SEMIGROUP COMPACTIFICATIONS BY GENERALIZED DISTAL FUNCTIONS AND A FIXED POINT THEOREM

The notion of “Semigroup compactification” which is in a sense, a generalization of the classical Bohr (almost periodic) compactification of the usual additive reals R, has been studied by J. F. Berglund et. al. [2]. Their approach to the theory of semigroup compactification is based on the Gelfand-Naimark theory of commutative C* algebras, where the spectra of admissible C*-algebras, are the semigroup compactifications. H. D. Junghenn's extensive study of distal functions is from the point of view of semigroup compactifications [5]. In this paper, extending Junghenn's work, we generalize the notion of distal flows and distal functions on an arbitrary semitopological semigroup S, and show that these function spaces are admissible C*- subalgebras of C(S). We then characterize their spectra (semigroup compactifications) in terms of the universal mapping properties these compactifications enjoy. In our work, as it is in Junghenn's, the Ellis semigroup plays an important role. Also, relating the existence of left invariant means on these algebras to the existence of fixed points of certain affine flows, we prove the related fixed point theorem.

Their approach to the theory of semigroup compactification is based on the Gelfand-Naimark theory of commutative C- algebras, where the spectra of admissible C*-aigebras, are the semigroup compactifications.
H.D. Junghenn's extensive study of distal functions is from the point of view of semigroup compactiflcations [5].In this paper, extending Junghenn's work, we generalize the notion of distal flows and distal functions on an arbitrary semitopological semigroup S, and show that these function spaces are admissible C*subalgebras of C(S).We then characterize their spectra (semigroup compactifications) in terms of the universal mapping properties these compactlficatlons enjoy.In our work, as it is in Junghenn's, the Ellis semlgroup plays an important role.Also, relating the existence of left Invariant means on these algebras to the existence of fixed points of certain affine flows, we prove the related fixed point theorem.
Let S be a semitopologlcal semigroup (binary operation separately continuous) with a Hausdorff topology, and C(S) denote the C*-algebra of all bounded complex valued continuous functions on S (all topologies are assumed to be Hausdorff).f(ts) (f e C(S) and t S).
A subspace F of C(S) is left (right) translation invariant if LsF c__ F (RsF c_ F).It is translation invarlant if it is both left and right translation invarlant.A C*-subalgebra F of C(S) is called admissible if it is translation invariant, contains the constant functlons, and is left-m-introverted, i.e., Txf(.) x(L(.)f) is a member of F whenever f F and x belongs to the spectrum of F (the space of all nonzero continuous homorphisms on F).In this case, T :F F is called the left-m-introversion operator determined by x.A right x topological compactificatlon of S is a pair (X,), where X is a compact right topological semigroup (i.e., X is a compact semigroup with the mapping x/xy:X/X continuous for all y X), and :S/X is a continuous homorphlsm with dense image such that for each s e S, the mapping x/(s)x:X+X is continuous.If, in addition, a C(X) F where F is an admissible subalgebra of C(S) and :C(X)/C(S) is the dual mapping f/fo, then (X,) is called an F-compactiflcation of S.
A right topological compactlfication X,) of S is said to be maximal with respect to a property P if K,) has the pcoperty P, a,d whe,lever (Y,3) is a right t.opologica[ compact if [cat ion of S with the property P, then there exists a cont [,uous }omo,orphi,n :X+Y such that o8=.
The factorizatlon of the mapping by i.
referred to .a universal mapping property of X,).F-compactficatio,s are maxima[ with respet:t to the property that a CtX) c_ F [2, [[[ Theorem 2.4].This result will be used frequently without specific reference to it.For a fixed admissible subalgebra F of CtS), all F-compact[flcations of S are algebraically and topologically isomorphic, and hence, we speak of the F-compactlflcat[on of S. If F is a norm closed, conjugate closed subspace of C(S) containing constants, then a F We denote the set of all means [multlplicatlve means] on F by M(F) [MM(F)].Wi+h w-topology, MM(F) is compact and it is the w-closure of e(S), where e is the evaluation map {e(s)(f) f(s)}.
We note that (MM(F), e) is an F- compactification of S, and we call ths the canonical F-compactlflcatlon of S. We We note that the LMC(S)-compactlflcation is maximal with respect to the property that it is a right topological compactlficatlon of S [2, III Theorem 4.5].
A flow is a triple (S,X,), where S is a semitopologlcal semlgroup, X is a compact topological space, and :SX X is a continuous homomorphism such that t(s):X+X is continuous for each s e S.
4e oten write (S,X) for (S, X, ) and sx for (s)x.X X is a compact right topological semigroup (with respect to the product topology and function composition) of all self maps of K. We denote the Ellis X X semigroup, the clsoure of (S)in by E(S, X).E(S, X) is then a compact right topological semigroup.
