A Note on Locally Compact Subsemigroups of Compact Groups
aa r X i v : . [ m a t h . GN ] O c t A NOTE ON LOCALLY COMPACT SUBSEMIGROUPS OFCOMPACT GROUPS
JULIO C´ESAR HERN ´ANDEZ ARZUSA AND KARL H. HOFMANN
Abstract.
An elementary proof is given for the fact that every locallycompact subsemigroup of a compact topological group is a closed sub-group. A sample consequence is that every commutative cancellativepseudocompact locally compact Hausdorff topological semigroup withopen shifts is a compact topological group.
Mathematics Subject Classification 2010:
Primary: 20M10, 22A25,22C05; Secondary: 54B30, 54H10.
Keywords and phrases:
Topological semigroup, compact group, cancella-tive semigroup, precompact, pseudocompact.
The Basics
A nonempty subset S of a group satisfying SS ⊆ S is calleda subsemigroup. All topological spaces are considered Hausdorff spaces. Definition 1. a) A topological group is called
Weil-adapted if the closureof each subsemigroup is a group.b) For a subset A of a topological space X , a point x ∈ A is a conditionallyinner point of A , if x has an open neighborhood U such that x ∈ A ∩ U ⊆ A . The following lemma has a remarkably elementary, self-contained, andstraightforward proof: emma 2. Let S be a subsemigroup of a Weil-adapted topological group G .If S has a conditionally inner point, then S is a closed subgroup of G .Proof. Define C = S . Then the continuity of the multiplication implies that CC = S S ⊆ SS ⊆ S = C . So C is a closed subsemigroup of G and thereforeis a group since G is Weil-adpated. It is no loss of generality to assume that C = G . So we assume now that S is dense in G and we must show S = G .Let T be the interior of S , then T = ∅ since S has a conditionally inner point.Take s ∈ S and t ∈ T , since left translations of G are homeomorphisms, sT is an open neighborhood of st and sT ⊆ sS ⊆ S . Hence st is containedin the interior T of S , so T is a left ideal of S . Let D = T , then D is asubgroup of G since G is Weil-adapted. Now SD = ST ⊆ ST ⊆ T = D .Therefore S ⊆ D , and since S is dense in G , we have D = G , that is, T isdense in G . Now the mapping x x − from G to G is a homeomorphism,thus T − is the dense interior of S − . Let H = T ∩ T − , then H is openand dense in G , too, and in addition, is closed under both multiplicationand inversion. Hence H is an open dense subgroup of G . But any opensubgroup of a topological group is closed (as complement of the union of allother cosets) and so H = G follows. Then G = H = T ∩ T − ⊆ T ⊆ S finally shows S = G , completing the proof of the lemma. (cid:3) (cid:4) Lemma 3.
In a topological group G the following conditions are equivalent:i) G is Weil-adapted.ii) Each closed subsemigroup of G is a group.iii) For each g ∈ G , the closure { g, g , g , . . . } is a subgroup of G .Proof. Trivially i ) implies ii ), and since the closure of a subsemigroup of atopological group is a subsemigroup according to the first step of the proofof Lemma 1, also ii ) implies iii ). So we have to prove that iii ) implies i ). Indeed, let S be a subsemigroup of G and let s ∈ S and consider C = { s, s , s , . . . } . Then C is a subgroup of G by iii ) and is contained in S bythe definition of C . So s − ∈ C ⊆ S which shows that S is a subgroup,which we had to show. (cid:3) (cid:4) Lemma 4.
Any subgroup of a Weil-adapted topological group is Weil-adapated.Proof.
Let A be a subgroup of a Weil-adapted topological group G and let S be a subsemigroup of A . Since G is Weil-adapted, the closure S of S in G is a group by Definition 1(a). Then the closure S ∩ A of S in A is a groupas well. (cid:3) (cid:4) We recall Weil’s Lemma saying that for an element g in a locally compactgroup the subgroup { . . . , g − , g − , , g, g , . . . } is either isomorphic to thediscrete group Z , or else { g, g , g , . . . } is dense in a compact subgroup (seee.g. [4], 7.43). This explains the terminology of Definition 1(a). Weil’sLemma also holds in any pro-Lie group by [5], 5.3. Accordingly, a locallycompact group or a pro-Lie group is Weil-adapted if and only if it does ot contain infinite discrete cyclic subgroups. In particular, every compactgroup is Weil-adapted.Now a precompact group P has a compact completion. The latter isWeil-adapted, hence by Lemma 4, P is Weil-adapated. So Lemma 2 impliesthe following corollary. Corollary 5.
