Arens regularity for totally ordered semigroups

Let S be a semigroup. We shall consider the centres of the semigroup (βS,□)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\beta \,S, \,\Box \,)$$\end{document} and of the algebra (M(βS),□)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(M(\beta \,S), \,\Box \,)$$\end{document}, where M(βS)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M(\beta \,S)$$\end{document} is the bidual of the semigroup algebra (ℓ1(S),⋆)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\ell ^{\,1}(S),\,\star \,)$$\end{document}, and whether the semigroup and the semigroup algebra are Arens regular, strongly Arens irregular, or neither. We shall also determine subsets of S∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^*$$\end{document} and of M(S∗)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M(S^*)$$\end{document} that are ‘determining for the left topological centre’ (DLTC sets) of βS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \,S$$\end{document} and M(βS)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M(\beta \,S)$$\end{document}. It is known that, when the semigroup S is cancellative, ℓ1(S)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell ^{\,1}(S)$$\end{document} is strongly Arens irregular and that there is a DLTC set consisting of two points of S∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^*$$\end{document}. In contrast, there is little that has been published about the Arens regularity of ℓ1(S)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell ^{\,1}(S)$$\end{document} when S is not cancellative. Totally ordered, abelian semigroups, with the map (s,t)→s∧t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(s,t)\rightarrow s \wedge t$$\end{document} as the semigroup operation, provide examples which show that several possibilities can occur. We shall determine the centres of βS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \,S$$\end{document} and of M(βS)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M(\beta \,S)$$\end{document} for all such semigroups, and give several examples, showing that the minimum cardinality of DTC sets may be arbitrarily large, and, in particular, we shall give an example of a countable, totally ordered, abelian semigroup S with this operation for which there is no countable DTC set for βS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta S$$\end{document} or for M(βS)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M(\beta S)$$\end{document}. There was no previously-known example of an abelian semigroup S for which βS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta \,S$$\end{document} or M(βS)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M(\beta S)$$\end{document} did not have a finite DTC set.


Introduction
In this section, we shall recall certain standard definitions and results.
Let S be a semigroup, so that S is a non-empty set together with an associative binary operation, S × S → S; the operation is usually denoted by juxtaposition. In this case, we shall write for the left and right translation operators on S for each s ∈ S. The centre of S is the sub-semigroup Z(S), where Z(S) = {t ∈ S : st = ts (s ∈ S)} , and S is abelian when Z(S) = S, so that st = ts (s, t ∈ S). An element s ∈ S is idempotent if s 2 = s, and S is an idempotent semigroup if each element is idempotent. A semigroup is cancellative if each L s and R s is injective, and weakly cancellative if the equations xs = t and sx = t have only finitely-many solutions for x for each s, t ∈ S.
Let (S, ≤) be a non-empty, partially ordered set, and suppose that s ∧ t = min{s, t} exists for all s, t ∈ S. Then (S, ∧) is an abelian, idempotent semigroup, called a semilattice, and these are the particular semigroups that we shall consider here. Conversely, suppose that S is an abelian, idempotent semigroup. Take s, t ∈ S, and set s ≤ t if st = s. Then (S, ≤) is a semilattice and s ∧ t = st (s, t ∈ S) [8,Proposition 1.3.2]. Hence (S, ∧) is a semigroup that can be identified with S.
A semigroup S which is also a topological space is: a left (respectively, right) topological semigroup if L s (respectively, R s ) is continuous for each s ∈ S. In the case where the semigroup S is also compact (and Hausdorff) as a topological space, we say that S is a compact, left or right topological semigroup, respectively.
The Stone-Čech compactification β S of a (discrete) set S is a compact set containing S as a dense subset, and is characterized by the property that every continuous mapping from S into a compact space K has a continuous extension from β S into K . We regard the semigroup S as a subset of β S. Now suppose that S is a discrete semigroup. We start by defining two products and ♦ on β S such that (β S, ) and (β S, ♦ ) are also semigroups. Of course, the semigroups (β S, ) and (β S, ♦ ) are very well-known; they are the main topic of the monograph [7].
