Semigroup Forum Vol. 73 (2006) 159–174

c 2006 Springer


DOI: 10.1007/s00233-006-0623-4

RESEARCH ARTICLE

Local Topological Structure in the LU C -Compactification of a Locally Compact Group and its Relationship with Veech’s Theorem Talin Budak and John Pym Communicated by Jimmie D. Lawson

Abstract The paper begins by presenting a construction of the largest semigroup compactification GLU C of a locally compact group as a quotient of the Stone˘ Cech compactification of the discrete group βGd . This presentation is used in a proof of the local structure theorem for GLU C , which gives a topological description of neighbourhoods of each point, and some new extensions of this result. These immediately imply Veech’s Theorem. Finally a result is given which extends Veech’s Theorem: for σ -compact groups the map g → gx is injective for all x ∈ GLU C on a set larger than G .

1. Introduction The LU C -compactification, GLU C , of a topological group is the largest semigroup compactification for which the multiplication is jointly continuous as a map G × GLU C → GLU C . This compactification plays a significant role in topological dynamics, and within the more special situation in which the group is locally compact an important position is occupied by Veech’s Theorem: for every g ∈ G with g = 1 and each s ∈ GLU C , gs = s; to put it another way, the map g → gs is injective on G for each s ∈ GLU C . Veech’s Theorem was proved in [18] via a Local Structure Theorem: for each s ∈ GLU C there exists a neighbourhood V of 1 in G and a discrete subset P ⊆ G with s ∈ P such that the multiplication map V × P → V P ⊆ GLU C is a homeomorphism onto an ˘ open set, and the closure P of P in GLU C is homeomorphic to the Stone-Cech compactification βP . This result obviously implies a ‘local Veech’, namely that for each s ∈ GLU C , the map g → gs is injective on V . The present paper addresses some questions about the relationship between the two theorems. Is there a ‘large’ set K ⊆ G for which there is a set P such that K × P → KP is a homeomorphism? Theorem 3.9 says that K can be any compact set. This fact immediately implies Veech’s Theorem; is it equivalent to Veech’s Theorem? The answer depends on how long a proof establishing equivalence is allowed to be, but the immediately apparent equivalence is with the case in which K consists of two (arbitrary) points (see after 3.9). Are there sets K which are

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not relatively compact but for which we can find P with K × P → KP a homeomorphism? This seems unlikely, but when G is σ -compact there exist many sets W open in G∗ = GLU C \ G for which G × W → GW is a homeomorphism onto an open set in G∗ (Theorem 3.10). We also ask whether Veech’s Theorem is the best possible in this sense: is there a large set U with G ⊂ U ⊂ GLU C on which the map g → gs is injective for each s ∈ GLU C ? In Theorem 4.1 we show that such U can exist, again for some σ -compact groups. The Local Structure Theorem 3.2 is a little improved on that given in [18], and the proof of Veech’s Theorem is better. It is also observed that for SIN groups—groups in which each neighbourhood of the identity contains a neighbourhood invariant under all inner automorphisms (see [16] for information on these groups)—the latter proof can be made even simpler. A recent, significantly different, proof of Veech’s Theorem can be found in an appendix to [12]. At a few points our arguments require us to consider GLU C as the quoˇ compactification of the group G with the discrete tient of βGd , the Stone-Cech topology. To establish the details in the form we need it is convenient for us to add to the plethora of ways of producing GLU C by constructing it as a quotient of βGd (§2). Our proofs here are not surprising and are like those used in some other approaches (for example in Chapter 21 of [11], and especially in [13] where the constructions are valid in situations much more general than ours), but it is perhaps of interest that the equivalence relation which produces the quotient is very easy to handle. 2. GLU C as a quotient We start with a topological group G . We denote by Gd the same algebraic ˘ group with the discrete topology. It is quite easy to make the Stone-Cech compactification βGd into a semigroup by writing




pq = lim lim xi yj i

j

for p, q ∈ βGd and (xi ) , (yj ) nets in Gd with xi → p , yj → q . Of course it must be checked that pq does not then depend on the nets chosen. A simple way of doing this can be found in [11], §4.1. This multiplication can be seen to be associative, continuous in the p -variable, and continuous in the q variable when p ∈ Gd (and in fact only when p ∈ Gd ; proofs of this more subtle fact are in [11], and in the more general context of locally compact groups in, for example, [17]). The compactification GLU C of G itself in which multiplication is continuous in the left-hand variable and for which the natural map G × GLU C → GLU C is continuous (a condition which is satisfied by βGd since Gd is discrete) is known as the left uniformly continuous compactification. We shall show that it can be constructed easily as a quotient of βGd . Our first lemma presents two easy observations which we put here for reference.

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Lemma 2.1. (i) If f : X → Y is a continuous map between compact spaces and A ⊆ X , f (A) = f (A). (ii) If π: X → Y is a continuous surjective map between compact spaces, ϕ: X → X is continuous, and ψ: Y → Y is such that πϕ = ψπ , then ψ is continuous. Proof. (i) holds because both sides are compact and have f (A) as a dense subset. (ii) holds because for K ⊆ Y compact, ψ −1 (K) = π(ϕ−1 (π −1 (K))) is compact. There are two ways of looking at βGd . One is to say that it is a space of ultrafilters on Gd , and that the ultrafilter p ∈ βGd is determined by its members P ∈ p (where P ⊆ Gd ). The other is to say that p , as a point in the extremally disconnected space βGd , is determined by its clopen neighbourhoods, and these are of the form clβGd (P ) with P ⊆ Gd . These two statements are equivalent: P ∈ p if and only if p ∈ clβGd (P ) . We shall use each formulation when convenient. We produce GLU C as a quotient of βGd by an equivalence relation. In this section we denote by N (1) the set of open neighbourhoods of the identity 1 of G (but from §3, G will be locally compact and N (1) the set of relatively compact open neighbourhoods). We write, for p, q ∈ βGd , p ∼ q ⇐⇒ U P ∩ U Q = ∅ whenever U ∈ N (1), P ∈ p, Q ∈ q. (Readers may like to notice the obvious similarity with the near ultrafilters of [13].) Lemma 2.2. p ∼ q is equivalent to each of (i) U P ∈ q for each U ∈ N (1), P ∈ p ;  (ii) q ∈ {clβGd (U P ): U ∈ N (1), P ∈ p} . Proof. U P ∩ U Q = ∅ if and only if U −1 U P ∩ Q = ∅ . Since U −1 U forms a base of neighbourhoods of 1 for U ∈ N (1), this holds if and only if U P ∩ Q = ∅ for every U , and this is true for every Q ∈ q (an ultrafilter) if and only if U P ∈ q for all U . Finally, U P ∈ q is the same as q ∈ clβGd (U P ) . Lemma 2.3.

