SUB-RINGS OF C(R n) by CHEONG KUOON GAN B.Sc. (Hons), University of Malaya, Malaysia, 1973 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of MATHEMATICS We accept this thesis as conforming to the required standard November, 1974 In presenting th is thes is in par t ia l fu l f i lment o f the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make i t f r ee ly ava i lab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho la r ly purposes may be granted by the Head of my Department or by his representa t ives . It is understood that copying or p u b l i c a t i o n of th is thes is fo r f i n a n c i a l gain sha l l not be allowed without my wr i t ten permission. Department of MATHEMATICS The Univers i ty of B r i t i s h Columbia Vancouver 8, Canada Date November 2 5 , 1 9 7 ^ - i i -Supervisor : J.V. Whittaker Abstract The content of t h i s thesis contains a study of the rings C(R n), L c ( R n ) , C m ( R n ) , C°°(Rn), A(R n) and P(R n) . We obtain the result that no two of the rings above can be isomorphic : i n fact we prove the following : i f cj) : A —> B i s a r i n g homomorphism where A, B are any two of the rings and A c B, then cp(f) = f(p) for some p e R n . We also characterise C(R n), C m(R n) and C°°(Rn) as rings. - i i i ' -Table of Content page Abstract i i Acknowledgement . * v Introduction v 1. Preliminaries 1 2. Sub-rings of C(R n) (I) 5 3. Sub-rings of C(R n) (II) 9 4. Semi-group Structures 22 5. Characterisations of C(R n), C m(R n) and C°°(Rn) 30 Reference 52 - iv 4 Ackribwl ed ement I would l i k e to express my gratitude to Professor J.V. Whittaker for his generous help and patient guidance. I would also l i k e to thank Professor A. Adler for reading the manuscript and offering valuable comments. Introduction In one of the f i r s t systematic studies of results on the rings of continuous functions, L. Gillman and M. Jerison [5] considered C(X), the ring of continuous functions under pointwise addition and m u l t i p l i c a t i o n on an arbitrary topological space X and i t s sub-ring C*(X), the ring of bounded functions i n C(X). They showed among other things that when X i s compact, then i t i s uniquely determined as a topological space by the ring C(X) or C*(X); i n fa c t , X i s homeomorphic to the space of (fixed) maximal ideals i n C(X), with the Stone topology [ 5 , Theorem 4.9] and when X i s realcompact, X w i l l be uniquely determined by the space of r e a l maximal ideals on C(X) [5 , Theorem 8 . 3 ] . Later, i n 1966, i n her thesis [ 11 ] , L.P. Su obtained p a r a l l e l results on the rings C m(X), the ring of a l l m-times continuously d i f f e r e n -t i a b l e functions on a C m - d i f f e r e n t i a b l e n-manifold X, L c ( X ) the ring of a l l functions s a t i s f y i n g a Lipschitz condition on a metric space and A(x), the ri n g of a l l analytic functions on a subset X of the complex plane, using the notions of m-realcompactness and L c-realcompactness. As part of the thesis, we make a cross-section study on the ring C(R n) and consider i t s sub-rings C m ( R n ) , C°°(Rn), L,(R n), P(R n) and A(R n), defined i n 2 . 1 . We obtain quite i n c i d e n t a l l y that R n i s uniquely determined as a topological space by each of the sub-rings L c ( R n ) , C m ( R n ) , 0 0 n n n C (R ) and of course ( i n accordance with above), C(R ) since R i s realcompact. - v i -I n o u r s t u d y o f t h e s e s u b - r i n g s o f C ( R n ) , w e s h a l l s h o w i n S e c t i o n 2 t h a t t h e s e s u b - r i n g s e n j o y a s t r o n g c o m m o n a l g e b r a i c p r o p e r t y t h a t t h e s e t o f r e a l m a x i m a l i d e a l s ( D e f i n i t i o n 1 . 4 ) i s t h e s a m e a s t h e s e t o f p o i n t s i n R n . A n o t h e r i n t e r e s t i n g o b s e r v a t i o n i s t h a t t h e r e a r e n o t " m a n y " h o m o m o r p h i s m s f r o m o n e s u b - r i n g i n t o a n o t h e r . F o r e x a m p l e , h o w d o e s o n e d e f i n e a n o n - z e r o r i n g - h o m o m o r p h i s m f r o m C ( R ) i n t o C ( R ) ? W i t h t h e i n t e n t i o n o f d i s t i n g u i s h i n g t h e s e s u b - r i n g s a m o n g t h e m s e l v e s , w e p r o v e i n S e c t i o n 3 t h a t t h e r e c a n o n l y b e o n e t y p e o f n o n - z e r o h o m o m o r p h i s m f r o m o n e s u b r - r i n g A i n t o a n o t h e r s u b - r i n g B , w h e r e B C A ( s t r i c t l y ) , n a m e l y , t h o s e o f t h e f o r m (j>(f) = f ( p ) , f s A , p e R n . T h i s , t h e n , s e t t l e s t h e q u e s t i o n o f w h e t h e r a n y t w o o f t h e s e s u b - r i n g s c a n b e i s o m o r p h i c . I n f a c t , w e d i s t i n g u i s h t h e s e s u b - r i n g s ( a s r i n g s ) n o t b y a n y a l g e b r a i c p r o p e r t y b u t b y u t i l i s i n g t h e e x i s t e n c e o f c e r t a i n f u n c t i o n s i n o n e r i n g b u t n o t i n t h e o t h e r . m oo I n S e c t i o n 4 , w e c o n s i d e r C ( R ) , C ( R ) , L £ ( R ) , P ( R ) a n d C ( R ) a s s e m i - g r o u p u n d e r c o m p o s i t i o n o f f u n c t i o n s . T h o u g h t h e r i n g s t r u c t u r e s a n d s e m i - g r o u p s t r u c t u r e s o n t h e m a r e i n g e n e r a l v e r y d i f f e r e n t , w e s h o w t h a t a t l e a s t i n C ( R ) a n d P ( R ) , t h e s e t w o s t r u c t u r e s a g r e e i n o n e s e n s e , t h a t t h e g r o u p o f r i n g - a u t o m o r p h i s m s a n d t h e g r o u p o f s e m i - g r o u p - a u t o m o r p h i s m s a r e e s s e n t i a l l y t h e s a m e . F i n a l l y i n S e c t i o n 5 , w h i c h i s o n e o f o u r m a i n o b j e c t s o f t h e t h e s i s , w e d e v e l o p i n d e t a i l a m a c h i n e r y t h a t e n a b l e s u s t o c h a r a c t e r i s e C ( R n ) , C m ( R n ) a n d C ° ° ( R n ) a s r i n g s . - 1 -1. PRELIMINARIES 1.1 We begin by showing some standard results which w i l l be needed l a t e r . We are only interested i n real-valued function rings under pointwise addition and m u l t i p l i c a t i o n on R n which contain the constant functions and the projections. Though our sp e c i f i c interest l i e s i n sub-rings of C(R n), namely, C m ( R n ) , m= 1, 2, C°°(Rn) , P(R n), L (R n) and A(R n) c (see 2.1), most of the following results remain true for a ring of r e a l -valued functions on any topological space and i n some instances, even for an arbitrary r i n g with unity. Throughout t h i s section, we s h a l l l e t A be a sub-ring of r e a l -valued functions on R n containing the constant functions and the projections on the axes. The set of constant functions w i l l always be i d e n t i f i e d with the set of r e a l numbers. The projections are denoted by u^, u^, . . . » . 1.2 Proposition : The only non-zero ring homomorphism from R, the r e a l s , into i t s e l f i s the id e n t i t y homomorphism. Proof : See [5, §0.22]. 1.3 Proposition : For each p e Rn, = {f e A : f(p) = 0} i s a maximal id e a l of A . Proof : I f f i M , then f(p) ^ 0 . For any g e A, g - g(p)f/f(p) e M , P P hence g = g(p)f/f(p) + g - g(p)f/f(p) e ( f , M p), implying ( f , Mp) = A . Therefore M i s a maximal i d e a l . P - 2 -1.4.1 De f i n i t i o n : A maximal i d e a l M i n A i s called a r e a l i d e a l i f A/M, the residue class ring of A modulo M, i s isomorphic to R . 