An Application of Compactifications: Some Theorems on Maximal Ideals (*). PA0-S~V.~(~ ttSV (Millbrook, New York, U.S.A.)

Summary. - The first paragraph is the summary.

Introduction.

I n this paper, we analyze a Stone-Cech type compactiIication, by replacing the real numbers ~ by a topological division ring A, discuss its relationship with the Banaschewski eompactification under suitable conditions, and apply these systematically to obtain as applications some results cf. GOLDEA]~:X and Wo~x [6], CoI~I¢~L and H ~ s ~ [3], and ST~u~I [13], on circumstances under which Stone's Theorem is valid. The treatment of the compactifications adopted (section 3) is that used in the paper (( l~ings of Continuous Functions with ¥alues in a Topological Field ~, by BACH~L&N, BECKE~NSTE]~N,~i[ARICI and W A ~ E R [1], in which continuous functions with values in a non-archimedean rank one valued field k are considered. IIere, k is replaced by a hausdorff zero-dimensional topological division ring A, as their applications extend to A-valued function. S is a topological space and A a topological division ring. C(S, A) denotes the set of all continuous functions from S to A, and C~(S, A) the set of all continuous functions from S to A with relatively compact range. C~(S, A) is the set of all continuous bounded functions on S, i.e., for all ] e Cb(S, A), ](S) is a bounded set in A, in the sense of KAPLA:NSXY[8]. I n the framework of compactifications, it is shown that a necessary and sufficient condition for every maximal ideal in Cb(S, k) (where k is a non-archimedean rank one valued field)to be of the form M~= {]eCb(S, k): ](s)----0 for some s o S } is that S is mildly compact (S~Av~) and that Stone's theorem holds for C~(S, k) (i.e., the only quotient field in Cs(S, k) is k} if k is locally compact or S is mildly coautably compact (StAv~0. A classical result states that if S is a completely regular topological sp~ce, R the real numbers, then Stone's theorem holds for Cb(S, R). The theorem also holds if R is replaced by the complex numbers~ the quaternions, or a finite field, i.e., by (*) Entrata in Redazione il 10 novembre 1975.

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a locally compact connected division ring. GOLD~A]~E]¢ and WOLK [6], ConnE]~ and IIE~I~IKSEN [3] investigated into conditions~ either necessary or sufficient, for the theorem to hold on C~(S, A) or C~(S, A), when A is a topological division ring. A version of some results of Correl and IIenriksen is shown here to be an immediate consequence of considerations of compactifications. I t is p r o v e d t h a t if A is a hausdorff zero-dimensionM division ring, and S is A-completely regular, t h e n Stone's t h e o r e m holds for C~(S, A).

1. - Z e r o - d i m e n s i o n a l and ultraregular spaces.

S is a topologicM space. S is said to be zero-dimensio~'tal if ~Shas a basis consisting of sets which are b o t h open and closed, and which shall be cMled clopen henceforth. S is said to be ultrahausdor]] if for e v e r y pair of distinct points, there are disjoint clopen sets containing each of t h e m respectively. S is said to be ultraregular if S is hausdorff and each point in S has a f u n d a m e n t a l system of neighbourhoods consisting of clopen sets. Hence a topologicM space S is hausdorff an([ zero-dimensioonM if and only if S is ultraregular. A field 1~with a r a n k one non-archimedean valuation I I, topologised b y ]], is ultraregular, as a sphere in k is clopen. A zero-dimensional space is completely regular, since the characteristic functions on clopen sets are continuous. Suppose 5 is a family of functions from a set T to a topological space B, and T is given the weak topology associated with 5 . I f B is zero-dimensional, t h e n so is T, as a subbasis for S consists of sets (]-~(_P~): ] e 5}, where (Pa} is a clopen basis for B. I n this paper, A denotes a topological division ring. A set / ) c A is said to be bounded if, for a n y neighbourhood E of zero, there exist neighbourhoods V and V' of zero such t h a t V D c E and DV~cE, where V D = { v d : v e V a n d tieD}. I f A has a r a n k one valutation ] ], t h e n D c A is bounded if and only if there exists a real n u m b e r c > 0 such t h a t ldl < c for all d in D. Suppose t h a t K(S, A) denotes the set of all continuous characteristic functions on S, t h e n K(S, A ) c Cb(S, A): F o r a n y neighbourhood E of zero, let V = E, t h e n V K ( S ) = 1~{0, 1} c E. PROPOSITIO:N 1. -- A is hausdorff zero-dimensionM division ring, and (S, ~) is a topologicM space. Then the following are equivalent: 1) (S, 5) is zero-dimensionM. 2) ~ =-5*: weak topology associated with C~(S, A). P~ooF. - 1) implies 2) : 5" c 3 in general. F o r the other inclusion, let 0 be open in ~ with P c 0, where P is clopen in 5. The characteristic function Kp on P is in C~(S, A), and _P is clopen in 5*.