∼ is a closed equivalence relation.

Proof. Reflexivity and symmetry are clear. If p ∼ q and q ∼ r then for U ∈ N (1), P ∈ p , R ∈ r , 2.2(i) gives U P, U R ∈ q so that U P ∩ U R = ∅; thus p ∼ r. Since βGd is compact, we can prove ∼ closed by showing its graph is closed [4] (Proposition 8, Chapter 1, §10.4). If pi ∼ qi for all i, pi → p and qi → q , then for P ∈ p , Q ∈ q we have eventually P ∈ pi and Q ∈ qi . Thus for U ∈ N (1) , U P ∩ U Q = ∅ , so that p ∼ q . We can immediately conclude from the theory of equivalence relations that βGd / ∼ is a compact Hausdorff space. We use E to denote the closure of E

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in βGd / ∼, and h: βGd → βGd / ∼ to denote  the quotient map. Lemma 2.2(ii) shows us that the equivalence class of p is {clβGd (U P ): U ∈ N (1), P ∈ p} . If W is a neighbourhood of this equivalence class in βGd , then h(W ) is a neighbourhood of h(p) in βGd / ∼ (see [4], especially the remark which ends Chapter 1, §5). In particular, h(clβGd (U P )) = h(U P ) (see 2.1(i)) is a neighbourhood of h(p) for each U ∈ N (1) , P ∈ p . In [13], Lemma 13, it is shown in a more general context that these neighbourhoods actually form a base; we give a proof for our situation. Lemma 2.4. (i) For p ∈ βGd , the sets h(clβGd (U P )) = h(U P ) form a neighbourhood base of h(p). (ii) h is injective on Gd ⊂ βGd . The topology induced on Gd / ∼ is the topology of G . Proof. (i) We have just seen the sets h(U P ) are neighbourhoods. If W −1 is any open neighbourhood  of h(p), then h (W ) is an open neighbourhood of the equivalence class {clβGd (U P ): U ∈ N (1), P ∈ p} . Since {U P } is directed by ⊇, the finite intersection property shows that there are U and P with clβGd (U P ) ⊆ h−1 (W ) , whence h(U P ) ⊆ W . (ii) To show that h is injective on Gd , for p ∈ Gd we take P = {p} . When U ⊆ Gd we have clβGd (U ) ∩ Gd = U , so that clβGd (U {p}) ∩ Gd = U p. The conclusion follows. Of course we want βGd / ∼ to be a semigroup, not just a compact space. Since we now know that Gd is injectively embedded in βGd / ∼, we shall consider that G ⊆ βGd / ∼ algebraically; thus for g ∈ G and p ∈ βGd we shall write gh(p) = h(g)h(p) = h(gp) when convenient. Lemma 2.5. βGd / ∼ is a semigroup in which multiplication is continuous in the left-hand variable and the product map G × (βGd / ∼) → βGd / ∼ is continuous. Proof. To show the quotient is a semigroup we show multiplication in βGd is compatible with ∼, and we do this in two steps, first for multiplication on the right, then for multiplication on the left. Take r ∈ βGd . We show right multiplication by r sends equivalence classes into equivalence classes. Let p ∈ βGd , and put t = pr . For T ∈ t continuity of multiplication on the right by r shows that there is P ∈ p with clβGd (P )r ⊆ clβGd (T ) . Then for any u ∈ Gd , continuity of multiplication on the left by u shows clβGd (uP )r ⊆ clβGd (uT ) , and consequently for any U ∈ N (1) , clβGd (U P )r ⊆ clβGd (U T ) (use 2.1(i)). Intersecting over all P and then U, T shows that r maps the equivalence class of p into the equivalence class of t . We now show that multiplication by g ∈ Gd on the left also preserves equivalence classes. With p ∈ βGd and t = gp , arguing as before shows that clβGd (U gP ) ⊆ clβGd (U T ) . Then, using continuity of multiplication on the left by g , observe that gclβGd (g −1 U g.P ) = cl βGd (U gP ) , and that as U runs −1 through N (1) , so also does g U g . Thus g {clβGd (U P ): U ∈ N (1), P ∈  p} ⊆ {clβGd (U T ): U ∈ N (1), T ∈ t} as required.

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Now we can show that multiplication on the left by s ∈ βGd is compatible with ∼. Take gi ∈ Gd with gi → s. If p ∼ q then we know gi p ∼ gi q and because ∼ is closed we conclude that sp ∼ sq . Thus multiplication is compatible with ∼. Multiplication on the right in βGd / ∼ is the quotient of multiplication on the right in βGd , so its continuity follows from that in βGd (Lemma 2.1(ii)). Since G is a topological group, to show G × (βGd / ∼) → βGd / ∼ is continuous we need only show continuity at each point of the form (1, h(p)) . Let h(U P ) be a basic neighbourhood of h(p) (Lemma 2.4(i)). Take V ∈ N (1) with V 2 ⊆ U . Then h(V P ) is another neighbourhood of h(p), and since h is a continuous homomorphism we find V × h(V P ) → V h(V P ) ⊆ h(V 2 P ) ⊆ h(U P ) . In the special case in which G is locally compact, for a compact neighbourhood V we can now observe from 2.1(i) that V h(P ) = h(V P ) . Of course, here every neighbourhood contains a compact neighbourhood and so we—as did Kocak and Strauss [13]—conclude Proposition 2.6. Let G be a locally compact group. Then the sets V h(P ) with V an open (or alternatively a compact) neighbourhood of 1 in G and P ∈ p form a neighbourhood base of h(p) in βGd / ∼. The compactification GLU C is characterised as a compact semigroup which has the joint continuity property, that G × GLU C → GLU C is continuous, and which is the largest with that property, in the sense that any other compactification with the joint continuity property is a natural quotient of GLU C . This criterion allows us to identify βGd / ∼ with GLU C . Theorem 2.7.