1.4.2 Proposition : Every i d e a l of \the form i s a r e a l i d e a l . Proof : We note that <f>(f) = f (p) i s a ring homomorphism from A onto R since A contains R, the constant functions. So, A/ker <(> i s isomorphic with R . Therefore M = ker <f> i s r e a l . • P 1.5 Proposition : I f <f> : A —> R i s a non-zero real-valued homomorphism on A, then <j) (r) = r . Proof : Observe f i r s t that <|>(g) = <Kg)<Kl) for a l l g e A . Since <f)(g) £ 0 for some g e A, we have 4>(1) = 1 • So the r e s t r i c t i o n of $ to R i s a non-zero homomorphism on R ; by Proposition 1.2, <j>(r) = r . This completes the proof. 1.6 We see that every non-zero real-valued homomorphism on A i s i n fact onto, therefore, A/kercjxJ), the residue class r i n g of A modulo ker <j> i s isomorphic to R . Since ker (j) i s necessarily maximal, the above shows that i t i s a r e a l i d e a l . Conversely, i f M i s a r e a l maximal i d e a l , then for each f e A, we can i d e n t i f y M(f), the residue class of f modulo M, with a r e a l number. One can eas i l y check that <j> : A —> R given by <}>(f) = M(f) i s a non-zero homomorphism and M(r) = cf>(r) = r . E x p l i c i t l y , we have 1.6.1 Proposition : There exists a one-to-one correspondence between r e a l ideals of A and non-zero real-valued homomorphisms on A . - 3 -1.7.1 De f i n i t i o n : We say that a maximal ide a l M i n A i s fixed i f there exists p s R n such that M = Mp = {f E A, f(p) =0} . 1.7.2 Proposition : Suppose A has the additional property that for -1 n f e A, 1/f = f E A whenever Z(f) = {x e R : f(x) =0} i s empty. Then a maximal ide a l M i n A i s fixed i f there exists f e M whose zero set Z(f) i s compact. Proof : Z(M) = {Z(g) : g e M} has the f i n i t e intersection property : for - n i f there exist g. e M, i = 1, 2 n and H Z(g.) = 0 then 1 1=1 1 and so r n 2) n ^ n -j —1 I g ± = 0 Z(§n-) = 0 implying \ g 2 i = i x; 1=1 1 t i = i e A , 1 = ( n > -1 r n V 2 V 2 •\ 8 i *~i=l L 8 ±. i i = l ^ e M which i s not possible. Obviously i f there exist f e M whose zero-set Z(f) i s compact then f\ Z ( g ) 3 f} {Z(g)"fi Z(f)} / 0 and there i s an geM geM p e R n such that g(p) = 0 for a l l g e M , i . e . M C M^ . By maximality, we have M = M 1.8 Proposition : I f every r e a l i d e a l i n A i s fixed, then every non-zero real-valued homomorphism cp on A i s an evaluation, i . e . there exists p E R n such that <p(f) = f(p) for a l l f e A . Proof : From 1.6, ker cp i s a r e a l i d e a l . So - 4 -ker <j> = Mp = {f e A, f(p) = 0} for some p e R n . For f e A, f - <j>(f) E ker <f> since <f>(f) e R and <(>((()(f)) = <j>(f) (Proposition 1.5), so f - <|>(f) e M and cj)(f) = f(p) . This completes the proof. We remark that p i s unique, for i f <j>(f) = f(q) for some q e R n then u^p) = $ ( u ± ) = u^q) , i = 1, 2, n, hence p = q . We also note that the set of non-zero real-valued homomorphisms i s the same as the set of evaluations. 1.9 Proposition : I f every re a l i d e a l i n A i s fix e d , then there i s a natural one-to-one correspondence between any two of the following : (a) R n ; (b) R , the set of r e a l ideals on A ; (c) ft(A), the set of non-zero real-valued homomorphisms on A . Proof : The proof follows from Propositions 1.6 and 1.8 and the fact that R n i s equipotent with the set of evaluations on A . Note that the correspondence i s given by <' p <—> M <—> <j> P yP where p e Rn, and <j>p(f) = f (p) = M (f) . - 5 -2. SUB-RINGS OF C(R n) (I) 2.1 We now focus our attention on the rings C(R n), C m ( R n ) , C (R n ) , P(R n), L c(R n) and A(R n) under pointwise addition and m u l t i p l i c a t i o n . We adopt the following d e f i n i t i o n s . Let f : R n —> R be a continuous function. ( i ) f i s said to belong to the class. C m i f a l l p a r t i a l d e r i -vatives of f of order (less than or equal to) m exi s t (m = 1, 2, . . . ) and are continuous. ( i i ) f i s said to belong to the class C i f a l l p a r t i a l d e r i -vatives of f of a l l orders e x i s t . ( i i i ) f* i s said to be an L c~function i f i t s a t i s f i e s a Lipschitz condition on each compact subset K of Rn, i . e . there exists a positive number for which |f(x) - f (y) | ^ .M^l |x - y| | whenever x, y e K . (iv) f i s said to be analytic i f f has a power series expansion about each point y e R n , i . e . n n f(x) = f(y) + I a (x - y.) + I a (x - y ) ( x . - y ) + i = l i , j = l n + i / k = l a i j k ( X ± ~ " Y j ) ( X k " 7 k > + for x i n a neighbourhood of y . In t h i s section we show some properties of '. C(R n), the ring of continuous functions - 6 -C m ( R n ) , the ring of functions of class C m CO ^ CO C (R ) , the ring of functions of class C P(R ), the ring of polynomials i n n indeterminates , x^, •« • , L (R n), the ring of L -functions c ° c A(R ), the ring of analytic functions which form a chain C D L D C 1 Z> C 2 Z> ... D C™ Z> ... D C°° O A 3 P. c 2.2 From Proposition 1.3, we know that the rings mentioned above contained r e a l maximal ideals of the form Mp = {f : f(p) =0} , one for each p e R n . We w i l l soon see that i n fact a l l r e a l ideals i n C, C m, oo C , P, L and A are of t h i s form, c —1 m 0 0 2.3 I t i s t r i v i a l l y true that f exists i n C, C or C whenever Z(f) i s empty, f e C, C m or C°° . In L c, i f Z(f) i s empty, fEe L c > then |f| i s bounded below on each compact subset K of R n . Therefore | f _ 1 ( x ) - f _ 1 ( y ) | = |f(y) - f ( x ) | / | f ( y ) f ( x ) | < £_2|f(y) - f ( x ) | < e ' ^ I J x - y|| for any x, y E K . Hence f ^ e L_ . - 7 -In A, i f Z(f) i s empty, then f has a power series expansion at each point of Rn, having non-zero constant term. We can find a unique power series g about the same point such that f(x)g(x) = 1 for a l l x i n some neighbourhood of the point. I t can be shown that g has positive radius of convergence about any point whenever f has [see 3, page 24] . So f - 1 = g e A . By v i r t u e of Proposition 1.7.2, to show that a r e a l i d e a l M i n m °° 1 C, C , C , L £ or A i s fixed we only have to exhibit an f E M whose zero-set Z(f) i s compact i n R n . Let M be a r e a l i d e a l i n C, C m, C L c or A , then M = ker cj> for some real^valued homomorphism on the 2 2 2 respective rings (Proposition 1.6). Consider g(x) = x^ + x 2 + ••• + x n » x = (x^, X£, x n) e R n which belongs to each of the rings mentioned 2 2 above. Let r = <j>(g) a i*d f = g - r . Evidently f e M = ker <p and Z(f) = j ( x 1 , x 2 , ... , x n) : x 2 + x 2 + ... + x 2 = I r I j-i s an n-1 sphere of radius |r| i n R n which i s compact. Unfortunately P does not s a t i s f y the hypothesis i n Proposition 1.7.2, nevertheless we can show that every r e a l i d e a l i n P i s fixed. Let M be a r e a l i d e a l i n P(R n). As before M = ker cp for some homomorphism cf) : R n —> R . Let r ^ = cp(u^), u^ being the i - t h projection u i ( x 1 , x 2 , ... , x n) = x i , i = l, 2, ...,n, and r = ( r ^ r 2 , r n ) . For each f e ker cp , f ( r ) = f(<p(u 1), cp(u n)). Since each f i s a f i n i t e sum of f i n i t e products of x^, x 2 , x n and cp maps r e a l