The weak compactification of locally compact groups
aa r X i v : . [ m a t h . GN ] F e b THE WEAK COMPACTIFICATION OF LOCALLY COMPACTGROUPS
MAR´IA V. FERRER AND SALVADOR HERN ´ANDEZ
Respectfully dedicated to Professor
Hans-Peter Kunzi
Abstract.
We further investigate the weak topology generated by the irreducibleunitary representations of a group G . A deep result due to Ernest [13] and Hughes[22] asserts that every weakly compact subset of a locally compact (LC) group G is compact in the LC-topology, generalizing thereby a previous result of Glicksberg[19] for abelian locally compact (LCA) groups. Here, we first survey some recentfindings on the weak topology and establish some new results about the preservationof several compact-like properties when going from the weak topology to the originaltopology of LC groups. Among others, we deal with the preservation of countablycompactness, pseudocompactness and functional boundedness. Introduction
In contrast to what happens with abelian groups, where the Bohr compactificationand Bohr topology display many nice features, the Bohr compactification presentsimportant shortcomings when applied to non-commutative groups. For example, itmay happen that the Bohr compactification of a locally compact group becomes trivial,what makes it useless in order to study the structure of those non-abelian groups. Onthe other hand, it is known, as a consequence of the celebrated Gel’fand and RaˇıkovTheorem, that the set of all irreducible unitary representations of a locally compactgroup G contains all the information necessary to recover the topological and algebraic Date : February 25, 2021.Research Partially supported by the Spanish Ministerio de Econom´ıa y Competitividad, grant:MTM/PID2019-106529GB-I00 (AEI/FEDER, EU).
Primary 22D05; 43A46. Secondary 22D10; 22D35; 43A40;54H11
Key Words and Phrases:
Locally compact group; I set; Sidon set; Interpolation set; Weak topology;Bohr compactification; Eberlein compactification. structure of the group (see [12]). Therefore, it seems appropriate to consider the weaktopology generated by the irreducible unitary representations of a group G as thegenuine weak topology of general not necessarily LC group G (as a matter of fact,they coincide for abelian topological groups, by the Schur’s Lemma). This is what wehave done in [15] and, here, we further develop this approach, which was initiated byHughes in [22].There is a plethora of results that concern the weak topology of abelian topologicalgroups and, although there still are interesting open questions in this setting, we haveplenty of information about the weak topology of abelian LC groups. The literaturein this regard is vast, so we only mention [7] and the references therein. On the otherhand, there also are some crucial findings that have been already established for generalLC groups, which are less known so far but form the basis for an in-depth study of weaktopologies in the setting of not necessarily abelian groups. It is pertinent to mentionhere the task developed by Ernest [13] and Hughes [22], who proved that every weaklycompact subset of a LC group G is compact in the LC-topology, generalizing thereby aprevious result of Glicksberg [19] for abelian locally compact (LCA) groups. The maingoal of this paper is twofold. First, we survey some recent results on the weak topologyof LC groups. On the other hand, we further develop this research line by studyingthe preservation of several compact-like properties when going from the weak topologyto the original topology of LC groups. Among others, we deal with the preservationof countably compactness, pseudocompactness and functional boundedness. We nowcollect some definitions and basic facts that will be used along the paper. HE WEAK COMPACTIFICATION OF LOCALLY COMPACT GROUPS 3 Basic facts
The Bohr topology.
The Bohr compactification of a topological group is awell known notion that has been widely treated in the setting of topological groups.Nevertheless, we remind here its most basic features for the reader’s sake. With every(not necessarily abelian) topological group G there is associated a compact Hausdorffgroup bG , the so-called Bohr compactification of G , and a continuous homomorphism b of G onto a dense subgroup of bG such that bG is characterized by the followinguniversal property: given any continuous homomorphism h of G into a compact group K , there is always a continuous homomorphism ¯ h of bG into K such that h = ¯ h ◦ b (see [21, V § bG and their properties is given). The Bohrtopology of a topological group G is the one that inherits as a subgroup of bG .In the sequel, following a terminology introduce by Trigos-Arrieta [30], if G is atopological group, we denote by G + the same algebraic group but equipped with theBohr topology. However, when the group G is discrete, we will use the symbolism G ♯ used by van Douwen in [19].As far as we know, the first main result about the Bohr topology is due to Glicksberg[19]. Theorem 2.1 (Glicksberg, 1962) . Let ( G, τ ) be a LCA group. It holds that A ⊆ G is τ -compact if and only if is τ + -compact. However, the systematic study of the Bohr topology was started by van Douwen(loc. cit.) in the setting of discrete abelian groups.