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2) implies 1): I f 3 = 5*, t h e n (]-~(P~): ] e C~(S, A)}, where {P~} is a elopen basis for A, is a clopen subbasis for S. [] /~ is a hausdorff space consisting of at least, two points. A topologicM space S is said to be E-completely regular if S is hansdorff a n d the given topology of S is the weak topology associated with C~(S, 12). P~Ol, OSI~IO~ 2. - If (S, 5) is ultraregular a n d / ~ is hausdorff and contains at least two points, t h e n (S, 5) is Y-completely regular. P~ooF. - L e t Y be the weak topology associated with C~(S, ~). 5' c 3 in general. 3 c Y: Suppose D is open in 3 with t ~ D, t h e r e exists a set P , elopen in 5, with t e 2P c D. The characteristic function Kp on /) is continuous with respect to 5, and therefore to Y. L e t 0~ and 0~ be disjoint open sets in ~ containing 1 and 0 respectively, t h e n teP-=K~{1)cK-~l(O~)cD, where K-i~(O~) is open in Y. H e n c e D is open in Y. [] I n their paper [3], COl~EL and HENI~IKSE~ dealt with the following spaces: A is a h~usdorff topological division ring. A topological space S is said to be completely regular with respect to A if S is hausdorff and for every s in S and every neighbourh o o a 2~~ of s, there is an ] e Cb(S, A) such t h a t ](s)----0 ~n4 ](Cl¥)=-a va O. A is a hausdorff zero-dimensionM division ring. I f S is a hausdorff zero-dimensionM space, t h e n a) S is completely regular with respect to A, and b) S isA-completely regular: F o r s e S and a neighbourhood iV of s, with s e P c 2V, where P is clopen, the characteristic function Ko~ on C 2 satisfies the requirement for a); and b) follows from Proposition 2. A necessary and sufficient condition for S to be completely regular with respect to A is t h a t S is zero-dimensionM and hausdorff: F o r any open set ~ , if s e N , t h e r e exists ]eC~(S,A) with ](s)-=0 and ](C.N)==a¢O. I f 2 is elopen in A containing a b u t not o, t h e n s e ]-~(Ct') c 2Y and ]-~(C2) is clopen; necessity follows. Sufficiency is clear. H e n c e if S is completely regular with respect to A a n d A is a hausdorff zero-dimensionM division ring, t h e n S is A-completely regular. PP~ot'osI~IO~ 3. - 1) I f (~, 5) is completely regular with respect to A, where A is a hausdorff division ring, t h e n S has the weak topology 5' associated with Ca(S, A). 2) I f S is completely regular with respect to A, where A is a hausdorff locally compact division ring, t h e n S is A-completely regular. P~oo~. - 1) Y c 5 in general. I f E is open in 5, and s e h- c E, where N is a neighbourhoo4 of s, t h e n there exists ] e Cb(S, A) such t h a t ](s) =- 0 and ](C2Y) ----a ~ O. L e t D be an open set in A containing 0 b u t not a~ t h e n sC]-I(1))c3TcE. Therefore E is open in 5'. 2) I n this case, C~(S, A)

=

C~(S, A).