βGd / ∼ is isomorphic with GLU C .

Proof. Let ψ: G → S present S as a compactification of G with the joint continuity property. Let f : βGd → S be the continuous extension of ψ regarded as a map from Gd to S . It is easy to deduce from the description of multiplication in βGd as an iterated limit at the beginning of this section that f is a homomorphism on βGd . Take p ∈ βGd . Let W be any closed neighbourhood of f (p) in S . Then there are neighbourhoods U ∈ N (1) and V of f (p) in S with ψ(U )V ⊆ W , by the joint continuity property. Since f is continuous, there is P ∈ p with f (clβGd P ) ⊆ V . Then f (U P ) ⊆ ψ(U )V , and because W is closed, f (clβGd (U P )) ⊆ W . We conclude that f maps the equivalence class of p to the point f (p) . Thus f is compatible with the relation ∼, and provides the required quotient map βGd / ∼ → S . Before we leave this section, we should point out that the construction just described is valid in a much more general context. Every uniform space X has an associated compactification C . It follows easily from the results of [6] that C is the quotient of βXd , regarded as the space of ultrafilters on the set X , by the equivalence relation p ∼ q if and only if U (P ) ∩ U (Q) = ∅ for every P ∈ p , Q ∈ q and every vicinity V .

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Although we now know that βGd / ∼ is GLU C , we shall still sometimes use the former notation as it fits more naturally into the context of the Local Structure Theorem (3.2). 3. Local Structure Theorems In this section we shall prove the Local Structure Theorem essentially in the form in which it appears in [18], and give some variants and extensions. Towards the end of the section we shall compare the results we have obtained with those of other authors. From this point on the group G will be locally compact and N (1) is the set of open relatively compact neighbourhoods of the identity. We shall use the notation ‘X ∼ = Y ’ to mean that the topological spaces X and Y are homeomorphic. We call P ⊆ G (with its group topology now) U -discrete for U ∈ N (1) if U p1 ∩ U p2 = ∅ when p1 = p2 in P . P ⊆ G is uniformly discrete if it is U -discrete for some U . We shall often consider a uniformly discrete set to ˘ be a discrete space in its own right so that we may speak of its Stone-Cech compactification βP . We may further think of βP as a subspace of βGd , and consider h as mapping βP into βGd / ∼; the image h(βP ) is P , the closure of P regarded as a subset of GLU C = βGd / ∼. Theorem 3.1. Let U ∈ N (1), and P ⊆ G be U -discrete. (i) (u, q) → uh(q), U ×βP → U P ⊆ βGd / ∼ is continuous and injective. (ii) If G is locally compact and U is compact, U P is open and homeomorphic with U × βP . Proof. (i) Continuity comes from 2.6. Take distinct points (u, q) , (v, r) in U × βP . If q = r there are disjoint subsets Q, R of P with Q ∈ q , R ∈ r . Choose W ⊆ U with W u, W v ⊆ U . We claim uQ ∪ vR is W discrete; indeed, W uQ ⊆ U Q and W vR ⊆ U R , and P itself is U -discrete. Therefore uq and vr are contained in disjoint ∼-equivalence classes, so that uh(q) = h(uq) = h(vr) = vh(r) . On the other hand if q = r we must have u = v . In this case we choose W ⊆ U symmetric with W u, W v ⊆ U and uv −1 ∈ W 2 . Then uP ∪ vP is W -discrete and as before uh(q) = vh(q). (ii) Here we first show that U P is a neighbourhood of any of its points ux with u ∈ U and x ∈ P . We take W ∈ N (1) with W u ⊆ U . We take p ∈ βGd with h(p) = x, and take P ∈ p , so that uP ∈ up . Then using (2.6), W uP = W uh(P ) = W h(uP ) is a neighbourhood of h(up) = ux contained in U P . So U P is open. Now take (u, p) ∈ U × βP . Let V be an open neighbourhood of u with V ⊆ U and let Q ⊆ P with p ∈ βQ. The map (v, q) → vh(q) , V ×βQ → V Q is a continuous bijection between compact spaces and therefore a homeomorphism. From the previous paragraph, it sends the open neighbourhood V ×βQ of (u, p) to the open neighbourhood V Q of up . The map U × βP → U P is thus a homeomorphism.

