natural condition to include is that E have an at most countably infinite order basis, but it is easily verified fact that, for a Banach lattice, this property is equivalent to having a weak order unit. We now turn to closures and adherences of associative subalgebras of a class of f-algebras with strong order units. For this, we need the following preparatory result. Lemma 6.14. Let E be a Banach lattice, and let A be a subset of E. Then a σo (A) = a σo (A) in E, where A denotes the norm closure of A.
Proof. We need to prove only that a σo (A) ⊆ a σo (A). For this, we may suppose that A = . Take x ∈ a σo (A) and a sequence (x n ) ∞ n=1 in A such that x n σo − → x in E. For n ≥ 1, take an y n ∈ A such that y n − x n ≤ 2 −n . For n ≥ 1, define z n by setting z n := ∞ m=n | y m − x m |, which is meaningful since the series is absolutely convergent. It is clear that z n ↓. Since z n ≤ 2 −n+1 , we have z n ↓ 0 in E. The fact that | y n − x n | ≤ z n for n ≥ 1 then shows that | y n − x n | σo − → 0 in E. From 0 ≤ | y n − x| ≤ | y n − x n | + |x n − x| σo − → 0, we then see that y n σo − → x in E. Hence x ∈ a σo (A), as desired. Theorem 6.15. Let be a Dedekind complete unital f-algebra with the countable sup property, such that its identity element e is also a strong order unit of , and such that it is complete in the submultiplicative order unit norm · e on . Suppose that admits a (necessarily unique) Hausdorff uo-Lebesgue topology τ . Let be an associative subalgebra of such that · e contains a strong order unit of . Then τ = a σo ( ) = a o ( ) = a σuo ( ) = a uo ( ) = · e τ = a σo ( · e ) = a o ( · e ) = a σuo ( · e ) = a uo ( · e ) (6.2) in .
Proof. We know from [15, Theorem 6.1] that · e is a Banach f-subalgebra of . Then Theorem 6.12 shows that all equalities in the second line of equation (6.2) hold. Furthermore, it is obvious that and that a σo ( ) ⊆ a σuo ( ) ⊆ τ .
Using that a σo ( ) = a σo ( · e ) by Lemma 6.14 and that-see the proof of Theorem 6.6-we also know that τ = · e τ , it then follows that all sets in equation (6.2) are equal.
The following is now clear from Theorem 2.1, Theorem 6.15, and [15, Proposition 6.5]. in Orth(E); here · denotes the coinciding operator norm, order unit norm with respect to the identity operator, and regular norm on Orth(E).
Acknowledgements. During this research, the first author was supported by a grant of China Scholarship Council (CSC). The authors thank Ben de Pagter for pointing out that a (not necessarily unital) positive algebra homomorphism between two unital Archimedean f-algebras is automatically a vector lattice homomorphism, which has led to Proposition 2.6, Corollary 2.7, and Theorem 6.2 being in a stronger form than in the original manuscript. They also thank the anonymous referee for their careful reading of the manuscript and the many detailed suggestions for improvement of the presentation.