numbers - 8 -id e n t i c a l l y onto themselves and preserves addition and m u l t i p l i c a t i o n , f(<('(u1), <j>(u2) <Kun)) = <f>(f). So f ( r ) = 0 for a l l f e ker <|> and ker d> = M = M T r A l l these show that the following proposition i s true. 2.3.1 Proposition : Every r e a l i d e a l i n C, C , C , P, L £ and A i s f ixed. m 0 0 2.4 Hence the only real-valued non-zero homomorphisms on C, C , C , L c, P and A are the evaluations, by Proposition 1.8. In the l i g h t of Proposition 1.9, the set of r e a l i d e a l i n each of these rings i s i n one-to-one correspondence with Rn, the correspondence being Mp <—> p , p e R n . - 9 -3. SUB-RINGS OF C(R n) (II) 3.1 We now know that C, C , C , P L c and A a l l enjoy a common character described by one of the following : (a) Every r e a l ideal i s fixed. (b) Every real-valued non-zero homomorphism i s an evaluation. (c) There i s a one-to-one correspondence between any two of Rn, R (the r e a l ideals) and Q (the non-zero real-valued homomorphisms). m 0 0 Later we show further that each of the rings C, C , C and L £ completely determines R n as a topological space. This n o n - t r i v i a l resem-blance among the rings considered leads us to ask i f any two of them can be isomorphic. m °° We show that no two of the rings C, C , C~, P, and A can be isomorphic; i n fact the only non-zero homomorphisms from A into B (where B c A s t r i c t l y , A, B are any of the rings mentioned above) are the evaluations. 3.2.1 Theorem : Let A and B be sub-rings of C(R n) containing R and u^, i = l , 2, . . . , n . Suppose every re a l i d e a l i n A i s fixed. Then every non-zero homomorphism <f> : A -—> B i s given by <f>(f) = f°f for some unique T : R n —> R n . Proof : We define T : R n —-> R n i n the following manner. For each - 10 -x E R", l e t 0 : g —> g(x) be a non-zero real-valued homomorphism on B. Since 0 °<p i s a non-zero real-valued homomorphism on A, Proposition 1.8 X applies and so 0 °<j>(f) = f ( y ) , f e A for some fixed y e R n (depending X on x). Define T(X) = y . Then (4>f)(x) = e (4>f) = e °<p(f) = f ( y ) = fo T(x) , x e R n , X X hence <p(f) = fox , f £ A . If T = (x^, x^, ...> T n)» w e see that T i s uniquely determined by <p(u^ ) = u^ox = T^» i = l, 2, . . . , n . This completes the proof. 3.2.2 Remark : Whenever such a T e x i s t s , i t must s a t i s f y the condition that for e B for a l l f £ A . In part i c u l a r x^ = U^OT E B, i = 1, 2, ...» 3.3 We consider now a homomorphism cp : C(R n) -—> B where B i s any of C , C , A and P . From Theorem 3.2.1, we have cp(f) = f o T for a l l continuous f . 3.3.1 Lemma : I f cp : C(R ) —> B i s a non-zero homomorphism, then cp(f) = fox for some constant function x : R n -—> R n . Proof : We need only to show x^ : R n —> R, i = 1, 2, ...» n are cons-tant functions where x(x) = (x^(x), X 2 W , x n ( x ) ) . Suppose x^ i s not a constant function for some i . Since x^ = u^oT e B C C"*"(Rn), there i s a point t = ( t , , t„, ..., t ) E R n at which 3x./8x. i s not zero x z n x 2 for some 1 < j < n . Let = i (t» , t„, t . 1 5 s, t . . , , t ) : s e — — J t • 1 2' ' j - l ' ' j+1' n' - 11 -then i s connected, hence T^(E^) i s an i n t e r v a l i n R containing x^(t) . Define a continuous function g : R g (^ 2. * ^2' * * * ' ^ n^ n r(x. - T. ( t ) ) s i n « R by — ^ , X. t T.(t) X. - T.(x) 1 1 V X. = T . ( t ) , 1 1 We are going to show that the j - t h p a r t i a l derivative of g°x does not exist. Let y = ( t 1 , t 2 , t ^ _ 1 , tj+h, t j + 1 , •, t ) e E ? , then £ (g°T(y) - gox(t)) = = (gC^Cy), T 2(y) T n ( y ) ) " sO^OO, r 2 ( t ) , T n ( t ) ) ) = £ (T.(y) - x.(t)) s i n T . ( y ) ^ T i ( t ) , x.Cy) * T.(t) . Since i s continuous we have x^(y) -—> x^(t) as h -—> 0, therefore ^ , | h | ! ^ ( T i ( y ) ~ T i ( t > ) S I n <iw - M « 3T. i 9x. (t) lim i n f J-(T. (y) - ft'(t)) s i n , . 1 — - j t - t -e-K),|h|<eh 1 1 T i ( y ) " T i C t ) * - -A-r l i m ^ f % j ( y ) - T (t)) 9T. ] 9x, •<t) implying that l i m r-(g°T(y) - g°x(t)) and hence 0)/3x.)(gox) does not h-K) 1 exist at t . This shows that gox i B , contradicting Remark 3.2.2. So (1 <_ i <_ n) must be a constant function, and the proof i s complete. 3.3.2 Lemma : I f <p : C(R ) —> L (R ) i s a non-zero homomorphism then <f) (f) = fox , f e C(R n) for some constant x : R n —> R n . Proof : Appealing again to Theorem 3.2.1, we just have to show that such a x must be a constant function. Assume the contrary that x(x) = ( x 1 ( x ) , x 2 ( x ) , T n(x)) where T± : R n —> R, i = 1, 2, n and there i s 1 <_ j <_ n for which x.. i s not a constant function. For each y E R n , l e t X^EE R n (k = 1, 2, ...) be any sequence converging to y . Consider the sequence CO f| Let mx = l i m sup 0^ . Since ( x ^ ^ - ^ U iy) i s a compact set i n R and k °° x.. e L c(R n) (Remark 3.2.2) there i s an M for which I x ^ x ^ - Tj(y)| < M|[x k - y|| for a l l k = 1, 2, ... . Hence the sequence {o^ } 1 S bounded above and 0 <_ < °° . Suppose now that m = 0 , then, 0 <_ lim i n f a f c ..<_ lim sup a f c = mx = 0 k -* °° k -> 00 implying l i m a f c exists and l i m a f c = 0 . Hence i f for every sequence { x ^ - 13 -converging to y , m =0, then 9x./9x. exists at y and x j 1 (9x./9x.)(y) = l i m cr, = 0 , i = 1, 2, ..'., n . Since T. i s non-constant there i s a point t e R n at which either 9x../9x^ does not exist for some i or (9Tj/9x^)(t) / 0 for some iC. So there i s a sequence x^ -—> t such that |T ( X K ) - T ( t ) | l i m sup — : :— = m > 0 . k + °° I 1^ " t | I We choose a subsequence ^ x^^ °f ^^y} s u c n that |x (x ) - T ( t ) | |T (X.) - T ( t ) | li m — J — J l i m sup — J — d = m > 0 A-*0 I K " t| | kk-> - | |x k - t| | and |x.(x ) - x.(t)|/||x - t | | >_j for a l l I. We now show that on the 3 J ^ /•_.. compact set ^ x£^ U 'i1*-} - K , h°x does not s a t i s f y a Lipschitz condition for the continuous function h(x) = |xj - T^Ct ) ! 1 ^ 2 , x = ( x 1 , x 2 , ..., x n) e R n . Indeed | h o T(x £) - h o T ( t ) | |T.(xp - T . ( t ) | 1 / 2 |T.(X £ ) - T.(.t)| l l * £ ~ t | | :! I Tj (x^ ) - x . (t) I : |x £ - t| . m 1 > — • _ 2 ' T j ^ - T j f t ) ! 1 ' 2 and the right hand side can be a r b i t r a r i l y large since x —> t , T. i s * J continuous and x. (x ) —> x.(t) as I oo . Hence ho T I L CRn) contra-j Jo J c dieting Remark 3.2.2. So r± must be a constant function for i = 1, 2, n. - 14 -3.3-3 Theorem : The only non-zero homomorphisms from C(R n) into L c ( R n ) , C m(R n), C°°(Rn), A(R n) or P(R n) are the evaluations. Proof : This follows immediately from Lemmas 3.3.1 and 3.3.2. m 0 0 3.3.4 Corollary : C i s not isomorphic to any of the rings L g, C , C , A or P . Proof : I t i s a consequence of Theorem 3.3.3. 3.4 We note that evidently there cannot be any isomorphism between C(R n) and P(R n) since P(R n) i s an int e g r a l domain and C(R n) i s not. We s h a l l use this algebraic property of P(R n) to give another proof that the only n o n - t r i v i a l homomorphisms from C(R n) into P(R n) are the evaluations. 