Theorem 2.2 (van Douwen, 1990) . Every A ⊆ G contains a subset D with | D | = | A | that is relatively discrete and C -embedded in G ♯ as well as C ∗ -embedded in bG . M. FERRER AND S. HERN ´ANDEZ
In addition to the previous main result, van Douwen studied the Bohr topologyof a discrete abelian group in depth. It is remarkable that, except for the standingabelian hypothesis, his proofs of results concerning ♯ -groups made no use whatsoeverof specific algebraic properties. This probably led him to ask whether two groups G and G with the same cardinality should have G ♯ and G ♯ homeomorphic. Someyears later Kunen [24] and, independently, Dikranjan and Watson [10], gave examplesof torsion groups with the same cardinality having nonhomeomorphic ♯ -spaces. Still,much remains unknown, even among groups of countable cardinality. One can getan idea of how involved the situation is by taking into account, see [9], that Q ♯ and(( Q / Z ) × Z ) ♯ are homeomorphic.Van Douwen’s work on the Bohr topology of discrete abelian groups was continued byTrigos-Arrieta, for locally compact abelian (LCA) groups, in his doctoral dissertation[30]. There, the author introduces the following notion: Definition 2.3 (Trigos-Arrieta, 1991) . Let G be a topological group and let P be atopological property. We say that G respects P if for any subset F of G the followingholds: the subspace F of G has P if and only if the subspace F of G + has P .For example, since G = G + for G compact, it is obvious that compact groupsrespects all topological properties. In [30], Trigos-Arrieta proves that LCA groupsrespect most compact-like properties: pseudocompactness, functional boundedness,and other topological properties: Lindel¨ofness and connectedness.Further properties concerning the Bohr topology of a LCA group can be found in[8, 9, 17, 20] HE WEAK COMPACTIFICATION OF LOCALLY COMPACT GROUPS 5
The weak topology of LC groups.
Given a locally compact group (
G, τ ),we denote by
Irr ( G ) the set of all continuous unitary irreducible representations σ defined on G . That is, continuous in the sense that each matrix coefficient function g
7→ h σ ( g ) u, v i is a continuous map of G into the complex plane. Thus, fixed σ ∈ Irr ( G ), if H σ denotes the Hilbert space associated to σ , we equip the unitary group U ( H σ ) with the weak (equivalently, strong) operator topology. For two elements π and σ of Irr ( G ), we write π ∼ σ to denote the relation of unitary equivalence and wedenote by b G the dual object of G , which is defined as the set of equivalence classes in( Irr ( G ) / ∼ ). We refer to [11, 16, 2] for all undefined notions concerning the unitaryrepresentations of locally compact groups.Adopting, the terminology introduced by Ernest in [13], set H n def = C n for n =1 , , . . . ; and H = l ( Z ). The symbol Irr Cn ( G ) will denote the set of irreducibleunitary representations of G on H n , where it is assumed that every set Irr Cn ( G ) isequipped with the compact open topology. Finally, define Irr C ( G ) = F n ≥ Irr Cn ( G ) (thedisjoint topological sum).We denote by G w = ( G, w( G, Irr ( G )) the group G equipped with the weak (group)topology generated by Irr ( G ). Since equivalent representations define the same topol-ogy, we have G w = ( G, w( G, b G )). That is, the weak topology is the initial topology on G defined by the dual object. Moreover, in case G is a separable, metric, locally com-pact group, then every irreducible unitary representation acts on a separable Hilbertspace and, as a consequence, is unitary equivalent to a member of Irr C ( G ). Thus G w = ( G, w( G, Irr C ( G ))) for separable, metric, locally compact groups. We will makeuse of this fact in order to avoid the proliferation of isometries (see [11]). In case thegroup G is abelian, the dual object b G is a group, which is called dual group , and the M. FERRER AND S. HERN ´ANDEZ weak topology of G reduces to the weak topology generated by all continuous homo-morphisms of G into the unit circle T . That is, the weak topology coincides with theBohr topology of G .2.3. The weak compactification of LC groups.Definition 2.4.