If X is a convex subset of a real or a complex vector space, and ,(s):X+X is afflne for each s in S, then (S, X) is ca[led an affine flow.A point x in X is called a fixed point of the flow (S, X) if s--X for each s in S. If Y is a closed Invariant subspace of X, then (S, Y) is a flow under the restricted action.A flow (S, X, ) is called distal if, whenever , y X such that lira six lira slY for some net (s i) in S, then x y.Let f e LMC(S) and Z be the closure of RSf in the topology of polntwise convergence on C(S).Define :SZ Z by (s) Rs Z'I Then Z is pointwlse compact [6], and (S, Z, )is easily seen to be a flow.
f is called a distal function if the flow (S, Z, ) is distal.H. D. Junghenn has shown that D(S), the set of all distal functions, is an admissible subalgebra of C(S) and that a function LMC(S) is distal iff uev(f) uv(f) for u, v in X and e E(X), the idempotents of X, where (X,a) is the LMC(S)-compactlfication of S.
Also, he has proved that the D(S)-compactification (Y,B)is maximal with respect to the property that xey=xy for all x, y in Y, e e E(Y) [5, Theorem 3.4].
Then E(S, X) n and E(S, X) n are both compact right topological semlgroups.We note that E(S, X) n is nonempty as compact right topological semlgroups have Idempotent elements [4].
$(y) for some F. e ES, X), then () (y) for every r. e E(S x ( l E(S, x)n).
A function f LMC(S) is said to be n-distal (-distal), if the flow (S, Z, ), where Z is the closure of Rsf in the topology of potntwise convergence on C(S), and (s) RslZ, is n-distal (-distal).We denote the set of all n-distal (-distal) functions by Dn(s} iDa(S)].Clearly, D(S)_ c DI(s) c_ D2(S) c_ c_DS).
A flow (S, X, n) is n-dlstal if and only if, whenever x, y e X such that llm six llm sly for some net (si) in S, then sx sy for sn-every s e {sis 2 s s S}.
Let x, y e X and (s i) S such that llm six llm sly.
Then, by hypothesis, (x) (y) for every E(S, X) n.Since (Sn) c_ E(S, X) n, it follows that sx sy for every s e S n.Sufficlency.Let x, y e X and llm (Sk) e E(S, X) such that (x) (y).(cl,n+2vO) h 2, s (cl, j) e S, one verifies that g e Dn(S) and g e (S).(ill) Let (N, +) be the semigroup of positive integers with discrete topology.We have a more general result corresponding to generalized distal flows.PROOF.We first prove the n-case.Necessity.Let Z E(S, X)n.
It suffices to prove that pe p for all p e Z and e e E(Z).
Let x e X and e e E(Z).
Therefore by delnlt[on of n- distal, p(x) p(e(x)) for all p Z, and hence, p pe. Sufficiency.Let x, y X such that p(x) p(y) for some p E(S, X).
Then pn(x) pn (y) where Pn E E(S, X) n.As Z is left simple, Z zp n.For any q Z, q rp n where n n r g Z, and q(x) (rpn)(x) r(p (x)) r(p (y)) rpn(y) q(y).
Hence, the flow is n-distal.
The proof of the -case is similar.We omit the necessity part and supply the sufficiency part.Sufficiency.
Let x, y X and p E EiS, X) such that p(x) p(y).
Then, pn(x) pn(y) for every n.As pn g ES, X), a compact space, n there exists a subsequence of (p), call it (qn), such that qn/q0 in E(S, X).It is readily verified that q0 E fiE(S, X) n and q0(x) q0(y).Since Z N E(S, X) n is left simple, Z Zqo.Let g Z. Then lq0 for some I in Z. Now, ( l(qo(X)) l(q0(y)) lq0(y) (y), and this completes the proof.
Let S be a semitopological semigroup, (X, a)the canonical LMC(S)- compactlflcatlon of S, and f LMC(S).
i) The following statements are equivalent.
a) f E Dn(s).N xn), v E X, and e E E(X).
PROOF.For x X, let T x be the left-m-introversion operator determlned by x.
Then, Z the closure of Rsf in the topology of pointwise convergence on C(S) {Tx f: x E X} [2, Lemma 4.19].
Defining k: X/E(S, Z) by k(x)(Tyf) Txyf, one verifies that k is a ontinuous homomorphism of X onto E(S, Z) satisfying koa .i) a) > b) Let u E Xn, v e X, and e E(X).Then, k(u) E(S, Z) n, and k(e) is an idempotent of E(S, Z)n.