For each s ∈ S, the map L s : S → β S has a continuous extension to a map L s : β S → β S; for each v ∈ β S, define s v = L s (v). Next, each map has a continuous extension to a map R v : β S → β S. For u, v ∈ β S, set Then is a binary operation on β S, and the restriction of to S × S is the original semigroup product. Further, for each v ∈ β S, the map R v : β S → β S is continuous, and, for each s ∈ S, the map L s : β S → β S is continuous.
Similarly, we can define a binary operation ♦ on β S by exchanging L s and R s . For u, v ∈ β S, we see that whenever (s α ) and (t β ) are nets in S with lim α s α = u and lim β t β = v in β S. The maps and ♦ agree with the maps defined in several different ways in [7].
In the case where S is an abelian semigroup, we have It is immediately checked that both and ♦ are associative operations on β S, and so we obtain the following fundamental result; see [7, §4.1] for more details. Theorem 1.1 Let S be a discrete semigroup. Then (β S, ) and (β S, ♦ ) are semigroups containing S as a sub-semigroup. Further: In fact, (β S, ) is the largest compactification of S which is a compact, right topological semigroup, in the sense that any other such compactification is a continuous homomorphic image of (β S, ) [7,Theorem 4.8]. We set S * = β S \ S, the growth of S.
The following definitions are well-known; see [ t (β S) = S, and strongly Arens irregular if it is both left and right strongly Arens irregular. A non-empty subset V of β S is determining for the left topological centre as defined in [7,Definition 2.4]. Thus S is Arens regular if and only if (β S) = β S. In the case where S is an abelian semigroup, we have the centre of both the semigroups (S, ) and (S, ♦ ); we refer to a 'DTC set' when the semigroup S is abelian.
In [4,Example 6.12], there is an example of a semigroup S that is right, but not left, strongly Arens irregular.
The following theorem is [4,Theorem 12.20 In §2, we shall determine the centre of β S for certain totally ordered, abelian, idempotent semigroups S with respect to the operation (s, t) → s ∧t (such semigroups S are usually not weakly cancellative), and shall consider their DTC sets in §3. We shall give in Example 2.16 an example of an infinite semilattice (that is not weakly cancellative) that is Arens regular.
We set D = {ζ ∈ C : |ζ | < 1}. The absolutely convex hull of a subset C of a linear space is denoted by aco C. This is the set of all elements of the form The centre of an algebra A is denoted by Z(A). Let E be a Banach space with a subset F. Then the closed unit ball of E is E [1] and F [1] = E [1] ∩ F. The dual and bidual spaces of E are denoted by E and E , respectively; we regard E as a subset of E , so that E [1] is dense in E [1] with respect to the weak- * topology, σ (E , E ).
There are two products, and ♦, on the Banach space A , called the first and second Arens products, that extend the module actions on A . Indeed, take λ ∈ A and M ∈ A , and define λ · M and M · λ in A by a, λ · M = M, a · λ , a, M · λ = M, λ · a (a ∈ A) , and then, for M, N ∈ A , define The basic theorem of Arens (see [1, §2.6] and [5, §2.3]) is that (A , ) and (A , ♦ ) are Banach algebras and that the embedding of A in A is an isometric algebra monomorphism in both cases. A Banach algebra A is Arens regular if the two products and ♦ agree on A ; for a commutative Banach algebra A this holds if and only if (A , ) is commutative. We shall write just A for (A , ).
For M, N ∈ A , we see that whenever (a α ) and t (A ), and that the empty set is a DLTC set when A is Arens regular.
In the case where the Banach algebra A is commutative, A is Arens regular when Z(A ) = A and strongly Arens irregular when Z(A ) = A, and we refer to DTC sets.