We denote by P ( G ) the set of continuous positive definite functionson ( G, τ ). If σ ∈ Irr ( G ) and v ∈ H σ , then the positive definite function: ϕ : g
7→ h σ ( g ) v, v i , g ∈ G is called pure , and the family of all such functions is denoted by I ( G ). When G isabelian, the set I ( G ) coincides with the dual group b G of the group G .The proof of the lemma below is straightforward. Lemma 2.5.
Let G be a locally compact group. Then G w = ( G, w( G, I ( G ))) . Definition 2.6.
Let G be a locally compact group and consider the following naturalembedding: w : G ֒ → Y ϕ ∈ I ( G ) ϕ ( G ) with w( g ) = ( ϕ ( g )) ϕ ∈ I ( G ) The weak compactification w G of G as the pair (w G, w), where w G def = w( G ).This compactification has been previously considered in [5, 6] using different tech-niques. Also Akemann and Walter [1] extended Pontryagin duality to non-abelianlocally compact groups using the family of pure positive definite functions. Again, incase G is abelian, the compactifican (w G, w) coincides with bG , the Bohr compactifi-cation of G . HE WEAK COMPACTIFICATION OF LOCALLY COMPACT GROUPS 7
A better known compactification of a locally group G which is closely related to w G is defined as follows (cf. [14, 29]): let B(G) k·k ∞ denote the commutative C ∗ -algebraconsisting of the uniform closure of the Fourier-Stieltjes algebra of G . Here, the Fourier-Stieltjes algebra is defined as the matrix coefficients of the unitary representations of G . Following [26] we call the spectrum eG of B(G) k·k ∞ the Eberlein compactification of G . Since the Eberlein compactification eG is defined using the family of all continuouspositive definite functions, it follows that w G is a factor of eG and, as a consequence, in-herits most of its properties. In particular, w G is a compact involutive semitopologicalsemigroup.We now recall some known results about unitary representations of locally compactgroups that are needed in the proof of our main result in this section. One main point isthe decomposition of unitary representations by direct integrals of irreducible unitaryrepresentations. This was established by Mautner [25] following the ideas introducedby von Neuman in [27]. Theorem 2.7 (F. I. Mautner, [25]) . For any representation ( σ, H σ ) of a separable lo-cally compact group G , there is a measure space ( R, R , r ) , a family { σ [ p ] } of irreduciblerepresentations of G , which are associated to each p ∈ R , and an isometry U of H σ such that U σU − = Z R σ [ p ] d r p. Remark . The proof of the above theorem given by Mautner assumes that the rep-resentation space H σ is separable but, subsequently, Segal [28] removed this constraint.Furthermore, it is easily seen that we can assume that σ [ p ] belongs to Irr C ( G ) locallyalmost everywhere in the theorem above (cf. [23]). M. FERRER AND S. HERN ´ANDEZ
A remarkable consequence of Theorem 2.7 is the following corollary about positivedefinite functions.
Corollary 2.9.
Every Haar-measurable positive definite function ϕ on a separablelocally compact group G can be expressed for all g ∈ G outside a certain set of Haar-measure zero in the form ϕ ( g ) = Z R ϕ p ( g ) d r p, where ϕ p is a pure positive definite functions on G for all p ∈ R . The following proposition is contained in the proof of Lemma 3.2 of Bichteler [3, pp.586-587]
Proposition 2.10.
Let G be a locally compact group. If H is an open subgroup of G ,then each continuous irreducible representation of H is the restriction of a continuousirreducible representation of G . Locally compact groups respect compactness
In this section we prove an old result by Ernest [13] (cf. [23]) and [22], assertingthat the weak topology of locally compact groups respects compactness, that is it holdsthat every weakly compact subset is compact for the original locally compact grouptopology. In fact, Ernest first proved that, for separable metric locally compact groups,every weakly convergent sequence is convergent for the locally compact topology andthis result was subsequently extended by Hughes by proving that the weak topologyof locally compact groups respects compactness.Unfortunately, even though the formulation of Hugues’ result quoted above can befound in [22], its full proof only appears in his Doctoral dissertation but has never been
HE WEAK COMPACTIFICATION OF LOCALLY COMPACT GROUPS 9 published later. Therefore, we have decided to include it here for the reader’s sake.Our proof is complete since it contains both Ernest’s and Hughes’ results. We firstneed a further definition and several previous lemmas.
