An answer to A.D.Wallace’s question about countably compact cancellative semigroups

Robbie, Desmond A.; Svetlichny, Sergey

doi:10.1090/s0002-9939-96-03418-1

Cited by 28 publications

(13 citation statements)

References 7 publications

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“…This means that it is, in fact, a closed subgroup: see e.g. [13,Lemma B.1] for (a strengthening of) the well known result to the effect that compact cancellative semigroups are groups.…”

Section: Translationsmentioning

confidence: 96%

Translations in quantum groups

Chirvăsitu

2020

Banach Center Publ.

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Let H be the Hopf C * -algebra of continuous functions on a (locally) compact quantum group of either reduced or full type. We show that endomorphisms of H that respect its right regular comodule structure are translations by elements of the largest classical subgroup of G.Furthermore, we show that for compact G such an endomorphism is automatically an automorphism regardless of the quantum group norm on the C * -algebra H. Introduction.The starting point for the present note is the observation that for a compact group G the map sending g ∈ G to the translationinduces an isomorphism between the group G and the monoid of C * -algebra endomorphisms α : C(G) → C(G) that respect the right comodule structure of C(G) in the sense that

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“…This means that it is, in fact, a closed subgroup: see e.g. [13,Lemma B.1] for (a strengthening of) the well known result to the effect that compact cancellative semigroups are groups.…”

Section: Translationsmentioning

confidence: 96%

Translations in quantum groups

Chirvăsitu

2020

Banach Center Publ.

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“…Conditions implying that a cancellative topological semigroup S is a topological group is another widely studied topic in Topological Algebra (see, [11,24,33]). The following corollary of Propositions 4.10 and 6.4 contributes to this field.…”

Section: Polyboundedness and Its Applicationsmentioning

confidence: 99%

Categorically Closed Topological Groups

Банах

2017

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Let C be a category whose objects are semigroups with topology and morphisms are closed semigroup relations, in particular, continuous homomorphisms. An object X of the category C is called C-closed if for each morphism Φ ⊂ X × Y in the category C the image Φ(X) = {y ∈ Y : ∃x ∈ X (x, y) ∈ Φ} is closed in Y. In the paper we survey existing and new results on topological groups, which are C-closed for various categories C of topologized semigroups.

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“…In 1996, Robbie and Svetlichny [10] answered Wallace's question in the negative under CH. A counterexample to Wallace's question has been then called a Wallace semigroup.…”

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confidence: 99%

Countably compact group topologies on the free Abelian group of size continuum (and a Wallace semigroup) from a selective ultrafilter

2019

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To Ofelia T. Alas on the occasion of her 75th birthday.Abstract. We prove that the existence of a selective ultrafilter implies the existence of a countably compact Hausdorff group topology on the free Abelian group of size continuum. As a consequence, we show that the existence of a selective ultrafilter implies the existence of a Wallace semigroup (i.e., a countably compact both-sided cancellative topological semigroup which is not a topological group).

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An answer to A.D.Wallace’s question about countably compact cancellative semigroups

Cited by 28 publications

References 7 publications

Translations in quantum groups

Translations in quantum groups

Categorically Closed Topological Groups

Countably compact group topologies on the free Abelian group of size continuum (and a Wallace semigroup) from a selective ultrafilter

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