Let S be a semigroup, and consider the Banach space ( 1 (S), · 1 ). There is a continuous product , called convolution, on the space 1 (S). Indeed, for f , g ∈ 1 (S), define (If there are no elements r , s ∈ S with rs = t, then ( f g)(t) = 0.) Then the structure ( 1 (S), · 1 , ) is a Banach algebra, called the semigroup algebra on S. This algebra is commutative if and only if S is abelian. Let S be a semigroup. A semi-character on S is a map θ : S → D such that θ(st) = θ(s)θ (t) (s, t ∈ S) and θ = 0. The space of semi-characters on S is denoted by S ; it is a locally compact space with respect to the topology of pointwise convergence.
Let S be an abelian semigroup. Then the algebra 1 (S) is semisimple if and only if S separates the points of S, in the sense that, for each s, t ∈ S with s = t, there exists θ ∈ S with θ(s) = θ(t) [6,Theorem 3.5], and this holds if and only if S is separating, in the sense that s = t whenever s, t ∈ S with st = s 2 = t 2 [6, Theorem 5.8]. Thus 1 (S) is algebraically isomorphic to a Banach function algebra, A( S ), on S in this case, and all characters on the Banach function algebra A( S ) are given by evaluation at a point of S . Certainly an abelian semigroup that is either cancellative or an idempotent semigroup is separating. See [5] for an account of Banach function algebras, including this example.
The commutative Banach algebra of all continuous functions that vanish at infinity on a non-empty, locally compact space K is denoted by C 0 (K ), taken with the uniform norm, | · | K . By the Riesz representation theorem, the dual space of the Banach space where M d (K ) and M c (K ) are the discrete and continuous measures on K , respectively, and we identify M d (K ) with 1 (K ).
Recall that the total variation of a measure μ is defined by where the supremum is taken over all finite partitions We write M R (K ) for the space of real-valued measures on K . Thus each μ ∈ M(K ) can be written uniquely as Let S be a discrete semigroup. The dual space of the Banach space 1  In the case where S is an abelian semigroup, the Banach algebra 1 (S) is Arens regular whenever Z(M(β S)) = M(β S) and strongly Arens irregular whenever Z(M(β S)) = 1 (S).
We regard S and β S as subsets of 1 (S) and M(β S) by identifying p ∈ β S with the point mass δ p ∈ M(β S). Let (s α ) be a net in β S that converges to p ∈ β S. Then δ s α converges to δ p in the weak- * topology of M(β S), and so convergence is consistent with the previous definitions. We also see that the notations and ♦ for products on β S and M(β S) are consistent. For example, we do not know a semigroup S for which the above inclusion is proper.
The augmentation character on (M(β S), ) is the map Analogues of the above theorem for 'weighted semigroup algebras' are given in [3] and [12].
We do not know whether the fact that 1 (S) is strongly Arens irregular implies that S is strongly Arens irregular, even for abelian semigroups.
We shall give in §3 examples of abelian, idempotent semigroups S such that the semigroup algebras 1 (S) are strongly Arens irregular. In one example, there is no finite DTC set for S, but M(β S) has a two-element DTC set. In another example, there is no countable DTC set for S or for M(β S).
Let S be a semigroup, and consider the centre Z = Z(M(β S), ). We write Z R for Z(M R (β S), ). The following result is immediate.

Totally ordered semigroups
In this section, we shall introduce a class of totally ordered, abelian, idempotent semigroups, and obtain some results about their Arens regularity and the Arens regularity of their semigroup algebras.
Throughout the section, T will denote an infinite, totally ordered space. For s, t ∈ T , we set so that T is a lattice and a semigroup with respect to the operations Thus T is an abelian, idempotent semigroup with respect to both these operations; we shall just consider the operation ∧. We further suppose that T has a minimum element, called 0, and a maximum element, called ∞, and that T is complete, in the sense that every non-empty subset of T has a supremum and an infimum. We give T its interval topology, so that the closed intervals provide a subbase for the closed sets, and the intervals of the form Certain preliminaries about the semigroup S are contained in the paper of Ross [14].