As E(S, Z) n is left simple (hypothesis), k(u)k(e) k(u), i.e., k(ue) k(u).In particular, k(ue)(Tvf) k(u)(Tvf) where Tv f e Z, i.e., Tuevf Tuvf.Sinc___e X is right topological with w topology, it follows that Tuevf Tuvf X n for all u v e X, and e e E(X).b) > c) Let u Xn+l, v e X, and e e E(X).Then, u UlU2, where u X, Let p E(S, Z) n and let d be an idempotent of E(S, Z) n.There X n exists u e e e E(X)such that k(u) p, and k(e) d.Such a choice of e is -I possible as k (d) is a compact subsemigroup of X.Let v e X.
For any w X, w(T f) wuev(f)= wuv(f) (hypothesis) W(Tuvf).Therefore, Tuevf Tuvf.Now Bey k(ue) (Tv f) Tuevf Tuvf k(u) (Tv f)' which implies that k(ue) k(u).Thus, As E(S, Z) n is right topological, it follows that pd p for all p E E(S, Z) n proving that E(S, Z) n is left simple.Consequently, the flow (S, Z, ) is n-distal, and thus, f g Dn(s).There exists an e e E(X) such that k(e) d.We prove the existence of an u in fiX n such that k(u) p.
Let n be fixed and p e E(S, Z) n Then, p lim Pl for some (pi) E(S Z) n X n X n For each Pi' there exists x E such that k(x i) Pi" Now (xi) (c_ has a convergent subnet (x j) converging to an element, call t x n, in Xn. pj k(xj)_whlch converges to k(x n) and hence p k(x ).We now have a sequence (Xn)C X, (x n xn), having a convergent subsequence (x' n) such that x' u in X.
One readily verifies n that u g0 X n and that k(u) p.We omit the rest of the proof whlch is smlar to the proof of c) ===> a) in i).

S.
Then, Hence, Dn(s) is left translation s s invariant.
In a similar manner, one verifies that Dn(s) is right translation invariant.
The fact that X is the set of all multiplicatlve means proves that Dn(s) is an algebra As uev(1) uv(1), Dn(S) contains all the constant functions Let Dn w MM(Dn(S)) and f (S).
Let 8:X MM(Dn(S)) be the restriction map.There exists a w in X such that 8(w) w' T f T f and uev(Tw,f) uev(Twf) uevw(f) W W uvw(f) uv(Twf --uv(Tw,f).Thus, Tw,f Dn(s)which proves that Dn(s) is left m introverted.
Thus, Dn(s) is an admissible algebra of C(S).The proof that D (S) is an admissible algebra is similar, b) We give the proof for the -case, and omit the proof for the n-case.
Let (X,a) denote the canonical LMC-compactiflcatlon of S.
Let 8:X/Y denote the restriction mapping.
The e is a continuous homomorphism of X onto Y such that eoa B. First, we prove that Y has the property (I).Let u e Y.O yn, v Y, and e e E(Y).___ u UlU2 where u e Y and u 2 yn.There exist Xl, y X, d e E(X), and x 2 eft X n such that (x i) u i (i I, 2), O(y) v, and 8(d)= e.
Therefore, for any f e Dm(S), uev(f) Hence, uev uv, and thus, Y has the property (I).
To prove that (Y, B) is maximal with respect to this property, it remains to show that B 0 C(Y0)c_ D(S) for any right topological compactificatlon (Yo' BO)of S having property (I), where Bo:C(YO) C(S) is the adJoint of B O.
Therefore, there exists a continuous homomorphism :X Y0 such that B 0 6oa.
Let u e X. O Xn, v X, and e e E(X).=op(e(x)).Hence, e(x) e x.If Xl, x 2 are two d[stlnct points of X, then by hypothesis, there exists e I such that (xl) $(x2).This implies that 8(i) 8(x 2) and hence, 8 is inJectlve.Let y e X.For , we prove that (P)) c-l(p(y)).Let z e (p(y)).Then, (z) p(y) and (z) o (z) (consistency of maps) o p(y) p(y) (since y -I Therefore, z g (p(y)).
(Y) 0 -I As t is continuous for each t and {yt} is closed, {wt (Yt):t e I} is a class of closed sets in X, a compact space, with every finite intersection being nonempty (by -l(yt )' O(x) y and (2)).Therefore, t s fl I l(Yt) # #" For any x e flteIt hence e is surjective.
Clearly 8 is continuous Since X is compact, 8 is a homeomo rphism.THEOREM 8. Let [F}eI indexed by a directed set I, be a family of admissible subalgebras of__C(S) such that Frl F( () and (X, e) is the F-compactificatlon of S. Then F U F is admissible and the F-compactification (X,e) is the inverse limit space of the s%ectrum {X:w} where n:X X ( ) is the restriction map ((p) uIF ).