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The local structure theorem is a version of 3.1 which describes neighbourhoods of points. We state it essentially as it appears in [18], except for the addition of a sentence which will enable us to exploit the properties of βGd in future proofs. The Local Structure Theorem 3.2. Let G be a locally compact group. Let x ∈ GLU C = βGd / ∼. Then there exist U ∈ N (1) and a U -discrete P ⊆ G with x ∈ P such that U P ∼ = U × βP is an open neighbourhood of x. Further, if p ∈ βGd is any element with h(p) = x we may take P (considered now as a subset of Gd ) with P ∈ p . Proof. Start with any symmetric V ∈ N (1) . Take any p ∈ βGd with h(p) = x. Let Q ∈ p , Q ⊆ G . Take any maximal V -discrete set Q ⊆ Q (thus q1 = q2 in Q means V q1 ∩ V q2 = ∅ ). Since Q is maximal, for any q ∈ Q 2 2 we have V q ∩ V Q = ∅ , whence q ∈ V Q . Thus p ∈ Q ⊆ V Q (using the 2 joint continuity property) and there exist v ∈ V and p ∈ Q with p = vp . Now vQ is vV v −1 -discrete ( q1 , q2 ∈ Q implies vV v −1 .vq1 ∩ vV v −1 .vq2 = v(V q1 ∩ V q2 ) = ∅ ). We take U = vV v −1 and P = vQ ∈ p in Theorem 3.1(ii). Then that Theorem tells us that U × βP → U P is a homeomorphism, that U P is open and contains h(p) = x, and so is a neighbourhood of x. We next draw attention to a special case. First comes a lemma which we shall need again. Lemma 3.3. Let P be uniformly discrete in the σ -compact locally compact group G . Then P \ P ⊆ GLU C \ G and P is countable. Proof. If K is any compact subset of G , P ∩ K must be finite because otherwise P would have a cluster point and could not be uniformly discrete. The conclusions follow. The following corollary to 3.2 is now immediate. Corollary 3.4. If G is σ -compact, every point in GLU C (resp. G∗ = LU C \ G ) has an open neighbourhood in GLU C (resp. G∗ ) homeomorphic G to U × βN (resp. U × N∗ ) for some neighbourhood U of 1 in G . We now wish to draw attention to the special case of IN groups. (A general discussion of the place of IN groups in the class of locally compact groups can be found in [16].) Such groups G have a relatively compact invariant neighbourhood V of the identity, that is gV g −1 = V for all g ∈ G . In the proof of 3.2 the U of which the theorem asserted the existence was of the form vV v −1 . If we had started with an invariant V , this neighbourhood would have been V itself. The proof of 3.2, together with 3.4, therefore already establishes part (i) of the next Corollary. Corollary 3.5. (i) Let V be an open relatively compact invariant neighbourhood of 1 in a σ -compact IN group G . Let x ∈ GLU C . Then for any V -discrete subset P of G with x ∈ P (and such sets exist) V P is an open neighbourhood

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of x. In particular, every point of GLU C has a neighbourhood homeomorphic with V × βN . (ii) Suppose in addition that V generates G . Then for any compact K ⊆ G there is a uniformly discrete P ⊆ G such that multiplication is a homeomorphism V K × βP → V KP onto an open neighbourhood of x.  Proof of (ii). Since K ⊆ n V n = G , there is n with K ⊆ V n . The set V n+1 is in N (1) , invariant, and contains V K . We may take P to be V n+1 -discrete in applying (i). For discrete groups there is a result obtained in a similar way which appears much less significant. We shall point out later (after 3.9) its precise standing in the theory. Corollary 3.6. Let G be discrete. Take any g ∈ G , g = 1, and x ∈ βG. Then there is P with x ∈ P such that {g, 1} × βP → gP ∪ P is a homeomorphism onto an open subset of GLU C = βG. Proof. We again modify the proof of 3.2. V = {g, 1} is in fact a compact neighbourhood of 1 . In taking U = vV v −1 in that proof, v is chosen from V 2 , and here all elements of V 2 commute with V . Thus as in the case of IN groups, V = U . Our conclusion follows. The last proof would have been (even more) trivial if we had used Veech’s Theorem, that gx = x, since then these two points would have had disjoint neighbourhoods and consequently gP ∩P = ∅ . We shall discuss the relationship between this Corollary and Veech’s Theorem after 3.9 below. Our next aim is to obtain a conclusion for general locally compact groups like that of Corollary 3.5(i). We shall use a result which is often employed in proofs of Veech’s Theorem. Its first use in the context of GLU C was probably in Ruppert’s paper [19]. We shall sketch a proof due to I N Baker [1] which is transparent—indeed almost trivial. We give no more than the form we need, so the result is not quite as general as those more frequently presented (see [11], Theorem 3.33). The Three Sets Lemma 3.7. Let P be a set, Q ⊆ P , f : Q → P an injective mapping with f (q) = q (q ∈ Q). Then there is a partition Q = F0 ∪ F1 ∪ F2 (Fi = ∅ for some i is permitted) with f (Fi ) ∩ Fi = ∅ ( 0 ≤ i ≤ 2 ). Proof. For q ∈ Q define the orbit of q by O(q) = {f r (q): f r (q) is defined and in Q} (here if r < 0 , f r (q) means the unique p for which f −r (p) = q if it exists). Because f is injective, orbits are either identical or disjoint. We fix one q in each orbit. As the orbits partition Q it will be enough to write each O(q) as F0 ∪ F1 ∪ F2 . O(q) might be a chain, finite or infinite, {. . . f r−1 (q), f r (q), f r+1 (q), . . .} ; in this case write F0 = {f r (q): r is even } , F1 = {f r (q): r is odd } , F2 = ∅. Then f (F0 ) ⊆ F1 ∪ (P \ Q) , f (F1 ) ⊆ F0 ∪ (P \ Q) . The conditions are satisfied. If it is not a chain then, because f