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2 . - The quasicomponent quotient space and the ultraregular kernel of S; some covering conditions. k denotes, throughout this paper, a field with a non-arehimedean rank one valuution. We adopt here Staum's t r e a t m e n t [].3] to identify continuous bounded function from a topological space S to k with a continuous hounded function from the quasicomponent quotient space of the ultraregular kernel QK(S) of S, which has desirable properties, to /c. A quasicomponen* of a topological space S is a minimal n o n - e m p t y intersection of elopen sets in S. F o r s in S, Q(s) denotes the quasicomponent eontainings, i.e., the intersection of all n o n - e m p t y elopen sets containing s. The set of quasieomponents of S forms a partition of S. E v e r y clopen set in a space S is a union of quasieomponents: in fact, P = U {Q(s): s e P } , for eveI~~ clopen set P in S. I n a zero-dimensional space, every open set 0 is ~ union of quasieomponents, because 0 ---- w -Pa~ where 2~ is clopen. E v e r y / e Cb(S, 7c) is constant on quasieomponents of S: I f / ( s ) =/= @/(t), /(s) e P and /(t) 6 P , where P is clopen in k, t h e n / - x ( P ) is a clopen set containing s b u t not t, and Q(s)=/=Q(t). The Qqzasicomponent Quotiant Space Q(S) of S is the space of quasieomponents of S, i.e., Q(S) = u {Q(s) : s e S}, with the quotient topology induced b y the mapping

Q: S -+Q(~) s -~Q(s)

A set B

c

.

Q(S) is open in Q(S) if and only i~ Q-I(B) is open in S.

P~oPosI~Io~ 4. - 1) If P is clopen in S, t h e n Q(P) is clopen in Q(S).

2) Q(S) is ultrahausdorff, or, eqldvalently, e v e r y quasicomponent of Q(S) is a singleton. 3) S ,'~Q(S) if and only if S is ultrahausdorff. Pl~ooF. - 1) I n this case, Q ( P ) = (J {Q(s): s e P } = P ,

and Q-l(Q(p))=.p.

2) Suppose Q(s)#Q(t) in Q(S), then Q(s)eQ(_P) and Q(t)~Q(2), where zo is a clopen set containing s b u t not t. 3) I f S is ultrahausdorff, Q(s) = {s}, t h e n Q is a bjiection which is continuous. Q-1 is continuous: 0 is open in S implies thatQ-~(Q-~)-l(O) = 0 (~ Q-~(S) = 0 is open. The converse is clear, m P~oPosI~Io5~ 5. - If S is zero-dimensional, then Q(S) is zero-dimensional. P]cooF. - If {P:} is a clopen basis for S, t h e n {Q(P~)} is a clopen basis for Q(S).

•

The Ultraregular kernel K(S) of S is the sp~ce with the same set of points as S and topolog~g generated b y etopen sets in S. K(S) is, b y definition, zero-dimensional,

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and hence has the weak topology associated with C~(S, k). P c S is clopen in K(S) if and only if P is clopen in S: P is clopen in K(S) implies t h a t _P is open in S and t h a t _P -- C(N C,) ~- ~ CC~, where C~ is clopen in S, so P is closed in S. The converse follows because a clopen set in S is open in K(S). Q(K(S)) is ultrahausdorff and zero-dimensionM. PI~oPoslTIO~ 6. - Q(K(S)) = .K(Q(S)), i.e., the two spaces and the topologies are identicM and are to be denoted b y QK(S). PI~ool~. - 0 is open in K(Q(S)) iff 0 is the union of clopen sets in Q(S). iff Q-I(O) is the union of clopen sets of S. iff Q - i ( O ) is open in K(S).

iff 0 is open in Q(K(S)).

•

P~OP0SITIO~ 7. -- There is a one-to-one correspondence between QK(S) and ideals M~ in Cb(S, k) of the form M~ = {/e C~(S, 7c):/(s) = 0}, with Q(s) = Q(t) if and only if M~ = M~.