Definition 3.1.
Let U be an open neighbourhood of the identity of a topological group G . We say that a sequence { g n } n<ω is U -discrete if g n U ∩ g m U = ∅ for all n = m ∈ ω . Lemma 3.2. (J. Ernest, 1971)
Let ( G, τ ) be a separable metric locally compact group.Then every convergent sequence { g n } n<ω in G w is also τ -convergent in G .Proof. Remark that there is no loss of generality in assuming that { g n } n<ω convergesto e G in G w . We must verify that { g n } n<ω is τ -convergent to e G .In order to do so, take a continuous positive definite function ψ ∈ P ( G ). Since G isseparable, by Corollary 2.9, there is a measure space ( R, R , r ), a family { ψ p } of purepositive definite functions on G , which are associated to each p ∈ R , such that ψ ( g ) = Z R ψ p ( g ) d r p for all g ∈ G. Therefore ψ ( g n ) = Z R ψ p ( g n ) d r p for all n < ω. Now, for each n < ω , consider the map f n on R by f n ( p ) def = ψ p ( g n ). Then f n isintegrable on R and, since { g n } n<ω is weakly convergent to e G , it follows that { f n ( p ) } converges to ψ p ( e G ) for all p ∈ R . Furthermore, if ψ p ( g ) = < σ p ( g ) v p , v p > for some σ p ∈ Irr ( G ) and v p ∈ H σ p , it follows that | f n ( p ) | = | ψ p ( g n ) | = | < σ p ( g n ) v p , v p > | ≤ k v p k . Thus defining f on R as the pointwise limit of { f n } , we are in position to applyLebesgue’s dominated convergence theorem in order to obtain that ψ ( e G ) = Z R ψ p ( e G ) d r p = Z R f ( p ) d r p = lim n →∞ Z R ψ p ( g n ) d r p = lim n →∞ ψ ( g n ) . In other words, the sequence { ψ ( g n ) } converges to ψ ( e G ) for all ψ ∈ P ( G ). Hence { g n } τ -converges to e G in G and we are done. (see [18] or [16, Prop. 3.33]). (cid:3) Theorem 3.3. (J.R. Hughes, 1972) Let (
G, τ ) be a locally compact group. Then(
G, τ ) and G w contain the same compact subsets. Proof.
Let B be a weakly compact subset of G . Remark that if B were τ -precompact,since B is τ -closed in G , it would follow that B is τ -compact in G .Thus, reasoning by contradiction, we assume that B is not τ -precompact in G . Thenthere exists an open, symmetric and relatively compact neighbourhood of the identity U in G such that B contains a U -discrete sequence { g n } n<ω .Consider the subgroup H def = < U ∪ { g n } n<ω > , which σ -compact and open in G .By Kakutani-Kodaira’s theorem, there exists a normal, compact subgroup K of H such that K ⊆ U and H/K is metrizable, and consequently separable (Polish). Let p : H → H/K be the quotient homomorphism and let p : w H → w H/K denote itscanonical extension to the weak compactifications. By Proposition 2.10, we have that H w G is canonically homeomorphic to w H . Therefore { g n } n<ω weakly converges to theneutral element in H . Hence { p ( g n ) } n<ω weakly converges to the neutral element in H/K , which is a separable, metrizable, LC group. Thus, by Lemma 3.2, the sequence { p ( g n ) } n<ω τ /K -converges to the neutral element in H/K . Then by a theorem ofVaropoulos [31], the sequence { p ( g n ) } n<ω can be lifted to a sequence { x n } n<ω ⊆ H converging to some point x ∈ H . This entails that x − n g n ∈ K for all n ∈ ω . Thus the HE WEAK COMPACTIFICATION OF LOCALLY COMPACT GROUPS 11 sequence { g n } n<ω would be contained in the compact subset ( { x n } n<ω ∪ { x } ) K , whichis a contradiction since { g n } n<ω was supposed to be U -discrete. This contradictioncompletes the proof. (cid:3) Weakly Cauchy sequences.