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is injective, O(q) = {q = f r (q), f 1 (q) . . . f r−1 (q)} must be a cycle (with r ≥ 2 since f (q) = q ). If r is even, define F0 , F1 , F2 as in the previous case. If r is odd, write F0 = {q, f 2 (q), . . . , f r−3 (q)} , F1 = {f 1 (q), f 3 (q), . . . , f r−2 (q)} , F2 = {f r−1 (q)} . Then f (F0 ) ⊆ F1 , f (F1 ) ⊆ F0 ∪ F2 and f (F2 ) = {q} ⊆ F0 . The next step is to get a topological version of Corollary 3.6. The Two-Point Local Structure Lemma 3.8. Let x ∈ βGd / ∼ , g ∈ G . Take p ∈ βGd with h(p) = x. Then there exist an open neighbourhood V of 1 in G and a V -discrete set P ⊆ G with P ∈ p such that (gV ∪ V ) × βP → gV P ∪ V P ⊆ βGd / ∼ is a homeomorphism onto an open set containing gx and x. Proof. g = 1 is Theorem 3.2, so we assume g = 1 . We take an open neighbourhood W0 of 1 for which there exists a W0 -discrete set P0 with P0 ∈ p (3.2). By taking W0 smaller if necessary we may assume g ∈ W0 . By 3.2, W0 P0 is an open neighbourhood of x. Again as in 3.2, gP0 is gW0 g −1 -discrete, so that also gW0 P0 = gW0 g −1 .gP0 is an open neighbourhood of gx ∈ gP . Our lemma would be immediate if gW0 P0 ∩W0 P0 = ∅ . We shall show we can achieve this by making W0 and P0 smaller. Now for P ⊆ P0 there is W ∈ N (1) with gW P ∩ W P = ∅ if and only if there is V with gP ∩ V P = ∅ (to see this note that for a symmetric W , gW g −1 .gP ∩ W P = ∅ if and only if gP ∩ gW g −1 W P = ∅ ). Take a symmetric open V with both V ⊆ W0 and g −1 V g ⊆ W0 ; notice g ∈ V . Write Q = {q ∈ P0 : gq ∈ V P0 } . Then x ∈ P0 = (P0 \ Q) ∪ Q. When x ∈ (P0 \ Q) , because V P0 is open, gx ∈ V P0 . In this case we can put P = P0 \ Q to find gP ∩ V P ⊆ gP0 \ Q ∩ V P0 = ∅, as required. Now we must consider the case x ∈ Q. We define a map π: V P0 → P0 by translating the projection V × βP0 → βP0 using the homeomorphism V × βP0 ∼ = V P0 . Define f : Q → P0 by f (q) = π(gq) . Then f (Q) ⊆ P0 , and f , being continuous, is determined by its values on Q. Notice π(x) = x. As a map of Q into P0 , f is injective (because f (q1 ) = f (q2 ) ⇔ π(gq1 ) = π(gq2 ) ⇔ gq1 ∈ V gq2 ⇔ q1 ∈ g −1 V g.q2 , and this means q1 = q2 since Q is W -discrete) and f (q) = q for q ∈ Q (because f (q) = q ⇔ gq ∈ V q , but g ∈ V ). We can therefore apply the Three Sets Lemma 3.7 to f and partition Q as F0 ∪F1 ∪F2 . Then the closures F0 , F1 , F2 partition Q and V F0 , V F1 , V F2 partition V Q (because of the homeomorphism with V × βQ). The point x is in one of the sets, Fi say. Put P = Fi . Then f (P ) = {π(gp): p ∈ P } is a subset of another set, Fj say. This means that gP ⊆ V Fj , and therefore that V P and gP are disjoint. The proof is finished. We now come to our most general local structure result. The Compact-set Local Structure Theorem 3.9. Let K ⊆ G be compact and let x ∈ GLU C . Take p ∈ βGd with h(p) = x. Then there is an open

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neighbourhood V of K and a uniformly discrete set P ⊆ G with P ∈ p (so x ∈ P ) such that the multiplication map V × βP → V P is a homeomorphism onto an open set V P containing Kx. Proof. We may assume that the identity of G is in K. By the Two-Point Local Structure Lemma 3.8 for each g ∈ G there exist an open neighbourhood Vg of 1 and a Vg -discrete Pg ∈ p such that (gVg ∪Vg )×βPg → (gVg ∪Vg )Pg is a homeomorphism onto an open subset of GLU C . (Notice that g = 1 is permitted here, and when g = 1 , gVg ∩ Vg = ∅ .) Let U be an open neighbourhood of K −1 with U compact. Cover the compact set U U by a finite number of open sets g1 Vg1 , . . . , gk Vgk . Put P = Pg1 ∩ · · · ∩ Pgk ; this finite intersection is in p since p is an ultrafilter. −1

Now if u, v ∈ U , then v −1 u ∈ U U , and therefore there is r with v −1 u ∈ gr Vgr . For any y, z ∈ P , we have y, z ∈ Pgr and therefore v −1 uy = z , that is uy = vz , if (u, y) = (v, z) . Thus U × βP → U P is injective. Now −1 U ⊆ U U ⊆ g1 Vg1 ∪ . . . ∪ gk Vgk . For each r the map gr Vgr × βPgr → gr Vgr Pgr is a homeomorphism onto an open set and βP is a clopen subset of βPgr . Therefore the map U × βP → U P is a homeomorphism onto an open set. Let us see how our results relate to Veech’s Theorem, the assertion that the map g → gs, G → GLU C is injective for each s ∈ GLU C . This is an immediate consequence of the Compact-set Local Structure Theorem 3.9 since that implies x → gx is injective on each compact subset of G and therefore on G itself. But in the same way it is a consequence of the Two Point Structure Lemma 3.8. On the other hand, Veech’s Theorem immediately implies that s and gs have disjoint neighbourhoods for s ∈ GLU C ; these can be taken to be of the form given in the Local Structure Theorem 3.2, and 3.8 is an easy consequence. Thus Veech’s Theorem is precisely equivalent to Two Point Structure Lemma. This explains why the Three Sets Lemma was used at that particular point in our development: something of this kind appears to be needed for the full Veech result. For some classes of groups Veech’s Theorem comes more easily, even trivially. The remarks in the last paragraph together with Corollary 3.6 show that this is true for discrete groups. Similarly Corollary 3.5(ii) shows it holds for IN groups generated by an invariant neighbourhood. Having achieved a result for every compact subset of G (3.9) it is natural to enquire whether there is a parallel result for the whole of G . Some changes must be necessary: for any P ⊆ G with more than one element multiplication is not injective as a map G × P → G . We can, however, get a version valid for a restricted class of groups. Theorem 3.10. Let G be locally compact and σ -compact. Take x ∈ G∗ = LU C G \ G . Then there is an open set W ◦ in G∗ with x ∈ W ◦ such that the multiplication map G × W ◦ → GW ◦ ⊆ G∗ is a homeomorphism onto an open subset of G∗ .