PR00F. - If M~=/=M. /eC~(S,k) with / e M ~ and / $ M ~ , t h e n / ( s ) = 0 and /(t) V=O. Q(s) V=Q(t), since / is constant on quasicomponents. Conversely, if Q(s) V=Q(t)~ P is a clopen set containing s b u t not t, t h e n the characteristic function K e on P

is in M,, b u t not in Mr.

•

LE~f:~A 1. - C~(S, k) ~_ Cb(QK(S), k), isomorphieMly and isometrically. P R o o f . - F o r / e Cb(QK(S), k), /Q e C~(S, k). Consider the map

Q': C~(QK(S), k) --> Cb(S, k)

! ~/Q. Cleaxly Q' is a ring homomorpkism, also injective. Q' is surjective: F o r / ' ~ C~(S, k), define / e C~(QK(S)~ k) as/Q(s) =/'(s), which is well-defined since /' is constant on quasicomponents. / is continuous on QK(S), because Q-1(]-1(0)) -- (/')-~(0) is open in S, for any open set 0 in L Since I[/II -~ Ill'H,/e Cb(QK(S), k) and Q'(/) =/Q -~/'. Q' is un isometry: ItQ'(])tl = sup I]Q(s)l, ll/ll = sup I/Q(s)I, and the two are the same.

•

sez

Q(s)e0(s)

A topological space S is said to be mildly compact if every clopen cover of S has a finite subcover, mildly co~tntably compact if every countable clopen cover has a finite subcover. PROP0SITIO~ 8. -- I f S is zero-dimensionM and mildly compact (mildly countably compact), t h e n S is compact (countably compact).

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PnooP. - A necessary and sufficient condition for S to be compact is that every cover of S by sets in a subbasis has a. finite subcover (11, p. 139). S is compact by this condition. A slight modification of Kelley's proof, considering finite inadequate countable families of open sets, would give the second statement. • As a result, a. mildly countably compact set in a zero-dimensional metric space is compact, since countably compactness is equivalent to compactness in a metric space. These covering conditions are reflected in the qnasicomponent quotient space and the ultraregular kernel of the space: PJ~OPOSITIO~ 9. - 1) If S is compact~ then Q(S) and K(S) are compact. 2) S is mildly compact (mildly countably compact) if and only if Q(S) is mildly compact (mildly countably compact). 3) S is mildly compact (mildly countably compact) if and only if K(S) is mildly compact (mildly countably compact). PROOF. - 1) Suppose Q(S)c(Jo~, where 0~ is open, then S cUQ-~(0~) has a finite subcover, the images of whose sets under Q constitute a finite subcover for Q(S). Suppose K(S) c U~0~, where 0~ is open in K(S), then 0~ := U P~ where P~ is clopen in S. S c [J P~, and the argument follows as before. 2) One part is similar to 1); the converse holds because when P is clopen in S, Q(P) is clopen in Q(S). 3) _P is clopen in S if and only if 2 is clopen in K(S).

•

3. - Two eompaetifications and their relation.

The following view [15] of the classicM Stone-Cech compactification is generMized: (S, 3) is a completely regular space. R is the set of reMnumbers. V is the weakest uniform structure on S with respect to which all functions in C(S, R) are uniformly continuous. V* is the weakest uniform structure on S with respect to which all functions in Cb(S, R) are uniformly continuous. Then 1) Both V and V* generate the topology ~ on S, i.e., 3 = 5v---- 5r*. 2) The completion fl(S) of the uniform space (S, V*) is a compactification of S in which S is C*-embedded, i.e., for each / e Cb(S,/t), there is an extens i o n / e cb(~(~), R). 3) fl(S) is the Stone-Cech compactification of S. Here, we consider a Stone-Cech type compacti/ication fl.dS of an A-completely regular space S, where, initially, only the properties that A is hausdorff and consists