3.4.1 Lemma : I f <p : C(R ) —> A i s a non-zero homomorphism where A i s a sub-ring of C(R n) containing R, then <p[C(Rn)] = C ( F ) for some closed set F C R n . Proof : There exists x : R n —> R n such that <p(f) = f o T , f e C(R n) . Let F = CI (x(R n)} and define a : <p[C(Rn)] —> C ( F ) by a(rpf) = f I f . R n I f <pf = <pg, f, g e C(R n), then fox(x) = g°x(x), x e R n implying f and g agree on x(R n), aand hence, on CI lx( R n ) } since R n i s a Hausdorff R n space. This shows that <j>f = rpg implies f |F = g|F so a i s well-defined. It i s easy to check that a i s a ring homomorphism. Now i f a(rpf) = 0 then f | F = 0 implying <pf = f o T = 0. And i f - 15 -g e C(F), then there exists f e C(R n) such that f|F = g and a(cpf) = f |F = g . We have proved that a i s an isomorphism. 3.4.2 Theorem : The only non-zero homomorphisms cp from C(R n) to P(R n) are the evaluations. Proof : For each f e C(R n), o = <K(f - | f | ) ( f + |f|)> = <p(f - |f|)cp(f + |f|) . Since P i s an int e g r a l domain <|>(f) = <f>(|f|) or cp(f) = - cp ([ f |) . From the fact that cj> . sends positive elements to positive elements [5, §1.6], cp[C(Rn)] i s a t o t a l l y ordered ri n g . By Lemma 3.4.1, cj>[C(Rn)] = C(F) for some closed set F = CI {x(R n)} i n R . F must consist of a single point R n p e Rn, for i f F has more than one point we can ea s i l y construct a continuous function on F which i s not comparable with 0 . Hence cp(f) = for = f ( p ) , f E C(R n) . 3.5 According to our t o o l , the absence of non-zero homomorphisms from one rfng into another, other than the evaluations depends not so much on the ri n g properties but on the existence of certain functions. We now observe that there exist functions which are L -functions but which are not c d i f f e r e n t i a b l e , e.g. f(x) = |x-^|; and there exist functions which are of class C m (m > 1) but not of class C m + r ( r ( r >^ 1) , e.g. f(x) = j x ^ . x ? . I t w i l l be shown that these functions serve to prove that the only non-zero homomorphisms from L £ into C m, C°°, A or P and from C m into C™**, C , A or P are evaluations. The same method of discrimination applies - 16 -CO . among C , A and P . 3.5.1 Lemma : Let B be any of the rings C m ( R n ) , C°°(Rn) , A(R n) and P(R ), $ : L (R ) —> B, a non-zero homomorphism. Then there exists a constant function x : R n -—> R n such that <|>(f) = fox , f e L c(R n) . Proof : The existence of x i s shown i n Theorem 3.2.1. We wish to prove that i f x(x) = (x^(x), x 2 ( x ) , x^Cx)) where x i : R n —> R ( i = 1, 2, .;., n) , then x^ ( i = 1, 2, . .. , n) are constant functions. Suppose x^ i s not constant for some 1 <^ i <^ n . Since x^ = U^OT = (J) (u^) e B , x^ has p a r t i a l derivatives. Moreover, for some j , 1 j< j <_ n, 8x^ /3Xj i s not zero at some point of R n . Choose t e R n for which (3x i/8xj)(t) ^ 0 and define Ej — " ^ ^ ] _ ' ~^2' **"' ^' ^j+1' ***' ^ "n^ ! s e R Since E^ i s connected, x^(E^) i s an i n t e r v a l i n R containing x^(t) as an i n t e r i o r point (x^(t) cannot be an end-point of the i n t e r v a l , for otherwise i t i s a l o c a l extremum point implying (3x i / 8 X j)(t) = 0 ). We define g £ L c(R n) by g(x^, x 2 x n) = | ax^--[r^(t) | . We s h a l l show that gox t C 1 . Let y = (t, , t O J t . -,'t.+h, t . I 1 } t ) E E^ , 1' 2' i - l i 1+1 n 1 then \ (g°T(y) - g°x(t)) = \ (gC^Cy), T n(y)) - gCx-j^Ct), x n ( t ) ) ) ! , , Ix.Cy) - x . ( t ) | x.(y) - x.(t) = E |T.(y) - x . ( t ) | = x. ( y) - T . ( t ) - 17 -We see that l i m g °T(y) - gox(t) = ! ! i ( t ) h-H) h 9 x j T i ( y ) - ^ T i ( t ) + l i m go T(y) - p o T ( t ) = _ ^ i ( t ) ^ h+0 h 9 x j T 1(y)->T i(t) This shows that the j - t h p a r t i a l derivative of gox does not e x i s t . Consequently g ° x I C"*" , i . e . got i B which i s a contradiction by Remark 3.2.2. Hence x i s a constant function. 3.5.2 Lemma ; Let B be one of C^CR 1 1) (r > 1) , C°°(Rn), A(R n) and P(R n), cp : C m(R n) —> B a non-zero homomorphism. Then cp(f) = f° T > f e C m(R n) for some constant x : R n —> R n . Proof : Following the method of proof i n Theorem 3.5.1, we define g e C™1 by g(x n, x 9, x ) = |x. - x.(t)|'(x. - x . ( x ) ) m . We wish to show that nH~l gox i C . F i r s t , we know that gox has a l l p a r t i a l derivatives of f i r s t order and , . . 3 (gox) \ .3T- 3 ( g ° x ) 3x„ .3 (gox) 3x 3x. 8 ° T ; " 3x, ' 3x. 3T 0, 3X. 3x 3x. 3 1 J 22 J n j 3(gox) 3x ± 8 (g °T) = — — • — — since — — =0 i f k ^ i . 3x. 3x. 3x, 3x = (m + 1)|T . (X ) - x.(t ) l - ( x i(x) - x i ( t ) ) m " 1 '-^ . 3 We can continue d i f f e r e n t i a t i n g with respect to x.. u n t i l we obtain, - 18 -.m 9x (goT) = (m + 1) ! | T ± ( X ) - T±(.t) + h(x) , x e R where h(x) is differentiable at a l l points in R n including x^(t) and is a polynomial in 9kx^/9x^ (k = 1, 2, ..., m) with coefficients of the form C*|T 1(X) - x ±(t)|•(x ±(x) - x ^ t ) ) (£ = 1, 2, m-1) . Now setting up the j-th partial differential quotient for 9 m —-(g°x) = 9m(gox) , 9x. J n we see that for y = ( t ^ t^, t._lt t.+s, t._^, tn) z E.. j - l ' j -j+l ! n 1 s ^ 3^ gox(y) - 9.. gox(t) = s-' t3 (nr+ 1).||.T.(yh-T (t\\ 3 T i ) m + h(y) - h(t) and = (m + 1) ! ^x^m |x i(y) - x i ( t ) | T i(y) - T ±(t) h ( y ) - h(t) x ±(y) - x ±(t) JL5.ia lim s-HD. x. (y)->x, (t) + 1. s 9j g«>x(y) - 9^ go-r(t) • ?3x.„-.nrf-: (:--) + =^(m)+'l)! 9x.^m+l l J lim s+0 x 1(y)^x i(t)* 9™ gox(y) - 9 m go T(t) = '-(m + 1)! ^9x>im+l , > , 3h , \ + ^ x T ( t ) - 19 -This shows that the j - t h p a r t i a l derivative of gox of order m+1 does not exist hence gox t C - B y the same argument as before, x i s a constant function. 3.5.3 Lemma : I f x : R n —> R n and fox e A(R n) for each f e C^CR11) then x^ ( i = 1, 2, ..., n) are constant functions, where x(x) = (x-^x), x 2 ( x ) , T n ( x ) ) . Proof : Suppose x^ i s non-constant, then x^ = u_^ ox e A has a power series at some point t e R n and for some j , ( S T ^ B X . . ) ( t ) ^ 0, i . e . , n n T.(X) = a Q + J a.Cx. - t.) + I a (x. - t )(x - t ) + i = l i»j=l |x - t | | < e . We now consider x e R n of the form x = (fci» t2 ^ - 1 ' t j + X j ' ' j + l ' "**' ^r?' then Let T.(X) = aQ + a j X . + a..x2. + . . . , |x.| < e± 2 a(x.) = T.(X) - a A = a.x. + a..x. + ... 3 i 0 3 3 33 3 where a.. = (8x^/3xj)(t) ^ 0 . Then a has a composition inverse a ^ such that era ^ (Yj) = v j • a ~ ^ a ^ x j ^ = x j a n c* CT ^ ^ j ^ ^ a s a P o w e r series expan-sion about 0 [ 3 , Proposition 7.1, 1.1.7. and Proposition 9.1, 1.2.9] . 2 Define g(x) = exp{-(x.j-a 0) } i f x. ^ a Q and g(x) =0 i f x ± = a Q , then g e C°°(Rn) , therefore gox e A(R n). Now - 20 -2 2 exp{-a(x..) } = exp{-(T ±(x)-a 0) } = g 0x(x) = b n + b . x . +b..x. +... , | x . | < e 0 . 0 3 3 33 3 3 2 By [3, Proposition 5.1, 1.2.5], composition of two convergent series has a positive radius of convergent, so, 2 -1 2 exp{- y..} = exp{-aa (y^) } = b Q + b.(a _ 1(y.)) + b j j ^ C y j ) ) 2 + ... , |y.| < e 3 2 which contradict the fact that exp{- y } has no power series expansion about 0 . Therefore T. i s a constant. I 3.5.4 Lemma : I f for a l l f e A(R n) (or C°°(Rn)), fox e P(R n) for some T : R n —> R n , then x i s a constant. Proof : Since g(x) = s i n x., x = (x,, ) e R , belongs to A(R n) (or C M ( R n ) ) , we have gox(x) = g ( x 1 ( x ) , x 2 ( x ) , T n(x)) = s i n x ±(x) e P(R n). Since a non-constant polynomial i n n variables i s unbounded s i n T^CX), being a bounded function must be a constant. Hence x^ ^ i s a constant for each 1 < i < n . 