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Proof. Take any p ∈ βGd for which h(p) = x. Take a sequence (Kn ) of compact sets with Kn  G , and such that there is an open neighbourhood U of 1 such that Kn U ⊆ Kn+1 for all n . From the Compact-set Structure Theorem (3.9) for each n there is a uniformly discrete set Pn ∈ p such that Kn × Pn → Kn Pn is a homeomorphism. Because p is an ultrafilter we can assume (Pn ) isdecreasing, and we also choose Pn with Pn ∩ Kn = ∅ for all n . Put W = n Pn ; then W ∩ G = ∅ . Also, since W is a zero set in βP1 ([21] 1.21), it follows from [21] 3.28 that p is in the closure of the interior (in P1∗ ) of W . From 3.9 we know that Kn × Pn → Kn Pn is a homeomorphism, and since Kn−1 U is open and contained in Kn the set Kn−1 U Pn is an open neighbourhood of x in GLU C (3.9). Therefore G × W = ( n Kn ) × W → GW is injective. We now prove this map is a homeomorphism. Take any net (gi , wi ) in G × W with gi wi → gw ∈ GW . Since g ∈ G we must have g ∈ Kn for some n . Then gw ∈ Kn W ⊆ Kn U Pn+1 ⊆ Kn+1 Pn+1 , and so for all large i, (gi , wi ) ∈ Kn+1 Pn+1 . By injectivity, gi ∈ Kn+1 , wi ∈ W for all such i. Using compactness we may assume gi → g0 , wi → w0 . The joint continuity property shows that gi wi → g0 w0 in G∗ , and then uniqueness of limits shows that g0 w0 = gw . Injectivity means that g0 = g , w0 = w . Thus we have proved that (gi , wi ) → (g, w) in G × W . Our map is a homeomorphism. We take W ◦ to be the interior in P1∗ of W . We know that Kn U Pn ∼ = Kn U × βPn is an open subset of GLU C . We intersect with G∗ to find Kn U Pn ∩ G∗ ∼ = Kn U × Pn∗ , and since W ◦ is open in Pn∗ we find that Kn U W ◦ is open in ∗ G . The union over n is just GW ◦ . As far as we know, a local structure theorem first appeared in a form like 3.2 in [14]. However, this was preceded by Filali’s paper [7], which gives a global structure theorem for the semigroup (Rk )LU C as essentially a direct product [0, 1)k × β(Nk ) . A result for much more general compactifications than ours (of uniform spaces) is given in [13]. Lemma 13 of that paper describes a neighbourhood base in their situation in the same general terms as 2.4 and 2.6 . Their notation would take too long to describe here, but in the special context of GLU C it is presented at the end of the introduction of [6]: for neighbourhoods U of the identity in G and sets P in a ‘near ultrafilter’ x (in our context, that is a point of GLU C ) the sets clGLU C (U P ) form a base of neighbourhoods of x. These neighbourhoods are just U clGLU C (P ) if U is compact. (See also Lemma 1.3 in [6].) However, [13] and [6] do not mention the homeomorphism with U × βP , though it is not hard to obtain this conclusion starting from theirs. Restricted versions of 3.2 hold in large subspaces of the weakly almost periodic compactification GW AP of many locally compact groups G . Filali [8] has given a direct proof that the conclusion of 3.2 holds in part for a class of groups including all SIN groups: there exist uniformly discrete sets P for which U × βP is homeomorphic with an open set in GW AP for suitable open neighbourhoods U of the identity in G . Moreover, there are usually a large number of such sets P ; in a SIN group, every set which is not relatively compact

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will contain one. Further light on this result is shed by the very remarkable main conclusion of Ferri and Strauss [6]: if G is a SIN group and π: GLU C → GW AP is the natural continuous homomorphism, then there is a dense open subset W of GLU C \ G for which π(W ) is a dense open subset of GW AP \ G and π: W → π(W ) is a homeomorphism. (In fact the conclusion of [6] is even stronger than this.) Since G is open and dense in both GLU C and GW AP — because in the one-point compactification of G , which is separately continuous, G is open—G ∪ W is open and dense in GLU C and G ∪ π(W ) is open and dense in GW AP . The corollary which follows is therefore immediate from [6] and 3.2. Corollary 3.11. Let G be a SIN group. Then there is an open set E in GW AP with G ⊂ E and E ∩ (GW AP \ G) dense in GW AP \ G , with the property that each x ∈ E has a neighbourhood homeomorphic with U ×βP ∼ = UP where U is a relatively compact neighbourhood of the identity in G and P is a uniformly discrete subset of G with x ∈ P . Of course the local structure theorem cannot hold for all points of GW AP . The minimal ideal of this compactification is the compact group GAP and any point in this ideal cannot have a neighbourhood homeomorphic to GAP ∩ V P ∗ for V a neighbourhood of 1 in G . 4. Extending Veech to a set larger than G For a set E in a semigroup S , let us write InjS (E) = {x ∈ S: y → yx is injective on E} . Veech’s Theorem says InjGLU C (G) = GLU C . When x ∈ InjS (S) , x is called right cancellative (for an obvious reason). According to a number of authors (including [5], [9]) there exists a large number of such elements in GLU C : for G locally compact, InjGLU C (GLU C ) contains a set which is open and dense. Recently results of this kind have been extended to the weakly almost periodic compactification. In [2], [6] and [8] it is shown that for SIN groups (and some others) InjGW AP (G) contains a dense open subset. In this section we ask a different kind of question. Is G the largest subset E of GLU C for which InjGLU C (E) = GLU C ? We obtain an answer for some σ -compact groups. Theorem 4.1. Let G be a locally compact, σ -compact group for which the natural homomorphism ϕ: G → GAP is not surjective. Then there is a set U open in GLU C with G ⊂ U and G = U for which u → us , U → U s ⊂ GLU C is injective for every s ∈ GLU C . Theorem 4.1 is related to Theorem 8.20 of [11]. That result says that for a countable discrete group G and s ∈ GLU C = βG, there is a set U (s) open in GLU C for which U (s) ∩ G∗ is dense in G∗ and u → us is injective on U (s) . This conclusion is stronger than ours in that U (s) is dense, but weaker in that our set works simultaneously for all s. Generally speaking, a locally compact group G is likely to satisfy the hypothesis of Theorem 4.1. However, a number of different examples can be