We shall denote by S an arbitrary, infinite subset of T , so that S is a sub-semigroup of (T , ∧) that is also an abelian, idempotent semigroup. For subsets A and B of S, we write A ≤ B if s ≤ t (s ∈ A, t ∈ B). The set S with the discrete topology is denoted by S d , and we set X = β S d and X * = X \ S. The closures of a subset A of S in T and X are cl T A and cl X A, respectively. The map π : X → T denotes the continuous extension of the inclusion map of S into T , so that π(X ) = cl T S. We shall write F t for the fibre {x ∈ X : π(x) = t} for t ∈ cl T S. We set We recall the standard facts, that, for every subset A of S, the set cl X A is clopen in X , and that, for every subsets A and B of S that are disjoint, the two sets cl X A and cl X B are disjoint in X .
We let E denote the set of accumulation points of S in T , so that E = ∅. Take Consider the space {0, 1} κ , where κ is a non-zero cardinal and a generic element in {0, 1} κ is (u α : α ≤ κ). Then {0, 1} κ is a complete lattice with respect to the product and a compact topological semigroup.
Suppose that S is a semi-lattice, and hence a separating semigroup, so that S separates the points of S and (S, ∧) is a partially ordered semigroup. Every semicharacter in S maps S into {0, 1}, and so (S, ∧) can be embedded as a semigroup in C := ({0, 1} κ , ∧), where κ denotes the cardinality of any subset of S that separates the points of S; we regard S as a sub-semigroup of {0, 1} κ . Note that we can always choose to have the cardinality |S| by using the fact that each s ∈ S defines an element θ ∈ S by setting θ(t) = 1 if and only if s ≤ t. In particular, when S = Q, we can set C = {0, 1} ℵ 0 , the Cantor set, whereas Q = c.
Consider the case where (S, ∧) is totally ordered, and define T to be the closure of S in C, so that T is also a complete lattice that is compact in its interval topology. We observe that every θ ∈ can be extended to a semi-character defined on C by putting θ(x) = sup{θ(s) : s ∈ S, s ≤ x} for every x ∈ C, where sup(∅) is regarded as the minimum element of C. We claim that the sub-semigroup (T , ∧) is also totally ordered. To see this, given s ∈ S, take T s to be Then T s is a closed subset of C that contains S, and so it contains T . The set T is contained in T s for every s ∈ S. Now take x ∈ T . Then, for every s ∈ S, we have θ(x) ≤ θ(s) (θ ∈ ) or θ(s) ≤ θ(x) (θ ∈ ). Hence, for each x, y ∈ T , we have x ≤ y or y ≤ x. This shows that (T , ∧) is also totally ordered, giving the claim.
Thus each totally ordered semigroup (S, ∧) can be embedded in a complete, compact, totally ordered topological semigroup (T , ∧), as described above; this fact is well-known.
Take t ∈ E. Throughout we shall write at least one of these sets is non-empty. It follows that and hence the sets F * t ∩ cl X A t and F * t ∩ cl X B t are disjoint, compact subspaces of F * t whose union is F * t .
Lemma 2.1 (i) Take a subset A of S, and suppose that μ ∈ M(X ) [1] with supp μ ⊂ cl X A. Then the measure μ belongs to the weak- * closure of aco{δ s : s ∈ A}.
(ii) Suppose that A and B are subsets of S such that A ≤ B. Take μ, ν ∈ M(X ) with supp μ ⊂ cl X A and supp ν ⊂ cl X B. Then (iii) Take p, q ∈ X with π( p) < π(q). Then p q = q p = p.
Proof (i) Certainly, by [2,Corollary 4.4.16], μ is in the weak- * closure of the set aco{δ u : u ∈ cl X A} , and each δ u for u ∈ cl X A is the weak- * limit of a net in {δ s : s ∈ A}.