3.5.5 Theorem : Let A, B denote any of the rings C, C™, L c, C°°, A and P, and B C A ( s t r i c t l y ) . Then the only non-zero homomorphisms <f> • A —> B are the evaluations. - 21 -Proof : Theorem 3.2.1 proves that tj> i s of the form <p(f) = f ° T for some T : R n -—> R n . Lemmas 3.3.1, 3.3.2, 3.5.1-3.5.4 show that such a x i s a constant function. I f we denote p = x(0) e Rn, then we have as required <j>(f) = f ( p ) , f e A . 3.5.6 Corollary : No two of the above-mentioned rings can be isomorphic. 3.6 Quite unlike the results i n 3.3 and 3.5, i f A C B, A, B any of the rings i n consideration, then there i s a wealth of homomorphisms <j> : A -—> B . In fact i f f^, i = 1, 2, n, are any n functions i n B, then x = (f-^ > r 2 ' •••» r n^ induces a homomorphism <p : A —> B given by (p(f) = fox , f e A . - 22 -4. SEMI-GROUP STRUCTURES 4.1 In this section, we would l i k e to consider the case n = 1, that i s , to r e s t r i c t our attention to C(R), L c(R), C m(R), C°°(R), A(R) and P(R). Of course a l l the results obtained previously remain true, with u^ taken to be the id e n t i t y function on R . Let End A denote the semi-group of ring-homomorphisms on A where A i s one of the groups mentioned above. Since each ring i s closed under composition of functions, we can regard A as semi-group (A, °) under composition. 4.1.1 Theorem : End A i s anti-isomorphic to the semi-group (A, °) . Proof : Each <f> E End A i s of the form <f>(f) = f ° T for some unique T = <j>(u^ ) £ A (Theorem 3.2.1). Conversely each T E A induces a r i n g -homomorphism <|> : A —> A given by <|>(f) = f°f • Hence cj> -—> <j>(u^ ) i s one-to-one and onto. Since ( j ^ o(|>2(f) = ( f > 1 ( f o T 2 ) = f o T 2 ° T 1 we have (<}>^ 0<j)2) (u^) = <f>2(u-^ ) o<j>^ (u^ ) arid the theorem i s proved. 4.2 L.E. Pur s e l l showed i n [9] that <J> i s an automorphism on P(R) i f and only i f there exist a 0) , b E R such that cf>(f) = f(ax + b) [9, Theorem 5'] and <j> —> <j>(x) i s an anti-isomorphism from the group of ring-automorphisms on P(R) onto the group of a l l non-singular a f f i n e - 23 -transformations on R [9, Theorem 6]. We can show that these results follow from the following theorem. 4.2.1 Theorem : Let A be any of the rings C(R), L c ( R ) , C m(R), C°°(R) , A(R) and P(R). Then Aut A, the group of ring-automorphisms of A i s anti-isomorphic with the group A' = {T e A : there exists cr e A such that rjox = x°a = u^ } under composition. , Proof : For each <j> e End A, there exist x, a e A i such that <p(f) = f°T and cp "*"(f) = foe . Since <f> ^ (CJOT) = O°T°O = crou^ - a = u^°a = cp "'"(u^ ) we have a°x = u^ . On the other hand u^ = fj> "'cp (u^) = u^°T°a = f a . Hence T = cp (u^) e A' . Conversely i f T e A1 , we can define a r i n g -endomorphism by cp(f) = fox . Since f°x = gox implies f = fou^ = f°x°a = g°x°a = gou^ = g and i f g e A', then <p(gocr<); = gorjox = g, t h i s endomorphism cp i s i n fact an automorphism. Since $y°(f,v ~— > ^v^uj} (u]) > $ — > <K.U]_) i - s a n a n t i -isomorphism. Remark : ( i ) In P(R), (P(R)) f i s the set of polynomials which have compositional inverses, so (P(R))' = {ax + b : a, b e R, a ^ 0 } . - 24 -( i i ) In C(R), (C(R))' i s the set of a l l homeomorphisms on R . ( i i i ) In A(R), (A(R))' i s the set of a l l functions 2 f (x) = a^x + a^x. + ... , x e R where a^ f 0 . 4.3 Very often, one encounters a set on which two d i s t i n c t algebraic structures can be defined. As an expected observation, the two algebraic e n t i t i e s always have different and unrelated characters. And t h i s i s the case with C(R), L c(R), C m(R), C°°(R) or P(R), considered as a rin g under pointwise m u l t i p l i c a t i o n and addition and as a semi-group under compo-s i t i o n . We show eventually i n th i s section that at least for C(R) and P(R), these two structures concur i n one instance, namely, the group of semi-group automorphisms and the group of ring automorphisms are ess e n t i a l l y the same. We f i r s t show a result on automorphisms of semi-group of a class of functions. We denote a semi-group of real-valued functions on R which contains the set of constant functions by A, while keeping i n mind that we m 0 0 are actually interested i n C, Lc» C , C and P, and i n pa r t i c u l a r C and P . 4.3.1 D e f i n i t i o n : An" element z e A i s said to be a l e f t zero i f z°f = z for each f e A . 4.3.2 D e f i n i t i o n : An automorphism <j> on A i s called inner i f there exists h e A whose compositional inverse h ^ e A exists and <j)Cf) = h°f °h for each f e A . - 25 -4.4.1 Lemma : The set of l e f t zeros i n A i s precisely the set of constant functions, R . Proof : Suppose h i s a l e f t zero, then for any arbitrary x , y e R, h(x) = ( h o y ) ( x ) = h(y). Therefore h i s a constant function. The fact that the constant functions are l e f t zeros i s t r i v i a l . 4.4.2 Lemma : I f cf> i s an automorphism on A, then there i s a b i j e c t i o n h : R —> R such that for each f e A, <j>(f) = h°f°h - 1 . Proof : I f x E R, then <Kx)«f = * (x) o<f4 _ 1(f) = K x ^ C f ) ) = <f> (x) implies that <j>(x) i s a l e f t zero. By Lemma 4.4.1, <f> (x) = y for some y E R . Define h : R —> R by h(x) = y, then h i s a b i j e c t i o n since <j> i s an automorphism. Moreover <j>(x) = h(x) for x e R and cj)foh(x) = <j>f°<f>x = <|>(f°x) = <f>(f(x)) = h(f ( x ) ) = h o f ( x ) . Hence <j>(f) = h°f°h _ 1 . 4.4.3 Lemma : I f h : R —> R i s a continuous b i j e c t i o n then, h i s s t r i c t l y monotonic. Proof : Immediate. 4.5.1 Theorem : Every automorphism of C(R) i s inner. Proof : By Lemma 4.4.2, <f>(f) = h°f°h 1 for some b i j e c t i o n h . We need to show that h i s continuous. Let X Q E R and E > 0 be given. Choose a number y Q 4 hCx Q ) , and define g E C(R) as follows : - 26 -( y o " e(yo " h(- kb ) )'* IX ~ h ( x o ) I ' ' x " h ^ I < e g(x) = -^ h(x Q) , |x - h(x 0) | >.:;£ . Since <j> i s an automorphism, there exists f e C(R) whose image <p(f) = g. Moreover V q = gh(x Q) = h f ( x Q ) implies f ( x Q ) ^ X Q > by continuity there exists 6 > 0 such that f(x) ^ x n whenever |x - x Q| < 5 . So gh(x) = hf(x) ^ n ' ( x n ) whenever |x - X Q | < <5, knowing that h i s one-to-one. From the d e f i n i t i o n of g, we conclude that |h(x) - h ( x Q ) | < e whenever [x - X Q | < <5. Since X q i s arbit r a r y , h i s continuous on R . 4.5.2 Theorem : Every automorphism cj> of C"*"(R) i s of the form <f>(f) = h o f o h ^ for some d i f f e r e n t i a b l e function h . Proof : By Lemma 4.4.2 and Theorem 4.5.1, <|>(f) = h<>f°h for some continuous h. We would l i k e to show that h indeed has a derivative on R. Lemma 4.4.3 states that h i s monotonic and so i s h ^. Since a monotonic function has a f i n i t e derivative almost everywhere [7, Theorem 4, p. 211], there i s a point X q at which h has a f i n i t e derivative. This l a s t fact w i l l allow us to conclude that h has a f i n i t e derivative at every other point of R . Let e R, then h(x) - h(x.) hf(y) - hf(x ) CD 1 " ° X - Xj^ X - Xj^ (<|>f)°h(y) - (<j,f)oh(xn) h(y) - h(x ) y - x h(y) - h ( x ) ' y - x " x - x ' ( y ^ X o ) o o 1 i f x = f(y) for some f e (^(R) such that f ( x Q ) = ^ . From (1) i t i s - 27 -clear that the derivative of h at x^ exists i f we have x —> x-^ implies y — > X q and (y - X Q ) / ( X - x ^ i s f i n i t e as x —> . We see that x = f(y) = y + x^ - X q does s a t i s f y the conditions, hence h i s di f f e r e n t i a b l e on R . This completes the proof. We remark that whether or not h has to have a continuous deriva-t i v e s t i l l remained unanswered by us. In fact whether every automorphism on C m(R) or C (R) i s inner remains hitherto an open question. 