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found in §4.12.2 of [3] for which it fails. One obvious class consists of the minimally almost periodic groups, those for which GAP consists of just one point. Amongst these are the non-compact connected semisimple Lie groups, and also the alternating subgroup of the discrete group of permutations of a countable set. Another example is the semidirect product of the additive group C of complex numbers with the circle group T, that is the product G = C × T with multiplication (z, eiθ )(w, eiφ ) = (z + eiθ w, ei(θ+φ ) ; here GAP = T. Then again there is the discrete group G discovered by Moran [15] which has the property that GAP is algebraically G but with a compact topology. In passing we might note that discrete groups are SIN, so this assumption would not help us to strengthen our conclusion. We need to turn the hypothesis of the theorem into something easier to work with. This is done in our next lemma. Lemma 4.2. group G :

The following are equivalent for a σ -compact, locally compact

(i) The natural homomorphism ψ: G → GAP is not surjective. (ii) There is a compact metrizable group GM and a continuous homomorphism ϕ: G → GM with ϕ(G) dense but ϕ(G) = GM . ∞ Proof. First suppose (i). Let G = 1 Kn with each Kn compact. By hypothesis there is x ∈ GAP with x ∈ ψ(G) . Then, with 1 the identity of GAP , 1 ∈ x−1 ψ(G). Therefore 1 ∈ x−1 ψ(Kn ) for every n . Since x−1 ψ(Kn ) is compact for every n , Un = GAP \ x−1 ψ(Kn ) is an open neighbourhood of 1 in GAP . From Theorem 8.7 of [10], find a compact normal subgroup H with ∞ H ⊆ n=1 Un and GAP /H metrizable. Then x−1 ψ(G) ∩ H = ∅ . Therefore xH ∩ ψ(G) = ∅ , and since H is a normal subgroup, xH ∩ ψ(G)H = ∅ . Thus in the quotient G/H the images of x and ψ(G) do not intersect, meaning that if ϕ is the composite G → GAP → GAP /H then the image of x is not in ϕ(G) , and so (ii) holds. Now suppose (ii). Let ψ: G → GAP be the canonical map. Given ϕ: G → GM , let ϕ:  GAP → GM be the extension of ϕ to GAP ; this map is surjective. Then ϕ = ϕ  ◦ ψ . If ψ is surjective then so will ϕ be. The next lemma is more routine. Lemma 4.3. Let S be a compact right topological semigroup. Let u, v ∈ S . If there is s ∈ S with us = vs then in each minimal left ideal L of S there is a minimal idempotent e for which ue = ve. Proof. Let us = vs. Take t ∈ L. Then ust = vst and st ∈ L. L is a union of groups; let e be the identity of the group containing st, and (st)−1 be the inverse in that group. Then ue = u(st)(st)−1 = v(st)(st)−1 = ve. Our final lemma is quoted without proof from Corollary 3.42(b) of [11]. It may also be found as Lemma 1 of [20].

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Lemma 4.4. Let A, B be countable subsets of βD , where D is a discrete topological space. If A ∩ B = A ∩ B = ∅ , then A ∩ B = ∅ . Proof of Theorem 4.1. We use Lemma 4.2 to obtain a continuous homomorphism ϕ: G → GM with ϕ(G) dense but ϕ(G) = GM . Denote the continuous extension of ϕ to GLU C again by ϕ , so that ϕ: GLU C → GM and ϕ(GLU C ) = GM . Fix a ∈ GM \ ϕ(G) . Fix an open neighbourhood V of 1 in G which has compact closure. Let (xn ) be a sequence in G for which ϕ(xn ) → a. Then (xn ) has a subsequence which is V -discrete. (If not, then (xn ) would lie in a finite union of translates of V and would therefore have a cluster point in G .) Replace (xn ) by this subsequence (in other words, assume (xn ) is V -discrete). Write X = {x1 , x2 , x3 , . . .} . Let x be any cluster point of X in GLU C . Then ϕ(x) = a ∈ ϕ(G) . The Local Structure Theorem (3.2) tells us that V X is an open neighbourhood of x in GLU C , homeomorphic with V × βX . We write U = G ∪ V X . Fix a minimal left ideal L in GLU C . We shall prove that u → ue is injective on U for every idempotent e ∈ L; this will establish our theorem (see Lemma 4.3). Take e ∈ L to be any idempotent. Notice that ϕ(e) = 1 , the only idempotent in GM . We now begin to show that the map u → ue is injective. We treat three cases. The first case is easy: Veech’s Theorem tells us that u → ue is injective on G . Next take g ∈ G and vx1 ∈ V X \ G = U \ G where v ∈ V ⊂ G , x1 ∈ X . Since vx1 ∈ G , in fact x1 ∈ X \ X . Then ϕ(vx1 e) = ϕ(v)ϕ(x1 )ϕ(e) = ϕ(v)a ∈ ϕ(G) because ϕ(v) ∈ ϕ(G) , whilst ϕ(ge) = ϕ(g)ϕ(e) ∈ ϕ(G). Thus vx1 e = ge. Our proof will be complete if we can show that when v1 x1 , v2 x2 ∈ V X \ G = U \ G , with v1 , v2 ∈ V , x1 , x2 ∈ X \ X and v1 x1 = v2 x2 , then v1 x1 e = v2 x2 e. Here we need two further cases. First, if x1 = x2 then we must have v1 = v2 . Because the map g → gx1 e is injective on G (Veech), we find v1 x1 e = v2 x1 e = v2 x2 e. We are left with the case x1 = x2 . We assume that v1 x1 e = v2 x2 e and we shall obtain a contradiction. Now there exist X1 ⊂ X , X2 ⊂ X with x1 ∈ X1 , x2 ∈ X2 and X1 ∩ X2 = ∅ (because X is homeomorphic with βX ). Using the Local Structure Theorem again, we take a neighbourhood W Q of v1 x1 e = v2 x2 e with v1 x1 e ∈ Q, Q taken to be V -discrete, and W taken to be a symmetric neighbourhood of 1 in G with W ⊆ V . Let π: W Q → Q be the canonical projection (coming from W × βQ → βQ).