(iii) We may suppose that there exists t ∈ [π( p), π(q)] such that p ∈ cl X A t and q ∈ cl X B t , and so this follows from clause (ii).

Lemma 2.2 Take p
Proof The element p is a point of accumulation of S in X , and so π( p) = t is a point of accumulation of S in T , giving the result.

Lemma 2.3
Take p, q ∈ X * , and t ∈ E.
(i) Suppose that p, q ∈ F * t ∩ cl X A t . Then p q = p. (ii) Suppose that p, q ∈ F * t ∩ cl X B t . Then p q = q. (iii) Suppose that p ∈ F * t ∩cl X A t and that q ∈ F * t ∩cl X B t . Then p q = q p = p.
Proof To prove (i), choose nets (s α ) α∈I and (s β ) β∈J in A t which converge to p and q, respectively, in X . Now π(s α ) = s α → π( p) = t and π(s β ) = s β → π(q) = t. So, for each α ∈ I , the net (s β ) β∈J ,s β >s α converges to q in X . Since p q = lim The proof of (ii) is similar, and (iii) is a special case of Lemma 2.1(ii).
We have seen that, for p, q ∈ X with p = q, we have p q = q p unless p and q both belong to sets of the form F * t ∩ cl X A t or of the form F * t ∩ cl X B t for some t ∈ E. These sets will play a crucial rôle in the description of Z(X ) and Z(M(X )).

Lemma 2.4 Take t ∈ E. Then F
Proof We may suppose that t ∈ cl T A t . We first note that, since T is totally ordered, there is an infinite limit ordinal τ and a strictly increasing net (s α : α < τ) in S ∩ [0, t) that converges to t in T .
Let (N k ) be a family of pairwise-disjoint, infinite subsets of N. For each k ∈ N, take E k to be the set of s α such that α = λ + n, where λ is 0 or a limit ordinal and n ∈ N k . The sequence {E k : k ∈ N} partitions the set {s α : α < τ} into an infinite number of disjoint subnets. The sets cl X E k are pairwise-disjoint in X and Since F * t is an infinite, compact subspace of X , we have F * t ≥ 2 c by [7, Theorem 3.59].

Theorem 2.5 The semigroup (S, ∧) is strongly Arens irregular, and the semigroup algebra ( 1 (S), ) is not Arens regular.
Proof Consider a point p ∈ X * , say p ∈ F * t , where t ∈ E. We may suppose that p ∈ cl X A t , and so there exists q ∈ F * t ∩ cl X A t with q = p by Lemma 2.4. By Lemma 2.3(i), it follows that p q = q p, and so p / ∈ Z(X ). Hence Z(X ) = S, showing that S is strongly Arens irregular.
It follows that, in the special case that we are considering, we have Z(M(X )) ∩ X = Z(X ).
We shall now consider when the semigroup algebra ( 1 (S), ) is strongly Arens irregular.
and these are disjoint subsets of X . It follows that This holds for each ε > 0, and so μ p = p μ.

Proposition 2.7
Let μ ∈ M(X ) [1] . Take t ∈ E, and set Suppose that p ∈ U and that μ p = p μ. Then μ | U = zp for some z ∈ D.
Suppose also that q ∈ U with q = p and that μ q = q μ. Then μ | U = 0.
Proof We fix t ∈ E as specified.
We again set U = F * t ∩ cl X A t , and we may suppose that μ | U = 0, using equation (2.1). By Lemma 2.4, there are two distinct points, say p and q, in U . It follows from Proposition 2.7 that either μ p = p μ or μ q = q μ, and hence μ / ∈ Z(M(X )).
Let E be a Banach space, and denote by B(E) the Banach algebra of bounded linear operators on E. P is a projection in B(E). Then the spaces P(E) and ker P are complemented in E and E = P(E) ⊕ ker P; both are 1-complemented if P = 1.