A subsemigroup of a precompact group is a group if it hasconditionally inner points.
In particular, any open subsemigroup of a precompact group is a group.For example, Z in its p-adic topology is precompact, hence is Weil-adaptedbut N fails to be closed.We say that a subspace of a topological space is conditionally open ifeach of its points is a conditionally inner point. Now recall that a locallycompact space is conditionally open in any Hausdorff space that contains it;the elementary proof is an exercise and is provided in [2], 3.3.9.Accordingly, the following conclusions are immediate: Proposition 6.
Any locally compact subsemigroup of a Weil-adapted topo-logical group is a closed subgroup.
Corollary 7.
Any locally compact subsemigroup of a precompact group is agroup.
Note that any locally compact subgroup of a Hausdorff topological groupis closed (see e.g. [4], Corollary A4.24). As a consequence we have thefollowing corollary.
Corollary 8 ( F. Wright , [8]) . Any locally compact subsemigroup of acompact group is a compact subgroup.
This concludes the essentially selfcontained part of this note. The follow-ing discussion makes references to other publications.
Some Consequences
The literature exhibits a variety of sufficientconditions for a cancellative topological semigroup S to be a topologicalgroup, Indeed this is true if S isi) compact ([4, Proposition A4.34]),ii) countably compact first countable ([7, Corollary 5])iii) sequentially compact ([1, Theorem 6]),iv) commutative and locally compact connected with open shifts ([3, The-orem 4]), orv) commutative feebly compact first countable regular with open shifts ([3,Theorem 3]).Here a space is called feebly compact if each locally finite open family is finite.What we call a shift in a semigroup is frequently also called a translation.Recall also that a space is called pseudocompact if every every real valued unction on it is bounded. The list can now be expanded if we first quoteCorollary 2 of [3] as follows: Proposition 9.
A commutative cancellative locally compact pseudocompacttopological semigroup with open shifts can be embedded in a compact topo-logical group as a dense open subsemigroup.
Now from Lemma 2 and Proposition 9 we obtain the following corollary.
Corollary 10.
Each commutative cancellative locally compact pseudocom-pact topological semigroup with open shifts is a compact topological group.
It is well known that every locally compact pseudocompact topologicalgroup is a compact topological group (see [6, Theorem 2.3.2]). Corollary 10now confirms this conclusion for a class of topological semigroups.It may be helpful to recall the example of the circle group T = R / Z inwhich the subsemigroup S = ( Z + √ N ) / Z is not a subgroup.From Proposition 6 we know that a genuine subsemigroup of a compactgroup (or indeed any Weil-complete group) cannot be locally compact. Forthe additive group R of reals, for any positive real number r and for any opensubset S such that ]2 r, ∞ [ ⊆ S ⊆ ] r, ∞ [, the subset S is an open subsemigroupof R . References [1] B. Bokalo and I. Guran,
Sequentially compact Hausdorff cancellative semigroup is atopological group
Matematychni Studii (1996), 39–40.[2] R. Engelking General Topology, Revised and completed edition
Heldermann Verlag,Berlin, 1989[3] J. Hern´andez Arzusa,
Commutative Topological Semigroups Embedded into TopologicalAbelian Groups , Axioms
DOI 10.3390/axioms9030087 (2020), 9p.[4] K. H. Hofmann and S. A. Morris,
The Structure of Compact Groups: a Primer for theStudent, a Handbook for the expert,
De Gruyter Studies in Mathematics , Walterde Gruyter, Berlin, 4th Ed. 2020.[5] K. H. Hofmann and S. A. Morris, The Lie Theory of Connected ProLie Groups
Tractsin Math. 2, Europen Math. Soc. Publishing House, Z¨urich, 2007.[6] M. Hrusak, A. Tamariz, and M. Tkachenko,
Pseudocompact Topological Spaces,
De-velopment in Mathematics , Springer, Morelia M´exico, 2018.[7] A. Mukherjea, and N. Tserpes, A note on countably compact semigroups , J. Austral.Math. Soc., (1972), 180–184.[8] F. B. Wright, Subsemigroups in compact groups , Proc. Amer. Math. Soc. (1956),309–411. Julio C´esar Hern´andez ArzusaPrograma de Matem´aticasUniversidad de CartagenaCampus San Pablo - Zaragocilla130014, Cartagena, [email protected] arl Heinrich HofmannFachbereich MathematikTechnische Universit¨at DarmstadtSchlossgartenstraße 764289 Darmstadt, [email protected] Heinrich HofmannFachbereich MathematikTechnische Universit¨at DarmstadtSchlossgartenstraße 764289 Darmstadt, [email protected]