PROOF. Since
is directed, it is easily seen that F is admissible.Define :X X as the restriction map.
Then, in view of theorem 7, it suffices to prove that o any two distinct points Xl, x 2 e X there exists n e such that (x) * tx2).
The fact that x[, x2e X and x * x 2 implies that there exists n n an e F such that x[) ,(x2 )" The fact that x I, x 2 e X and x x 2 impl[es that there eKists an f e F such that xl(f) x2(f).By continuity of x| and x2, there exist g I and g e F such that ](Xl)(g) xl g) x2 (g) n (x2)(g)" Hence, (x I) # (x2) and that completes the proof.The following theorem is an immediate corollary to theorem 8. THEOREM 9.
O Dn(s) is an admissible subalgebra of C(S) and [ts compactlflcatlon (X,) is the inverse limit space of the spectrum {Kn:nm} where (Xn, e n) is the Dn(s)-compactification of S and :X X (n a m) is the restriction map.It is nm m clear that Dn(s) c_D=(S).Now, we give an example of a function f e D(S) but D N)for any n.Let fn be defined as in Example ill.Defining f(t) as f(t) I/t, t e N, we see that (norm).Thus, f e U Dn(N) C_D(N).Clearly, n f .D N).We remark that at this point we do not know whether the containment in O Dn(s) c_D(S) is proper. 4. FIXED POINT THEOREM.
Let F be a norm closed, conjugate closed, [eft (right) translation invariant subspace of C(S) containing constants.
Then a mean on F is called left (right) invariant if for each f e F, s e S, (Lsf) (f) translation invariant subspace F of C(S) is said to be left (right) amenable if there is a left (right) invariant mean on F, and amenable if F is translation invariant and both left and right amenable.L. N. Argabright [I] has proved that F is left amenable if and only if every affine flow (S, X, ) such that {x e X:U A(X)C_F} # # has a fixed x point, where A(X) denotes the Banach space of all continuous complex valued afflne functions on X, and U :C(X) C(S) is defined as U h(s) h(sx), s S, h e C(X), and x e X.We make use of this result to prove the following fixed point theorem.Let us prepare a lemma for proving the theorem.Defining : S M(F) M(F) as , (s)(x) L x, where L denotes the adjolnt of L :F F, one verifies that, relative s s , s to the action (s, x) L x, (S, M(F), w) is afflne flow.
If in addition, F is an s algebra, then MM(F) is a closed invariant subspace of M(F), and relative to the restricted action, (S, MM(F), ) is a flow.These actions of S are called the natural actions of S on M(F) (MM(F)).let (Z MM(Dn(S), B) denote the canonical Dn(s) compactificatlon of S. Then relative to the natural action, (S, Z, ) is a flow.LEMMA 10.The flow (S, Z, 7) is (n+1)-distal.

PROOF.
Let z z 2 e Z and (si) c_ S such that lim siz lim siz 2.
THEOREM II.
(Fixed Point Theorem) Dn(S) is left amenable if and only if every affflne flow (S, Y, ) containng a closed invariant subspace Z such that (S, Z, ) is (n+1)-dlstal has a fixed point.
(-distal) if and only if E(S, X) n (0 E(S, X) n) is left simple.
X, and e e E(X).
ii) The proofs of a) ==> b) and b) ==> c) in i) are easily modified to prove the corresponding results in il).Let us prove the case c) ==> a).It suffices to show that f E(S, Z) n is left simple.

THEOREM 6 .
a) Dn(s) and D(S) are admissible subalgebras of C(S).b) The Dn(s) -(D(S) -) compactificatlon (Y,B)of S is maximal with respect to the property that uev uv for u e yn+l (u e Y. flyn), v Y, and e e E(Y) PROOF.a) Let (X,a) denote the canonical LMC-compactificaion of S. That Dn(s) is a linear subspace of C(S) is immediate from Lemma 5 (i).It is easily verified that xn+1 Dn(s) is norm closed Let f Dn(s), u v X, e E(X), and s n d(v) e YO and d(e) is an idempotent of Y0" Then ue(f) Clearly, d(u) Y0" OY0 g(d(uev)) g(d(u)d(e)d(v)) g(d(u)(v)) (since YO has property (I)) ( g(d(uv)) uv(f).Thus f e D S) and this completes the proof maps) Then by hypothesis, sx sy for s e S n.Let e E(S, X) .nThen, i o 2 o e (S, x)n,nand (x) llm i(x) lira i(y) (y).This completes the proof.(ll)Let S be the semlgroup of all strictly upper triangular matrices (elements on the diagonal and below are zero) of order n+2 with entries from reals.With discrete topology, it is a topological semlgroup and Dn(s) LMC(S) C(S).Defining g:S/R by Dn-I g(s)