Let E be a (complex) Banach lattice, so that E is the complexification of the real Banach lattice E R . A Banach lattice is Dedekind complete if every increasing net in E + = E + R that is bounded above has a supremum. Let K be a non-empty, compact space. Then M(K ) is a Dedekind complete Banach lattice. Let B ∈ B K , and set Since |μ ∨ ν| ≤ |μ| + |ν| (μ, ν ∈ M(K )), it follows that M B is a sublattice of M(K ). Also M B is Dedekind complete because, for any increasing, bounded net (μ α ) in M + B , we have μ := α μ α exists in M(K ) + , and clearly μ ∈ M B . Thus (We remark that, for each compact space K , the Banach lattice M(K ) is an ALspace; that each AL-space is order-continuous; and that, in an order-continuous Banach lattice, a set is order-closed if and only if it is norm-closed. Thus every norm-closed sublattice of M(K ) is Dedekind complete. For definitions and proofs of these statements, see [17], for example.)

Corollary 2.13 The space Z(M(X )) is 1-complemented in M(X ), and Z(M(X )) is a Banach sublattice of M(X ) that is Dedekind complete.
Proof Take μ ∈ M(X ). The set E 0 := {t ∈ E : |μ|(F * t ) > 0} is countable because {F * t : t ∈ E} is a pairwise-disjoint family of sets. Define P : The sum converges in M(X ), and P is a projection in B(M(X )) with P = 1. By Theorem 2.11, ker P = Z(M(X )), and hence Z(M(X )) is 1-complemented in M(X ). It follows from the above remarks that Z(M(X )) is a Banach sublattice of M(K ) that is Dedekind complete.
Conversely, suppose that cl T S is not scattered. Choose a continuous probability measure, say μ, on cl T S, and set so that F is a closed linear subspace of C(X ). We define a continuous linear functional ν on F by setting so that ν = ν(1 cl T S ) = 1. By the Hahn-Banach theorem, we can extend ν to be a continuous linear functional on C(X ), still with ν = ν(1 cl T S ) = 1; we regard ν as a probability measure on X . Since ν(F * t ) = 0 (t ∈ E), it follows from Proposition 2.9 that ν ∈ Z(M(X )). Since ν / ∈ 1 (S), the Banach algebra 1 (S) is not strongly Arens irregular.
The following corollary includes the case where S = (Q, ∧). Analogous results for the algebra L 1 ([0, 1]), where [0, 1] is a compact, totally ordered semigroup with respect to the operation ∧, are given by Saghafi in [15].

Corollary 2.15 Suppose that cl T S is not scattered. Then
and so the algebra ( 1 (S), ) is neither Arens regular nor strongly Arens irregular. Example 2. 16 We conclude this section by noting that, in Theorem 2.5, we cannot take T to be just partially ordered. Indeed, set T = {0, 1} ℵ 0 , as above, so that T is a partially ordered set that is a lattice. Then take S to consist of the elements of T that have at most one non-zero component, so that (S, ∧) is an abelian, idempotent semigroup, and S is not weakly cancellative. The zero sequence is denoted by 0. It follows from equation (1.1) that u v = 0 for u, v ∈ β S, save when u = v ∈ S. Thus S and 1 (S) are both Arens regular.

DTC sets
We recall from Theorems 1.3 and 1.6 that an infinite, weakly cancellative semigroup is strongly Arens irregular and has a two-point DLTC set, and an infinite, cancellative semigroup S is such that the semigroup algebra 1 (S) is strongly Arens irregular and has a two-point DLTC set contained in S * . We now consider some related results for the semigroups (S, ∧) considered above.
Let S, T , X , and E be as above. We can obtain a DTC set V for M(X ) that is contained in X * as follows.