In some special cases, Hughes’ theorem impliesthe convergence of weakly Cauchy sequences. Indeed, let us denote by inv (w G ) thegroup of invertible elements of w G . It is known (see [38, Proposition II.4.6.(i)]) thatevery maximal subgroup of a compact semitopological semigroup is a topological groupthat is complete with respect to the two-sided uniformity. In particular, this appliesto inv (w G ), which is a complete (for the two-sided uniformity) topological group. Proposition 3.4.
Let ( G, τ ) be a locally compact group and suppose that { g n } n<ω is aCauchy sequence in G w . If { g n } n<ω w G ⊆ inv (w G ) , then { g n } n<ω is τ -convergent in G .Proof. Assume that { g n } n<ω is a Cauchy sequence in G w . First, we verify that thesequence is a precompact subset of ( G, τ ).Indeed, we have that { g n } n<ω converges to some element p ∈ inv (w G ). If { g n } n<ω were not precompact in ( G, τ ), there would be a neighbourhood of the neutral element U and a subsequence { g n ( m ) } m<ω such that g − n ( m ) · g n ( l ) / ∈ U for each m, l < ω with m = l . On the other hand, since inv (w G ) is a topological group, we have that thesequence { g − n ( m ) · g n ( m +1) } m<ω converges to p − p , the neutral element in G w . This takesus to a contradiction because, by Proposition 3.3, it follows that { g − n ( m ) · g n ( m +1) } m<ω must also converge to the neutral element in ( G, τ ).Therefore, the sequence { g n } n<ω is a precompact subset of ( G, τ ). This implies that p ∈ G and we are done. (cid:3) Locally compact groups respect other compactness-like properties
Countably compactness.
In this subsection we prove that LC groups respectcountably compactness.
Theorem 4.1.
Let G be a LC group and let A ⊆ G be countably compact in G w . Then A is countably compact in G .Proof. First, we assume that G is a σ -compact LC group. If K := A G w is weaklycompact, by Hughes [22], K must also be compact in the LC topology. Therefore, theidentity maps is a homeomorphism of A w onto A . Therefore, A is countably compact inthe LC topology. On the other hand, if A G w is not compact, there must be some point p ∈ A w G \ G . Since the weak topology on G is weaker than the original LC topologyof G , we have that G w is also σ -compact. That is, there is a collection { K n : n ∈ ω } ofweakly compact subsets such that G = S n ∈ ω K n . For every n ∈ ω , take f n ∈ C (w G ) suchthat 0 ≤ f n ( x ) ≤ / n for all x ∈ w G , f n ( p ) = 0 and f n ( K n ) = { / n } . Set f = P n ∈ ω f n .Then p ∈ Z ( f ) and Z ( f ) ∩ G = ∅ . Thus, the map 1 /f is weakly continuous and,since p ∈ A w G \ G , is not bounded on A . This is a contradiction since A is countablycompact in G w . This completes the proof when G is σ -compact.In the general case, assume that A is a countably compact subset of G w . In case A is precompact in G , it follows that K := A G is compact in G and, as a consequence,is also compact in G w . Therefore both topologies, the weak topology and the LC-topology, coincide on K . Therefore, we may repeat the same argument used in theparagraph above to conclude that A is countably compact in the LC topology. Thuswe may assume that A is not precompact in the LC topology. As a consequence, thereis a compact neighborhood of the identity U and a sequence ( a n ) ⊆ A such that ( a n )is U -discrete; that is a n U ∩ a m U = ∅ if n = m . Set H the group generated by U ∪ ( a n ). HE WEAK COMPACTIFICATION OF LOCALLY COMPACT GROUPS 13
Clearly, H is a σ -compact open subgroup of G . By [15, Prop. 3.12] (cf. [3, Lem. 3.2]),it follows that H is closed in G w and the weak topology in H w coincides canonicallywith the weak topology that H inherits from G w . As a consequence, it follows that A ∩ H is countably compact in H w . Since the group H is σ -compact, it follows that A ∩ H is countably compact in H . This is a contradiction because ( a n ) ⊆ A ∩ H is U -discrete. This completes the proof. (cid:3) Functional boundedness.
