West semigroups as compactifications of locally compact abelian groups

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Elgun E.

SEMIGROUP FORUM, cilt.93, sa.1, ss.71-85, 2016 (SCI-Expanded) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 93 Sayı: 1
  • Basım Tarihi: 2016
  • Doi Numarası: 10.1007/s00233-015-9747-8
  • Dergi Adı: SEMIGROUP FORUM
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Sayfa Sayıları: ss.71-85
  • İstanbul Üniversitesi Adresli: Hayır

Özet

In this paper, we will identify certain subsemigroups of the unit ball of as semitopological compactifications of locally compact abelian groups, using an idea of West (Proc R Ir Acad Sect A 67:27-37, 1968). Our result has been known for the additive group of integers since Bouziad et al. (Semigr Forum 62(1):98-102, 2001). We will construct a semitopological semigroup compactification for each locally compact abelian group G, depending on the algebraic properties of G. These compact semigroups can be realized as quotients of both the Eberlein compactification , and the weakly almost periodic compactification, , of G. The concrete structure of these compact quotients allows us to gain insight into known results by Brown (Bull Lond Math Soc 4:43-46, 1972) and Brown and Moran (Proc Lond Math Soc 22(3):203-216, 1971) and by Bordbar and Pym (Math Proc Camb Philos Soc 124(3):421-449, 1998), where for the groups and , it is proved that and contain uncountably many idempotents and the set of idempotents is not closed.

In this paper, we will identify certain subsemigroups of the unit ball of L∞[0,1]L∞[0,1] as semitopological compactifications of locally compact abelian groups, using an idea of West (Proc R Ir Acad Sect A 67:27–37, 1968). Our result has been known for the additive group of integers since Bouziad et al. (Semigr Forum 62(1):98–102, 2001). We will construct a semitopological semigroup compactification for each locally compact abelian group G, depending on the algebraic properties of G. These compact semigroups can be realized as quotients of both the Eberlein compactification GeGe, and the weakly almost periodic compactification, GwGw, of G. The concrete structure of these compact quotients allows us to gain insight into known results by Brown (Bull Lond Math Soc 4:43–46, 1972) and Brown and Moran (Proc Lond Math Soc 22(3):203–216, 1971) and by Bordbar and Pym (Math Proc Camb Philos Soc 124(3):421–449, 1998), where for the groups G=ZG=Z and G=Z∞qG=Zq∞, it is proved that GeGe and GwGw contain uncountably many idempotents and the set of idempotents is not closed.