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The continuity of y → ye on GLU C allows us to find a neighbourhood N1 of v1 x1 with N1 e ⊆ W Q. Since v1 x1 ∈ V X ∼ = V × βX we may take N1 to be of the form W1 × Y1 where W1 is an open neighbourhood of v1 contained in V and Y1 is a subset of X1 with x1 ∈ Y1 . Similarly there is a neighbourhood N2 of the form W2 × Y2 of v2 x2 with N2 e ⊆ W Q. A fortiori Y1 e, Y2 e ⊆ W Q. Now we work with Y1 , Y2 as spaces rather than continuing with v1 x1 , v2 x2 . First for y ∈ Y1 \ Y1 and g ∈ G it is not possible that ye = ge because, since y ∈ (X \ X) , this would mean a = ϕ(y) = ϕ(g) , which is impossible because a ∈ ϕ(G) . This conclusion further implies that for such a y , π(ye) = π(ge) for any g ∈ G , for if this did happen there would be w1 , w2 ∈ W ⊆ G with w1 ye = w2 ge, or ye = w1−1 w2 ge, and this we have just seen is impossible. In particular then π((Y1 \ Y1 )e) ∩ π(Y2 e) = ∅ . Now suppose for the moment that π(Y1 e) ∩ π(Y2 e) = ∅ . Then there are y1 ∈ Y1 ⊂ G , y2 ∈ Y2 ⊂ G and w1 , w2 ∈ W with w1 y1 e = w2 y2 e (because there are z ∈ Q and some w1 , w2 ∈ W with w1 z = y1 e and w2 z = y2 e). Veech’s Theorem now tells us that w1 y1 = w2 y2 but since W ⊆ V this contradicts the fact that Y is in a V -discrete set. We conclude from the last two paragraphs that π(Y1 e) ∩ π(Y2 e) = ∅ . Similarly π(Y1 e) ∩ π(Y2 e) = ∅ . Now π(Y1 e) = π(Y1 e) by Lemma 2.1(i). Therefore we can apply Lemma 4.4 to conclude π(Y1 e) ∩ π(Y2 e) = π(Y1 e) ∩ π(Y2 e) = ∅. Now π(v1 x1 e) = π(x1 e) ∈ π(Y1 e) , and using also the corresponding observation with all the suffices taken to be 2’s, we deduce that v1 x1 e = v2 x2 e which definitely contradicts the assumption that these two are equal. References [1] Baker, I. N., Solution to problem 5077, American Math Monthly 71 (1964), 219–220. [2] Baker, J. W. and M. Filali, On the analogue of Veech’s Theorem in the WAP-compactification of a locally compact group, Semigroup Forum 65 (2002), 107–112. [3] Berglund, J., H. Junghenn and P. Milnes, “Analysis on Semigroups”, Wiley, New York, 1989. [4] Bourbaki, N., “Elements of Mathematics: General Topology Chapters 1– 4”, Springer, Berlin, 1989. [5] Ferri, S. and D. Strauss, Ideals, idempotents and right cancelable elements in the uniform compactification, Semigroup Forum 63 (2001), 449–456. [6] Ferri, S. and D. Strauss, A note on the WAP-compactification and the LUC—Compactification of a topological group, Semigroup Forum 69 (2004), 87–101.

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[7] Filali, M., The uniform compactification of a locally compact abelian group, Math. Proc. Cambridge Philos. Soc. 108 (1990), 527–538. [8] Filali, M., On the actions of a locally compact group on some of its semigroup compactifications, preprint, University of Oulu, Finland. [9] Filali, M. and J. Pym, Right cancellation in the LUC-compactification of a locally compact group, Bull. London Math. Soc. 35 (2003), 128–134. [10] Hewitt, E. and K. A. Ross, “Abstract Harmonic Analysis I”, Springer, Berlin, 1963. ˘ [11] Hindman, N. and D. Strauss, “Algebra in the Stone-Cech Compactification”, de Gruyter, Berlin, 1998. [12] Kechris, A. S., V. Pestov and S. Todorcevic, Fra¨ıss´e limits, Ramsey theory and topological dynamics of automorphism groups, Geom. Funct. Anal. 15 (2005), 106–189. [13] Kocak, M. and D. Strauss, Near ultrafilters and compactifications, Semigroup Forum 55 (1997), 94–109. [14] Lau, A. T., A. R. Medghalchi and J. S. Pym, On the spectrum of L∞ (G) , J. London Math. Soc. 48 (1993), 152–166. [15] Moran, W., On almost periodic compactifications of locally compact groups, J. London Math. Soc. 3 (1971), 507–512. [16] Palmer, T. W., “Banach Algebras and the General Theory of ∗ -Algebras”, Vol. II, Cambridge University Press, 2001. [17] Protasov, I. and J. Pym, Continuity of multiplication in the largest compactification of a locally compact group, Bull. London Math. Soc. 33 (2001), 279–282. [18] Pym, M., A note on GLU C and Veech’s Theorem, Semigroup Forum 59 (1999), 171–174. [19] Ruppert, W., On semigroup compactifications of topological groups, Proc. Royal Irish Acad. 79 (1979), 179–200. [20] Strauss, D., N∗ does not contain an algebraic and topological copy of βN , J. London Math. Soc. 46 (1992), 463–470. ˘ [21] Walker, R. C., “The Stone-Cech Compactification”, Springer, Berlin, 1974.

Matematik B¨ ol¨ um¨ u ¨ Bo˘ gazi¸ci Universitesi ˙ 34342 Bebek Istanbul Turkey [email protected] [email protected] [email protected]

Received June 3, 2005 and in final form March 20, 2006 Online publication December 8, 2006