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of at least two points are required of A. Parenthetically, a classical completely regular space S does have the weak topology associated with Cb(S, R) [5, p. 259]. P~OP0SITI0~ 10. - F is a hausdorff sp~ce containing at least two points, and (S, 3) is /?-completely regular. U is the weakest uniform structure with respect to which each / ~ C~(S, ~), considered as a function from S to the compact uniform space

(/(S), U~), is uniformly continuous (Us if unique [5, p. 261]). Then 1) U generates the topology ~, i.e., 3 U--- ~. 2) The completion fl~S of the uniform space (S, U) is a compaetifieation of S. P~oo~. - 1) ~ c ~ , since S is ~-completety regular. 3 v c ~: L e t G be open in ~ , and t e G. There is a set E in U with t e E[t] c G; and there is a finite set {/} in C~(S,/v), with an open symmetric set V~e U~ for each /~ such t h a t ~ (/×])-~(V~) ~ D c ~ , ~ein

and teD[t]cE[t]cG.

D[t]=~/-~:V~(/(t)) is open in ~ since ° V~(/(t)) is open in ~in

/(S). G is t h e n ~ neighboarhood of every t e G. 2) S is dense in fl~S [5~ p. 257]. fl~S is compact: Every ] e C~(S~ E) is uniformly continuous from S to the complete space (/(S), U~), and hence could be uniquely extended to ]~: fl;S-~-/(S) [5, p. 252]. The m a p

fe~(z,F)

s -+/(s)

is ~ homeomorphism from firs onto a complete and therefore closed subspace of a compact spac% so fi~S is compact. • The Banaschewski compacti/ication floS of an ultraregular space is considered: P~0P0SITIO~ 11. - (S, J) is ultraregular. :5 is the collection of sets of the form B ~ - - ~ ( V × V), where ~ is a finite open partitions of ~. (A partition of S is a collection of non-empty mutually disjoint subsets of S whose union is S.) Then 1) :5 forms a base for a uniform structure U* ~ ~50(S) on S, which generates the topology of S~ i.e., 3 = Jr.. 2) The completion floS of the uniform space (S, U*) is a compactification of S. I'~ooP. - 1) The family :B of non-empty sets forms a symmetric filter base on

S × S [57 P. 203]: For B1----~j([J V × V) and B~----~%~[J( V × V),let ~Ube t h e n o n - e m p t y sets of the form V1 (~ V~, where V~ e ¢U1 and V~ e ~U~, then B3 ---- [.J (V × V) c B1 n B~. The filter U* = (D e P(S × S) : D ~ B for some B e 3~} [5, p. 212] is a uniform struetare. ~ c ~v*: Suppose P is clopen in ~ with s e P. Consider ~U ---- (P, C-P}. B~j(s) = 8 -

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= {t: (s, t)E53~} = P, implying t h a t 2~ is open in ~v.. 3~. c ~: If cO== {/~, ..., V~}, t h e n B r ( s ) = V, if and only if s a V~, i = 1, ..., n.. Eavh B~(s) is open in ~. 2) A necessary and sufficient condition for /3oS to be compact is t h a t for every D e U*, there is a finite covering of S by D-small sets [2 I, p. 201]. I n this case, if D E U*, then there cxixts B~a = ~ ( V × V) c D, and ~U is the finite cover of S consisting of B~-small, therefore D-small, sets. [] The Banaschewski eompaetifieation affords a measure-theoretic t r e a t m e n t : Suppose S is an ultraregular space, and ~' the algebra of all elopen sets of S. The set of all two-valued non-trivial measures #: ~Y--> {0, 1}, such t h a t a) #(¢) = 0, and b) # is finitely additive, is topologised by using the family of sets of the form V(P) == = {#E~]: # ( P ) = 1}, for all P e if, as a basis for a topology ~. I t could be shown t h a t (~, ~), actually having the subspace topology of the product 17[ {0, 1} and being ¢ closed, is compact, and also hausdorff. # is homeomorphic to a subset ~L.of ~7, where ~1~ consists of those measurs it, with