4.5.3 Theorem : Every automorphism <J> on P(R) i s inner. Proof : Again by Lemma 4.4.2, § (Q) = ho.Q.oh ^ for some b i j e c t i o n h on R . To show h i s continuous we note that Q(R) i s either a l l of R or has the form [a, °°) or (-°°, b] . Since <f> (Q) (R) = <j)(Q)°h(R) = h (Q (R)) we see that h maps closed subbasic sets of R into closed subbasic sets. This shows that h i s continuous. By Lemma 4.4.3, h i s monotonic and as i n the proof of Theorem 4.5.2, h i s d i f f e r e n t i a b l e at some X q e R. We show that i t i s i n fact d i f f e r e n t i a b l e at any other point of R . Let x^ e R, x = Q(y) = y + x^ - x then h(x) - .h(x 1) hQ(y) - hQ(x Q) x - x^ x - x^ (<)>Q)°h(y) - (<j>Q)oh(xo) h(y) - h(x Q) y - X Q h(y) - h(x ) y - x x - x, J o o 1 and h' (x^) = (<j>Q)' (h(x Q))-h' ( X q ) . Now Q i s i n v e r t i b l e implies <j>Q i s i n v e r t i b l e ; but the only i n v e r t i b l e elements i n the semi-group of polynomials - 28 -are the linea r ones. So (<j>Q)' = c where 0 f c e R . The equation h' (x.^ = (<pQ) ' o h ( x ^ ) ' h 1 ( X Q ) then suggests that (<pQ) 1 =1 since h' cannot have a jump discontinuity. This shows that h' i s a constant function and therefore h i s a linea r polynomial. -1 m 0 0 4.6.1 Theorem : Let A denote the semi-group L c ( R ) , C (R) , C (R) or P(R). Then every automorphism cp on A can be extended uniquely to an automorphism on C(R). Proof : From the proofs of Theorems 4.5.1 and 4.5.3, i t i s clear that cj>(f) = h°f°h for some homeomorphism h on R . Defining <p*(f) = h°f°h for f e C(R) we see that <j>* i s an automorphism which extends <j>. I f if) i s another automorphism on C(R) which extends cp then if)(f) = k o f o k ^ for some homeomorphism k; and by d e f i n i t i o n of h and k, h(x) = <j>(x) = if)(x) = k(x) for each x e R . Hence h = k and <p* = if) . m 0 0 4.6.2 Theorem : Let A be the semi-group C, L £, C , C or P . Then every automorphism on A i s determined uniquely by i t s action on A', the sub-semigroup of i n v e r t i b l e elements i n A . Proof : We f i r s t show that i f an automorphism cp maps A' onto A' i d e n t i c a l l y , then cp i s the i d e n t i t y on A . We know that tp(f) = h 0 f o h ^ for some homeomorphism h . Suppose there exists X q e R such that h(x ) 4 x and define o ' o h(x o) - y Q _ r f (x) = — r — (x - x ) + y where y = h (x ) . h(x ) - x o Jo J o o o o - 29 -We see that f i s an i n v e r t i b l e element i n C, L c, C m, C and P . From the functional i d e n t i t y (<(>f)oh = hof we ar r i v e at x = h(y ) = hf (x ) =. (<|>f)h(x ) = fh(x ) = h(x ) o J o o o o o which i s a contradiction. Hence h(x) = x for a l l x e R . This shows that <|>(f) = f for each f e A . Suppose now that <f> and are two automorphisms on A which agree on A' . The composition if) i s an automorphism on A that maps A' i d e n t i c a l l y onto i t s e l f , so o^<j) i s the id e n t i t y on the whole of A, implying \> = (j> . 4.6.3 Theorem : Let A denote C(R) or P(R). Then the group of r i n g -automorphisms on A i s anti-isomorphicmwith the group of semi-group automorphisms on A . Proof : By Theorem 4.2.1, the group of ring automorphisms of A i s a n t i -isomorphic with A' , the group of i n v e r t i b l e elements of A . Hence we need only to show that the group of semi-group automorphisms on A i s isomorphic to A' . Indeed there exists a one-to-one correspondence given by <j>^ -—> T where <f>T(f) = T°'f°T . Since <f> °<j> (f) = < f > ' ( u ° f o y "*") = T°y°f°y = ( t ° y ) °f ° ( t 0 y ) = <|>' (f) , i t follows that the correspondence does define a group isomorphism hence the proof i s complete. - 30 -n m n co -n 5. CHARACTERISTIONS OF C(R ), C (R ) and C (R ) 5.1 In this section, we set out to characterise C(R n), C m(R n) and CO -j^ C (R ) as rings. Let X be a topological space and C(X), the ring of continuous functions on X . We s h a l l consider a sub-ring A of C(X) which contains the set of constant functions. 5.1.1 D e f i n i t i o n : A i s called a regular sub-ring of C(X) i f Z(A) = {Z(f) : f E A} forms a base for closed sets i n X . 5.1.2 D e f i n i t i o n : An ide a l M c A i s r e a l i f A/M == R . We denote the set of a l l r e a l ideals i n A by R. . J A 5.1.3 D e f i n i t i o n : A i s said to be a point-determining sub-ring of C(X) i f for each M e R. , M=M = { f e A : f(x) = 0 } for some x e X, i . e . A x i f every r e a l i d e a l i n A i s fixed. 5.1.4 Remark (i ) I f C(X) contains a regular sub-ring, then X i s necessarily completely regular [5, Theorem 3.7]. ( i i ) We wish to point out that since there exists a one-to-one correspondence between re a l ideals and non-zero homomorphisms (1.6), A i s point-determining i f and only i f every non-zero homomorphism i s an evaluation. For each M e R^ , the corresponding homomorphism i s denoted by f —> M(f) where M(f) i s i d e n t i f i e d with a re a l number. From Proposition 1.5, i f r e R, then M(r) = r for each M e RA . - 31 -( i i i ) I f A i s a regular sub-ring of C(X), then any sub-ring of C(X) containing A i s also regular. (iv) As a note, L,(Rn)> C m ( R n ) , C°°(Rn) and P(R n) are a l l point-determining sub-rings of C(R n), by Proposition 2.3.1. 5.2 Here we would l i k e to show that L £ ( R n ) , C m(R n) and C°°CR n ) are regular sub-rings of C(R n) . By Remark 5 . 1 . 4 ( i i i ) i t i s s u f f i c i e n t to 0 0 n n n show that C (R ) i s a regular sub-ring of C(R ), i . e . , for any p e R and closed set F C R n such that p I F , there exists f e C (R n) whose zero set Z(f) z> F but p i Z ( f ) . In fact something more can be shown. 5.2.1 Theorem : Let F be an arbitrary closed subset of R n . Then F = Z(f) for some f e C (R ) . Proof : See [11, Theorem 2.2]. 5.2.2 Corollary : L £ ( R n ) , C m(R n) and C°°(Rn) are regular sub-rings of C(R n) . Proof : Follows from Theorem 5.2.1 and Remark 5 . 1 . 4 ( i i i ) . 