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Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - Arens regularity for totally ordered semigroups
AU - Dales, H.G.
AU - Strauss, D.
PY - 2022/8/31
Y1 - 2022/8/31
N2 - Let S be a semigroup. We shall consider the centres of the semigroup (βS,□) and of the algebra (M(βS),□), where M(βS) is the bidual of the semigroup algebra (ℓ1(S),⋆), and whether the semigroup and the semigroup algebra are Arens regular, strongly Arens irregular, or neither. We shall also determine subsets of S ∗ and of M(S ∗) that are ‘determining for the left topological centre’ (DLTC sets) of βS and M(βS). It is known that, when the semigroup S is cancellative, ℓ1(S) is strongly Arens irregular and that there is a DLTC set consisting of two points of S ∗. In contrast, there is little that has been published about the Arens regularity of ℓ1(S) when S is not cancellative. Totally ordered, abelian semigroups, with the map (s, t) → s∧ t as the semigroup operation, provide examples which show that several possibilities can occur. We shall determine the centres of βS and of M(βS) 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 or for M(βS). There was no previously-known example of an abelian semigroup S for which βS or M(βS) did not have a finite DTC set.
AB - Let S be a semigroup. We shall consider the centres of the semigroup (βS,□) and of the algebra (M(βS),□), where M(βS) is the bidual of the semigroup algebra (ℓ1(S),⋆), and whether the semigroup and the semigroup algebra are Arens regular, strongly Arens irregular, or neither. We shall also determine subsets of S ∗ and of M(S ∗) that are ‘determining for the left topological centre’ (DLTC sets) of βS and M(βS). It is known that, when the semigroup S is cancellative, ℓ1(S) is strongly Arens irregular and that there is a DLTC set consisting of two points of S ∗. In contrast, there is little that has been published about the Arens regularity of ℓ1(S) when S is not cancellative. Totally ordered, abelian semigroups, with the map (s, t) → s∧ t as the semigroup operation, provide examples which show that several possibilities can occur. We shall determine the centres of βS and of M(βS) 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 or for M(βS). There was no previously-known example of an abelian semigroup S for which βS or M(βS) did not have a finite DTC set.
KW - Stone–Čech compactifications of semigroups
KW - Totally ordered sets as semigroups
KW - Arens products on the second dual of a Banach algebra
KW - Topological centres
KW - DTC sets
U2 - 10.1007/s00233-022-10299-x
DO - 10.1007/s00233-022-10299-x
M3 - Journal article
VL - 105
SP - 172
EP - 190
JO - Semigroup Forum
JF - Semigroup Forum
SN - 0037-1912
IS - 1
ER -