#~(-P) =

1,

if s E P

O,

if s 6.P ;

the latter is also dense in ~1. The relative uniformity for ~1, (relative to the unique uniformity on ~1) turns out to be identical as the uniformity of finite partitions on S. Hence, by identification, ~/ is the Banaschewslfi compactification of S. I n the presence of a hausdorff zero-dimensional division ring A, [3~S and tics coincide: Lv,~t~1A 2. - A is a hausdorff zero-dimensional division ring, and S is A-completely regular. U and U* are uniformities defined in Propositions 10 and 11 respectively. Then 1) S is ultraregallar. 2) U = U*. 8) 4) Cr(S, A) ~ C(fl.dS , A) == C(fioS, A) (A-isomorphic). Furthermore, if A is replaced by a field k with a rank one non-archimcdean valuation, we have C~(S, It) :~ C(fikS , k) -= C(floS, k), isomorphically and isometrically.

5) tics is ultraregular. Pl~oo~. - 1) The sets {]-l(p):] e C'(S~ A), _P clopen in A) form a clopen subbasis for S.

2) U c U*: L e t D ~ U, there are finite sets {]} in C~(S, A) and {Vse Us} such

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For each f, Uf---- :5o(](S)), since f(n) is compact. There

f~n

is a set B / ~ c Vs, where fly is a finite open paxtition of f(n). Then W~_~(~)= ---- U (]-l(v) x/-~(v)) = (] x])-~(B,~) e U*; FI (f x ] ) - ~ ( B ~ ) c ~ (] xf)-l(V,)cD; and veq3

fin

D e U*. U * c U: U* is actually the weakest uniform structure with respect to which

K(S,A), with uniformity of neighbourhoods of the diagonal A for {0, 1}x{0, 1}, are uniformly continuous: Suppose K e K ( S , A), K(S)= {0, 1} has a unique uniformity W consisting of neighbourhoods of the diagonal A of {0, 1} X {0, 1}. A e W; and (K(s), K(t)) e zl if and only if s and t are both in K-l{0} ~ 21 or in K-l{1} ~= P~. fly = {P1, P2} is a finite open partition of n, and B~T= U (P~ x P,) -- (K x K ) -~ (d). In this case, for E e W , (KxK)-~(E) is either B ~ above or n x n , both in U*; hence k is uniformly continuous relative to U* and W. If K(S, A) are uniformly continuous with respect to some U' on n and W, t h e n ( K x K ) - I ( A ) e U'~ for all K e K ( S , A ) . If Bo~= U (V, xV~), let K, be the characteristic function on I~, then B ~ = [~ (K, x K , ) -~ (A), implying t h a t B ~ e U' and U* c U'. i

Since K(S, A) c C~'(S,A), U* c U. 3) The hausdorff completion is unique [11]. 4) E v e r y ] e C'(S, A) could be extended uniquely to a uniformly continuous function p on /sAn [11, p. 195]. The mnp

¢,(n, A) + ¢(~An, A) = ¢(~0n, A) /-+p, is an A-isomorphism: For example, (]-~ g)~ and (P-F g~) agree on a dense set n in /5on, hence are identical. If A is replaced by k, the map is an isometry: 111tt< IIPtt since t](s)l< llPli for all s e n . For a n y s effort, there is a net {s~} e n converging to s, with (p(s~)} converging to p(s). Then for a n y e > 0, Ip(s)]< I](s~)] + e < ]I/]1 -F e, if ~ > 7 for some y;

and lIPli < i]lil. 5) If P c n is clopen in n, t h e n the continuous characteristic function K on _P has an extension K~ on fion. 1~ (closure in fion) -- (Ka) -1 {1} and S - - P = (Ka) -1 {0} ; both are clopen in [3on. {P, P - n } is non-intersecting, with Pun-P=

Pw (n-P)=3=~0n,

hence forms an open partition of floS. I f fly = (VI, ..., V.} is a finite open partition of S, then ~ = {F~: V~e fly} forms a finite open partition of floS; and {B~} forms a base for a uniformity in /~oS. For sefioS, s is in some open set G in floS, and there exists P clopen in S with

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P c G n S.