For each t ∈ E, we choose two distinct points in F * t ∩ cl X A t and two distinct points in F * t ∩ cl X B t whenever the respective sets are non-empty. The collection of these points is called V . Now take μ ∈ M(X ), and suppose that μ p = p μ for each p ∈ V . It follows from Proposition 2.7 that μ | F * t ∩ cl X A t = μ | F * t ∩ cl X B t = 0 for each t ∈ E. Thus μ | F * t = 0. By Theorem 2.11, this implies that μ ∈ Z(M(X )), and hence V is a DTC set for M(X ). It follows that, if E is infinite, M(X ) has a DTC set consisting of at most 2 κ points of X , where κ = |S|; this is a small subset of X because |X | = 2 2 κ .
Suppose that E is finite. Then the above DTC set V is also finite.
Suppose that E is infinite, so that |V | = |E|. Then there cannot be a finite DTC set for the semigroup S. For suppose that V is a finite subset in X , and choose t ∈ E \π(V ), and then choose p ∈ F * t . We have Since S is strongly Arens irregular by Theorem 2.5, this shows that V is not a DTC set for S. Similarly, there is no countable DTC set for S when E is uncountable.
Thus we obtain the following result.

Proof
The set E is scattered because it is a countable, compact space, and so cl T S is scattered. By Theorem 2.14, 1 (S) is strongly Arens irregular. Set (One of these two sets might be empty, but at least one is non-empty.) Take {s n : n ∈ I } and {t n : n ∈ J } to be enumerations of A and B, respectively, where I and J are subsets of N. For each n ∈ I , choose two distinct points, say p 1,n and p 2,n , in F * s n ∩ cl X A s n , and, for each n ∈ J , choose two distinct points, say q 1,n and q 2,n , in F * t n ∩ cl X B t n . Consider the two measures μ j = n∈I 1 2 n δ p j,n ( j = 1, 2) , defined when A = ∅, and similarly μ 3 , μ 4 , defined when B = ∅. We obtain at most four measures in M(X * )+, forming a set V .
We claim that V is a DTC set for M(X ). Indeed, suppose that μ ∈ M(X ) and that μ ν = ν μ (ν ∈ V ).
Proof Assume that V is a DTC set for M(X ) with |V | < κ. For each ν ∈ V , the set {t ∈ E : ν | F * t = 0} is countable because {F * t : t ∈ E} is a family of pairwise-disjoint, non-empty, compact sets. Hence there exists t ∈ E such that ν | F * t = 0 (ν ∈ V ). Choose p ∈ F * t . It follows from Lemma 2.6 that p ν = ν p (ν ∈ V ). By the assumption, p ∈ Z(M(X )), a contradiction of Proposition 2.10.
It follows that there is no DTC set for M(X ) with |V | < κ.
It follows that any two distinct points in N * form a two-point DTC set for (N, ∧). Further, 1 (N, ∧) is strongly Arens irregular and any two distinct points in N * form a DTC set for M(β N).
(ii) Consider the semigroup (Q, ∧). By Proposition 3.1, there is no countable DTC set for this semigroup. By Corollary 2.15, 1 (Q, ∧) is neither Arens regular nor strongly Arens irregular, and, by Corollary 3.5, there is no countable DTC for M(X ).
(iii) Consider the subset S of T := {−∞} ∪ R ∪ {∞} that consists of numbers of the form n − x, where n ∈ Z and x ∈ {1/2, 1/4, 1/8, . . . }. Then the corresponding set E is equal to {−∞} ∪ Z ∪ {∞}, a countable set, and so, by Theorem 3.2, the semigroup algebra 1 (S) is strongly Arens irregular. By Proposition 3.1, there is no finite DTC set for the semigroup S, but, by the argument of Theorem 3.2, M(X ) has a two-element DTC set in M(X * ) + .
(iv) Consider the semigroup S = T = ([0, κ], ∧), where κ is a cardinal with κ ≥ ℵ 1 , so that the corresponding set E has cardinality κ. Since T is scattered, the algebra 1 (S) is strongly Arens irregular by Theorem 2.14. By Proposition 3.4, there is no DTC set for M(X ) with cardinality strictly less than κ. This shows that the cardinality of a DTC set can be arbitrarily large, even when 1 (S) is strongly Arens irregular. and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.