A subset A of a topological space X is said to be functionally bounded when f | A is bounded for every f ∈ C ( X ). The topological space X is a µ -space when every functionally bounded subset of X is relatively compact. Inthis subsection, we prove that G w is a µ -space for all LC group G . Lemma 4.2.
Let A be functionally bounded subset of G w for an LC group G . If p ∈ A w G , then p belongs to the G δ -closure of G in w G and δ p is t p ( G ) -continuous oneach countable subset F ⊆ I ( G ) .Proof. Remark that Z ( f ) ∩ G = ∅ for all f ∈ C (w G ) such that p ∈ Z ( f ). Indeed, if Z ( f ) ∩ G = ∅ , then 1 /f would be weakly continuous and, since p ∈ A w G , we wouldhave that 1 /f ought to be unbounded on A , a contradiction. In general, given anarbitrary G δ -open subset N in w G containing p , it is readily seen that there is a zero-set Z ( f ) ⊆ w G such that p ∈ Z ( f ) ⊆ N , which implies that N ∩ G = ∅ . This verifiesthat p belongs to G δ -closure of G in w G .Now, given an arbitrary countable subset F of I ( G ), set N := ∩{ ϕ − ( ϕ ( p )) : ϕ ∈ F } .Then N is a G δ open subset in w G and p ∈ N . As a consequence N ∩ G = ∅ . Take g ∈ N ∩ G . Then ϕ ( p ) = ϕ ( g ) for all ϕ ∈ F , which yields the continuity of δ p on( F, t p ( G )). (cid:3) Proposition 4.3.
Let G be an LC group and let A be a countable subset of G . If p ∈ A w G and δ p is t p ( G ) -continuous on each countable subset of I ( G ) , then δ p iscontinuous on every compact subset of I ( G ) .Proof. We define the following equivalence relation on I ( G ): ϕ ∼ ϕ if and only if ϕ | A = ϕ | A . Let K be a compact subset of I ( G ). Take the quotient map π : ( I ( G ) , t p ( G )) → ( I ( G ) ∼ , t p ( A ))that is clearly continuous. This means that π ( K ) is compact in ( I ( G ) ∼ , t p ( A )). Thereforedensity( π ( K )) ≤ weight( π ( K )) = | A | = ω . Let D be a countable subset of K suchthat π ( D ) is dense in π ( K ). Since D is countable, and p belongs to the G δ -closure of G in w G , there is g ∈ G such that ϕ ( p ) = ϕ ( g ) for all ϕ ∈ D . Furthermore, using thecontinuity of δ g and δ p on each countable subset of I ( G ), it follows that we can extend δ p = δ g to a continuous map on D I ( G ) (indeed, δ p is continuous on D ∪ { ϕ } for all g ∈ D I ( G ) and by [4, I.57.5], this implies the continuity of δ p throughout D I ( G ) .Consider the following diagram I ( G ) π / / δ p ! ! ❇❇❇❇❇❇❇❇ I ( G ) ∼ ¯ δ p ~ ~ ⑥⑥⑥⑥⑥⑥⑥⑥ D where ¯ δ p is defined by ¯ δ p ( π ( ϕ )) = δ p ( ϕ ).Remark that ¯ δ p is properly defined because δ p ( ϕ ) = δ p ( ϕ ) whenever ϕ ∼ ϕ .Furthermore, it holds that ¯ δ p is continuous on π ( D I ( G ) ). Indeed, in order to verifythis, it will suffice to prove that for every ϕ ∈ D I ( G ) and each net ( ϕ i ) in D I ( G ) such HE WEAK COMPACTIFICATION OF LOCALLY COMPACT GROUPS 15 that π (( ϕ i )) converges to π ( ϕ ), there is a subnet ( ϕ m ) such that δ p (( ϕ m )) converges to δ p ( ϕ ).Now, since D I ( G ) is compact, it follows that there is a subnet ( ϕ m ) converging to ϕ ′ ∈ D I ( G ) . By the continuity of π the net π (( ϕ m )) must converge to π ( ϕ ′ ) but, byour previous assumption, will also converge to π ( ϕ ). This means that ϕ ∼ ϕ ′ and, asa consequence, δ p ( ϕ ) = δ p ( ϕ ′ ). The continuity of δ p on D I ( G ) implies the convergenceof δ p (( ϕ m )) to δ p ( ϕ ) = δ p ( ϕ ′ ). Bearing in mind the definition of ¯ δ p , the continuity ofthis map has been proved.Now, since π ( D ) is dense in π ( K ), it follows that π ( D I ( G ) ) = π ( K ). Thus, we haveproved the continuity of ¯ δ p on π ( K ). The commutativity of the diagram K π / / δ p (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ π ( K ) ¯ δ p | | ③③③③③③③③ D implies the continuity of δ p on K , which completes the proof. (cid:3) Now follows the main result of this paper. It extends to non-necessarily abeliangroups a previous result of Trigos-Arrieta [23] for locally compact Abelian groups.