5.3.1 Theorem : I f A i s a point-determining, regular sub-ring of C(X), then A determines X uniquely. Proof : We would l i k e to show that X i n fact i s homeomorphic with R^ with a suitable topology. Let {M e R^ : M(f) = 0 } , f e A b e a -base for the closed sets i n R . Since f(x) = 0 i f and only i f M (f) = 0, the - 32 -correspondence Z(f) <—> {M : M(f) = 0 } i s one-to-one from Z(A) onto the base for the closed sets i n R. , so. x —> M i s c l e a r l y a homeomor-phism between X and R^ . This completes the proof. The topology on R^ described above i s actually the Stone topology on R^ and i s the same as the hull-kernel topology [5, p. 111]. We denote this space by sR^ . For f e A, we define f* : R^ —> R by f*(M) = M(f) . The family {f* : f e A} of functions on R^ induces a weak topology on R^ which we denote by wR^ . It i s easy to see that the weak topology i s f i n e r than the Stone topology, since from the equality {M e R A : M(f) = 0} = ( f * ) _ 1 { 0 } , f e A , every member of the base i n sR i s weakly closed. In case A i s a point-determining regular sub-ring of C(X), we can show that the two topologies coincide : Let {My : |M x(f) - M^(f)| >_ E} , f E A, be a subbasic weakly closed set i n R^ . Since {y e X : |f(x) - f ( y ) | ±_ e} i s closed i n X (note that f e A implies f i s continuous on X), by Theorem 5.3.1, i n which i t i s proved that x —> M x i s a homeomorphism, {My : |M x(f) - My(f) | >_-e} i s closed i n the Stone topology. Thus we have : 5.3.2 Corollary : I f A i s a point-determining, regular sub-ring of C(X), then sR. = wR. . A A - 33 -5.3.3 Corollary : L £ ( R n ) , C m(R n) and C°°(Rn) determine R n as a topological space. Proof : Since a l l the rings mentioned are point-determining regular sub-rings of C(X) (Remark 5.1.4(iv) and Corollary 5.2.2), the corollary follows from Theorem 5.3.1. isomorphic to the r i n g of r e a l numbers, R . (Note that A need not be a sub-ring of any ring of continuous functions). We s h a l l always i d e n t i f y t h i s sub-ring with R. Such a ring A i s the same as an algebra A over R where A i s an algebra containing unity. However since such an algebra has no different algebraic properties, we s h a l l only concern ourselves with ring structures i n a ring A (containing R). and w i l l be denoted by R. . This ;set with the Stone topology and weak 5.4 We now consider an arbitrary ring A containing a sub-ring The set of r e a l ideals on A i s defined as i n D e f i n i t i o n 1.4.1 5.4.1 D e f i n i t i o n An ( arbitrary );rfing A with unity i s said to be regular i f flR. = JX {f : f e M} = 0 and for every M e R and a e M, there i s a b e M such that N{(b-1) (1-a)} > 0 for each N £ R We point out that the d e f i n i t i o n of regularity for a sub-ring A of C(X) (for some X) (Definition 5.1.1) and the-'definition.,of regularity for an arbitrary ring A (Definition 5.4.1) are d i f f e r e n t . Nevertheless, - 34 -these two d e f i n i t i o n s are equivalent i n case A i s a point-determining sub-ring of C(X) for some completely regular topological space X . This i s to be j u s t i f i e d i n Theorem 5.5. Furthermore, and of more importance, we s h a l l see shortly that i f A i s a regular r i n g , then wR^ = • (Note : this fact i s proved i n Corollary 5.3.2 when A i s a point-determining sub-ring of C(X) ). To th i s end we f i r s t prove the following. 5.4.2 Lemma : I f A i s any r i n g , then for each M E R. contained i n a o A weakly open set U i n wR. , there exists an a E A such that a*(M ) = 0. A O a*(N) > 1 for N if U Proof : By d e f i n i t i o n of the weak topology on R , there exist a. e A , £ £ > 0 , i = l , 2, . . . , n , such that M e o f| { M : |a*(M) - a * ( M Q ) | < C U 1 n 2 Let a = min { e 2 e n } and a = -j £ { a ± - M Q(a i)} . Then a i = l a*(MQ) =0 and for N t U , |a*(N) - a|(M Q)| > for some j , sc a*(N) >^ a • ± 2 |a*(N) - a*(M )| > 1 5.4.3 Theorem : Suppose D R A = 0 > t h e n A i s a regular ri n g i f and only i f wR. = sR. . A A Proof : (Necessity) Assume that A i s a regular ri n g ( i . e . , D e f i n i t i o n 5.4.1 i s true). From the equality - 35 -{ M e R : M(a) = 0 } = (a*) """{O} , a e A , A we see that the Stone topology i s contained i n the weak topology for any ring. To show sR^ = wR^ , we only need to show that every closed set i n wR^ i s closed i n the Stone topology. Let F be a closed set i n wR. and M e R. such that M i f . A o A o By Lemma 5.4.2, there exists an a e A such that a*(MQ) = 0 ( i . e . a e M Q ) , a*(N) >_ 1 for N e F . Regularity of A now ensures the existence of a b e A such that b e M and o M f ( b - l ) 2 ( l - a ) } ^ 0 for every M e R ; i . e . M{(b-1)% • M{(l-a)} > 0 for every M e R^ . As N ( l - a) = N(l) - N(a) = 1 - a*(N) <_ 0 for a l l N e F , we obtain from 2 the inequality above that N{(b-1) } = 0 for any N e F . Moreover, M Q{(b-l) 2} = 1 i 0 so, M Q i K , where K = { M : (b-l)*(M) = 0 } and K e Z(A) i s closed i n sR A . We have K D F but M Q i K . This shows that F i s closed i n the Stone topology. So sR^ = wR^ . (Sufficiency) Let M e R. and a e M , we wish to find a o A o b e A for which D e f i n i t i o n 5.4.1 i s s a t i s f i e d . Choose U = { M : (l-a)*(M) > 0 } to be an open set i n wR. containing M . By assumption U i s open i n sR^ , and there exists c e A such that M Q t { M : c*(M) = 0 } and M^ e { M : c*(M) = 0 } - 36 -for a l l i U . Hence for any N e R^ , I f N e U then N{c 2(l-a)} = {N(c) } 2-N(l-a) ^ 0 i f N i U then N{c 2(l-a)} = {N(c)} 2-N(l-a) = O-N(l-a) = 0 . Let b = 1 - (c/c*(M ) ) , then b e M and for a l l N e R. o o A N{( b - l ) 2 ( l - a ) } = c*(M o)" 2 N{c 2(l-a)} >_ 0 . Therefore A i s a regular r i n g / 5.5 Theorem : Let A be an arbitrary ring (containing R). A i s regular i f and only i f i t i s isomorphic to a point-determining regular sub-ring of C(X) for some topologically unique completely regular space X . Proof : (Sufficiency) Let f e 0 R^ > x e X . By Proposition 1.4.2, M e R. and so f e M , implying f(x) = 0 . Therefore f = 0 on X , X A X f l ^ A = 0 . From Theorem 5.3.1 and Corollary 5.3.2, X i s homeomorphic to sR^ = wR^ . Theorem 5.4.3 then states that A i s a regular r i n g . (Necessity) For each a e A , define a* : R^ —> A by a*(M) = M(a) and endow R^ with the weak topology induced by A* = {a* : a e A} . Because f) R^ = 0, we can show that A and A* are isomorphic as rings. Defining a*b* = (ab)* , a* + b* = (a + b)* , we see that a —> a* i s a ring homomorphism, and we only have to show that this homomorphism i s one-to-one. Let a* = b* , then a*(M) = b*(M) for a l l M E R A , i . e . - 37 -M(b - a) = M(b) - M(a) = b*(M) - a*(M) = 0 implying b - a e M for a l l M e . Hence b - a = 0 and b = a showing that A and A* are isomorphic. Now A i s regular implies wR^ = sR^ (Theorem 5.4.3). Since by d e f i n i t i o n , Z(A*) = {Z(a*) : a e A} forms a base for sR^ where Z(a*) = {M e R A : a*(M) =0} = {M e R A : M(a) = 0} , i t also forms a base of wR^ . Hence A* i s a regular sub-ring of C(wR^) by D e f i n i t i o n 5.1.1. Next, we show that A* i s point-determining. Let M* E R .J . , A* denote {a e A : a* e M*} by M . From the isomorphism a —> a* i t i s clear that M e R A . We have M* = {a* : a*(M) = 0} = M* C C ( R A ) b y d e f i n i t i o n of M . Hence A* i s a point-determining, regular sub-ring of C ( R A ) . Uniqueness of R A follows from Theorem 5.3.1 and from Remark 5.1.4(i), R A i s completely regular. 5.6 We are now i n a position to proceed with the characterisation of C(R n), C m(R n) and C (R n) as regular rings with certain properties on the re a l i d e a l space that can be used to i d e n t i f y R A with R n . 5.6.1 Lemma : Suppose A i s a regular ring and there exist u., u« u e A such that 1 2 n (i) for r. e R, i = 1, 2, n , u.