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Research output: Contribution to Journal/Magazine › Journal article › peer-review

Published

In: Semigroup Forum, Vol. 105, No. 1, 31.08.2022, p. 172–190.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Dales, HG & Strauss, D 2022, 'Arens regularity for totally ordered semigroups', *Semigroup Forum*, vol. 105, no. 1, pp. 172–190. https://doi.org/10.1007/s00233-022-10299-x

Dales, H. G., & Strauss, D. (2022). Arens regularity for totally ordered semigroups. *Semigroup Forum*, *105*(1), 172–190. https://doi.org/10.1007/s00233-022-10299-x

Dales HG, Strauss D. Arens regularity for totally ordered semigroups. Semigroup Forum. 2022 Aug 31;105(1):172–190. Epub 2022 Jul 18. doi: 10.1007/s00233-022-10299-x

@article{17443128af014a518fb6c3e274a34a5f,

title = "Arens regularity for totally ordered semigroups",

abstract = "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 {\textquoteleft}determining for the left topological centre{\textquoteright} (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. ",

keywords = "Stone–{\v C}ech compactifications of semigroups, Totally ordered sets as semigroups, Arens products on the second dual of a Banach algebra, Topological centres, DTC sets",

author = "H.G. Dales and D. Strauss",

year = "2022",

month = aug,

day = "31",

doi = "10.1007/s00233-022-10299-x",

language = "English",

volume = "105",

pages = "172–190",

journal = "Semigroup Forum",

issn = "0037-1912",

publisher = "Springer New York",

number = "1",

}

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 -