¢U .... {_P, S - - P ) and ~U == (P, S ~ P ) are finite open partitions of S and The sets {B~(s) : s e rio S, ~ is a finite open partition of S) form a basis of neighbourhoods consisting of clopen sets in fioS. •

floS respectively, s e P or s e S - - P , i.e., s ~ B ~ ( s ) = S - - P or s e B ~ ( s ) = P .

4. - Fixed maximal ideals and Stone's theorem: k-valued functions.

W e say t h a t an ideal in a subring C' of Ms--~ { ] ~ C ' : ](s)---0 for some s~S}, and t h a t e v e r y maximal two-sided ideal M of C', C'/M

C(S, A) is I]ixed if it is of the form Stone's Theorem holds for C' ff for ~ A (isomorphic).

P~OP0SITIO~ 12. -- 1) S is mitdly compact if and only if

QK(S) is compact and

fioQK(S) := QK(S). 2) I f S is mildly compact, t h e n isometrically.

Cb(S, k) ~ C(floQK(S), k), isomorphieally and

P~ooF. - 1) I f S is mildly compact, t h e n K(S) and therefore QK(S) are mildly compact. The ultraregular space QK(S) is t h e n compact. The converse is clear. 2) I n this case,

C~(QK(S), ~ ) ~ C(floQK(S), k). Apply temma 1. •

T ~ n o ~ M 1. - S is mildly compact if and only if every maximal ideal in is fixed.

Cb(S, k)

P~ooF. - a) I f S is mildly compact, Cb(S, k)~_ C(floQK(S), k), where fioQK(S) is compact~ hausdorff, and zero-dimensionM. E v e r y maximal ideal in C(floQK(S), k), therefore in Cb(S, ]¢), is fixed [12, p. 154]. b) Suppose all maximal ideMs in Cb(S, k) axe fixed, t h e y con'espond to points Q(s) in Q(S) (Proposition 7); or, the fixed maximal ideals in Cb(QK(S), k) correspond to points Q(s) in QK(S), b y virtue of lemma 1. QK(S) is ultraregular: C~(QK(S), k) ~ C(fioQK(S), k) b y lemma 2. Therefore maximal ideals in C~(QK(S), k) are fixed and correspond to points in floQK(S). I n general, C~(QK(S), k)c Cb(QK(S), k), since compa~ctness implies boundedness. I t follows from Proposition 7 t h a t there is a ono-to-one correspondence between fixed maximal ideals in C~(QK(S), k) and those in C~(QK(S), k). Hence there is a one-to-one correspondence between points in floQK(S) and those in QK(S)~ or floQK(S): QK(S) and S is mildly compact. • Co~o1~A~v. - If S is mildly compact, t h e n Stone's theorem holds for

C~(S, k) =

= Cb(S, 7¢). PROOF. - C~(S~k) -~ Cb(S~ k): I n fact, for every ]e C(S, k)~ ](S) c k is mildly compact and therefore compact. The fact t h a t all maximal ideals in C'(S, k ) = -~ C~(S, k) are fixed implies t h a t Stone's theorem holds for t h e m [12, p. 150]. •

PAO-SttER~G HSU: An application o/ compacti]ications,

etc.

117

THEO~E~I 2. -- I f a) k is locally compact, or b) S is mildly eountably compact, t h e n Stone's t h e o r e m holds for Cb(S, k). P~ooF. - a) Here, C~(S, k) ~ C~(QK(S), k) = C~(QK(S), k) ~_C(floQK(S), k). b) S is mildly countably compact, t h e n so is QK(S). Again, Cb(QK(S), k)-= = C~(QK(S), k): F o r every ] e Cb(QK(S), k), ](QK(S)) c k is mildly countably compact in a zero-dimensional metric space, therefore compact.

5. -

Stone's

theorem:

A-valued

[]

functions.