Theorem 4.4.
Let G be a LC group. Then the group G w is a µ -space.Proof. We must verify that every closed functionally bounded subset of G w is compactin G w . Let A be a closed functionally bounded subset of G w . If A is countably compactin G w then, by Theorem 4.1, it follows that A is closed and countably compact in G .Hence, we have that A is compact in G and, as a consequence, in G w . Therefore, from here on, we assume that A is not countably compact in G w withoutloss of generality. This implies that there is some sequence ( a n ) ⊆ A that has no closurepoints in G w . Let p ∈ ( a n ) w G \ G . By Lemma 4.2, p belongs to the G δ -closure of G inw G and δ p is t ( G ) -continuous on each countable subset of I ( G ). Then, by Proposition4.3, we deduce that δ p is t p ( G )-continuous on each compact subset of I ( G ).The proof now requires a case-study approach on the structure of the group G .(1) G is σ -compact and metrizable and, therefore, a Polish LC group. In this case, thespace ( I ( G ) , t k ( G )) is metrizable and, as a consequence, a k -space. Since we haveverified that δ p is t p ( G )-continuous on each compact subset of I ( G ), it follows that δ p is continuous on I ( G ). Applying Akenmann duality thorem [1], it follows that p ∈ G and, as a consequence, that A w G ⊆ G .(2) Suppose first that ( a n ) is not precompact in G . Then we may assume, with somenotational abuse, that ( a n ) is U -discrete for some compact neighborhood of theidentity U . Therefore, by Kakutani-Kodaira’s theorem, there exists a normal,compact K of G such that K ⊆ U and G/K is metrizable, and consequentlyPolish. Let p : G → G/K be the quotient map, which is also continuous for the theweak topologies p : G w → ( G/K ) w , since every pure positive definite map on G/K can be lifted canonically to G . Therefore p (( a n )) is a functionally bounded subsetin ( G/K ) w . By (1) p (( a n )) is relatively compact in G/K , which is a contradictionbecause ( a n ) was assumed to be U -discrete and K ⊆ U . Thus, we may assume,without loss of generality, that ( a n ) is precompact in G . This means that ( a n ) isrelatively compact in A , which is a closed subset of G . Therefore, we have againthat A w G ⊆ G . HE WEAK COMPACTIFICATION OF LOCALLY COMPACT GROUPS 17 (3) G is LC. Again, we suppose first that ( a n ) is not precompact in G and, with somenotational abuse, that ( a n ) is U -discrete for some compact neighborhood of theidentity U . Consider the subgroup H generated by U ∪ { g n } n<ω , which σ -compactand open in G . Since H w G is canonically homeomorphic to w H , we may identify { g n } n<ω w H with { g n } n<ω w G . Hence we may assume, without loss of generality,that G is σ -compact and the proof follows from (2). The case ( a n ) is precompactfollows also from (2).Thus, we have proved that A w G ⊆ G for evey functionally bounded subset A of G w ,which proves that G w is a µ -space. (cid:3) Corollary 4.5.
Every LC group respects pseudocompactness.
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Universitat Jaume I, Instituto de Matem´aticas de Castell´on, Campus de Riu Sec,12071 Castell´on, Spain.
Email address : [email protected] Universitat Jaume I, Departamento de Matem´aticas, Campus de Riu Sec, 12071 Castell´on,Spain.
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