A version of the theorem proved b y CORRE]~ and H ] ~ R I K S ~ [3] is shown to follow immediately from the identification of functions in C'(S, A) with those in ~(/~0 S, A) • THEOREM 3. -- A is a hausdorff zero-dimensional division ring, and S is A-completely regular. Then Stone's theorem holds for C~(S, A). PROOF. - Apply L e m m a 2. Modified versions of Theorem 3 on p. 154 and Theorem 1 on p. 150 in [12], in which the c o m m u t a t i v e algebra C~(S, k) is replaced b y the ring C~(S, A), could be applied to give the result. COROLLARY. - If a) A disconnected division ring, metric division ring, and Cb(S, A) -: C(S, A)), t h e n

is a hausdorff, non-discrete, locally compact, and totally and S is A-completely regular, or, b) A is a zero-dimensional S is pseudo-compact (i.e., S is A-completely regular and Stone's theorem holds for C~(S, A).

P R o o f . - a) A h~usdorff, locally compact, and totally disconnected space is zerodimensional [12, p. 151]. Since A is hausdorff locally compact, C~(S, A) = C~(S, A). The s t a t e m e n t follows from the theorem. b) F o r e v e r y ] e C~(S, A), ](S) is also pseudo-compact: I f G: ](S) c A -->A is continuous, t h e n G] is continuous on S, and G](S) is a bounded set in A. I n the metric space A, pseudo-compactness is equivalent to comltably compactness [4, p. 232], which is equivalent to compactness. H e n c e C~(S, A ) = C~(S, A). i W i t h the structure of hausdorff locally compact non-discrete division rings known, the corollary completes a generalization of the classicM result: I f A is ~ non-discrete hausdorff locally compact division ring, and S is completely regular with respect to A, t h e n Stone's t h e o r e m holds for C~(S, A).

BIBLIOGRAPHY [1] G. BACI~MAN - E. B]~CK]~NST~IN - L. NA~ICI - S. WA~N~R, Rings o/continuous/unctions with values in a topological ]ield, Trans. Amer. Math. Soc., 294 (1975), pp. 91-112. [2] N. BOURBAKI, General Topology, 2 vols., Reading, Massachusetts, Addison-Wesley Publishing Co. (1966).

118

]?A0-SttE~G HSU: A n application o] compacti]ications, etc.

- M. HENRIKSEN, On rings o] bounded continuous ]unctions with values in a division ring, Proc. Am. Math. Soc., 7 (1956), pp. 194-198. [4] J. DUGUNDJI, Topology, Boston, Allyn and Bacon, Inc. (1966). [5] W. W. FAIRCHILD - C. I. TULCEA, Topology, Philadelphia., W. B. Saundcrs Co. (1971). [6] J. K. GOLDHAB~R - E. S. WOLK, Maximal ideals in rings o] bounded continuous ]unctions,

[3] E. C O ~ L

Duke Math. J., 29 (195~), pp. 565-569. [7] J. G. HOCKING - G. S. YOVNG, Topology, Reading, Massachusetts, Addison-Wesley Publishing Co. (1961). [8] I. KAPLANSK¥, Topological rings, Am. J. Math., 69 (1947), pp. 153-183. [9] I. KAt'LAI~SK¥, Topological methods in valuation theory, Duke Math. J., 14 (1947), pp. 527-541. [10] I. KAPLA~SKY, Topological rings, Bull. Am. Math. Soe., $4 (1948), pp. 809-826. [11] J. L. KELLEY, General Topology, Princeton, D. Van Nostrand Co. Inc. (1955). [12] L. NA~ICI - E. BECKENSTEIlV - G. BACHi~AN, "~unctionat Analysis and Valuation Theory, New York, Marcel Dekker, Inc. (1971). [13] R. STAV~, The algebra o] bounded continuous ]unctions into a nou-archimedean ]ield, Pacific Journal, to appear. [14] W. J. TENON, Topological Structures, New York, Holt, Rinehart and Winston (1966). [15] S. WA~NE~, Ulam's measure problem, Notes from lectures given at Reed College, Oregon (1970-71).