-r., u 0 - r 0 , ..., u - r l l ± i. L n n are contained i n one unique M e R A ; - 38 -( i i ) i f a e A and M e R^ such that M(a) ^ 0,. then there exist a e R, b e A for which I (u. - M(u.)) 2 + a 2 = a 2 + b 2 . i=l 1 1 Then R i s homeomorphic with R n . Hence A i s isomorphic to a point-determining, regular sub-ring of C(R n). Proof : Define y : R ^ — > R n by y (M) = (M^), ,M(u 2), M(u n)) . Suppose for £ R^ , y(M) = y(N), then we have for i = 1, 2, ..., n , M(u i) = N(u ±). Let r± = M(u ±) = N(u ±), i = 1, 2, n . Then u. - r. e M and u. - r. £ N for 1 < i < n , by condition ( i ) , we have i i i i — — J M = N . And for s = (s^, s^, ..., s^) e Rn, l e t Mg be the unique r e a l ideal containing u^ - s^, u 2 - s 2 , u n - s^, then y(M g) = (M s( U ; L), M s(u 2) M s(u n)) — (s-^, ^ 2 > • • • , s^) - s . This shows that y i s one-to-one and onto. We topologise R^ by the weak topology induced by A* = {a* : a e as i n Theorem 5.5. Since y = (u*, u*, u*) and each u* i s continuous 1 2 n i by d e f i n i t i o n of weak topology, we see that y i s also continuous. We would l i k e to show that y(F) i s closed i n R" for any closed set F i n wR^ . Knowing that A i s regular and by Theorem 5.4.3, we need to consider only closed sets i n R^ of the form {F = M e R^ : M(a) = 0} , a e A. Suppose - 39 -,s i y(F) = j (M( U ; L), M(u 2) M(u n)) : M(a) = 0 j then M (a) ± 0 where s = y(M ) = (M (u.,) , M (u 0) , M (u ) ) . By ( i i ) there exist a e R, b E A such that 1 1 2 , 2 _ ; 2 . , 2 then for N e R, , A I (u. - M g ( u . ) r + a = a 2 + b i = l O o <•> J t < a + N(b) Z — • ' J (N(u.) - M ( u . ) ) 2 + N(a) 2 / / i = l 1 s 1 Let B a / 2 ( s ) = { t £ R n : | | t - s | | ( t . - s i ) 2 < f } . -1 Then for N e y ( B a / 2 ( s ) } , i.e. for N e R^ s a t i s f y ! ng /n 2 . a X (N(u.) - M s(u.)) Z < 2 i = l 2 2 2 we have, by the inequality above a < a /4 + N(a) implying N(a) ^ 0 or N t F . Since y ^{B^^Cs)} fl F 4 0 and y i s one-to-one and onto, we must have B^^Cs ) Pi y(F) = 0 . This argument showsvthat y(F) i s a closed subset i n R n and hence y : R — > R n i s a homeomorphism. Under conditions of the Lemma, an application of Theorem 5.5 w i l l enable us to conclude that A i s isomorphic to a point-determining, regular sub-ring of C(R n). This completes the proof. Now i f we impose a maximal condition, we obtain the following characterisation. - 40 -5.6.2 Theorem : Let A be a ring containing R . Then A = C(R n) i f and only i f A i s regular and there exist u^, u^, ...» such that ( i ) for r. E R, i = 1, 2 n, u^"^, u2~ r2> •*•» u n " r n are contained i n one unique M £ R^ ; ( i i ) i f a e A and M e such that M(a) 4 0, then there exist a E R, b £ A for which I (u, - M(u,)) 2 + a 2 = a 2 + b 2 ; 1=1 ( i i i ) A has no ring extension that i s regular and s a t i s f i e s conditions ( i ) and ( i i ) above for u^, u^, ..., . Proof : (Necessity) From 2.4, there i s a one-to-one correspondence between R n and R given by x —> M = {f e C(R n) : f(x) = 0 } . C ( R n ) 6 ' x Suppose f e f i R = fi { M : M E R }, then for each x e Rn, C(R n) C(R n) M e R and f e M implies f(x) = M(x) = 0 . Hence f = 0. Now the x c ( R n ) x complete regularity of R n w i l l t e s t i f y that C(R n) i s a regular r i n g . I f u^ (1 < i < n) i s taken to be the i - t h projection on R n , then for r ^ e R, i = 1, 2 n , l e t r = ( r ^ r 2> r ) £ R n then u ± - r ± E Mr = {f E C(R n) : f ( r ) = 0 } , i = 1, 2, ...,n and i s unique, since the correspondence r -—> M i s one-to-one. And i f f e C(R n), s E R n are such that f(s) / 0, then - 41 -I (u. - s.) + f > 0 on R n i = l 1 1 i n p a r t i c u l a r on S = { t e R n : | | t - s | | < _ l } . Let min / Y (x. - s . ) 2 + f ( x ) 2 V •> 0 xeS t A 1 1 J 2 1 and choose a = minlB, 1} . Then i = l Defining n I (u. - s.)" + f - a" > 0 on R" = { (x. - s . ) 2 + f ( x ) 2 - a 2 we see that g e C(R n) and ^ 2 2 2 2 y ( u . - s . ) + f = a + g where s. = M (u.) . ,L- x l 6 x s x i= l We have thus proved that conditions ( i ) and ( i i ) are s a t i s f i e d . Now l e t B be a regular ring containing C(R n) s a t i s f y i n g ( i ) and ( i i ) . By the proof of Eemma 5.6.1, there i s a homeomorphism u : wRB —> R n . We define a function cfi : B —> C(R n) by c|)(b) = b * o l i ~ 1 . —1 n Since y and b* are continuous, (j)(b) i s also continuous on R , moreover i f <f>(b) = <f>(c) then b* = b * ° y "'"oy = cj)(b)oy = <j)(c)oy = c * « y ^ o y = c* and since a -—> a* i s one-to-one (see proof of Theorem 5.5), b = c. To - 42 -show that <f> i s a monomorphism, for x c R , l e t y (x) = M e Rg, then <p(b 0 c)(x) = (b D c ) * o y _ 1 ( x ) = (b 0 c)*(M) = M(b fl c) = M(b) B.M(c) = b*(M) Q c*(M) = b*oy - 1(x) Q c*oy - 1(x) = (<p(b) D <Kc))(x) . Hence <p(b Q c) = <j>(b) 0 <j>(c) where II represents pointwise addition or mu l t i p l i c a t i o n , so cp i s indeed a monomorphism. Let if) = <J)|c(Rn) be the r e s t r i c t i o n of cp to C(R n) , then 0Jif)(x) = f * o y _ 1 ( x ) = f*(M ) = M v(f) = f ( x ) , x e Rn, f e C(R n), implying l/if = f for f e C(R n). Now l e t b e B , then <j>(b) = f for some f e C(R n) and cp(b) = f = ^ ( f ) = <p(f). Since cp i s a monomorphism b = f e C(R n). This shows that B = C(R n) as a r i n g . (Sufficiency) By Lemma 5.6.1, we know that A i s isomorphic to a point-determining, regular sub-ring of C(R n). By ( i i i ) , A = C(R n) , since i t has been proved above that C(R n) does s a t i s f y the stated conditions. 5.7 As proved i n Theorem 5.5, i f A i s a regular ring then a —> a* i s a ring isomorphism between A and A* = {a* : a e A} where a* : R^ —> R i s given by a*(M) = M(a). In what follows, R A, the r e a l i d e a l space of a regular r i n g A i s granted the weak topology induced by A* (which coincides with the Stone topology by Theorem 5.4.3). So A* i s a sub-ring of ^CR^), the ring of continuous functions on R A. Letting - 43 -M* = {a* : a e M} for M e C(R A)-(u*-r) = {g.(u*-r) : g e C(R A)} for u e A C(RA)-M* = {g-a* : g e C(R A), a* E M*} and noting the M* C C(R A), we f i r s t prove this lemma. 5.7.1 Lemma : I f A i s a regular ring containing elements u^, u2> u n such that (i) A has ring extensions A = A C A .. C . . . C A.. C A = C(R n) m m-i l o which are regular.; ( i i ) I f M^. e R A , k = 1, 2, m, then M k c Jx c ( R A k ) ( u i - r i } c c<i\K for some r,, r„..... r E R . 1 2 ' n ( i i i ) For each k, i f a £ A^ and e are such that M^ .(a) 4 0, then there exist a e R, b e A^ for which I (u. - \ ( u . ) ) 2 + a 2 = a 2 + b 2 . (iv) For each k (1 <_ k <_ m) , there exist l i n e a r mappings k * + 3 i : A£ —> A£ , i = 1, 2, n s a t i s f y i n g 3*(rf) = r9^(f) r e R, f e Aj 9^(fg) = f.9^(g) + g-9^(f). , f, g e A^ - 44 -3 i < V -0 i f i ^ j , 1< j < n ^ 1 i f i = j . Then A i s isomorphic to a regular sub-ring of C m(R n) Proof : Let e A^^ .' then for r ^ , r 2 , r n E R we have by ( i i ) , ( u i " r i ) = g' a* ( i = 2' n ) for some g £ C(R A ) and a* E . But a* E implies a E M, and k k 1c a*(Mfc) = M^a) =0 . Therefore (u* - r ± ) (M^ .) = 0, hence (u* - r ± ) E M£ or u i " r i e \ * ^ o w ^ with these properties i s unique, for i f there exists N E RA^ such that u^ - r ^ e N, i = 1, 2, n, and i f f N, then we can assume without loss of generality that there exists an' a E Mj^\ N, that i s M^ .(a) = 0, but N(a) 4 0. However since by ( i i ) n I i = l we have a* = _X g i ' ( u i " r i } ' g l ' 8 2 ' gn e C ( V n n N(a) = a*(N) = J g ±(N) • (uj-r.) (N) = £ g.(N) ^ ( u . - r ^ = 0 i = l i = l which i s a contradiction. Therefore u^ - r^, 1 = 1 , 2, n, belong to a unique E R^ . Now with ( i i i ) , Lemma 5.6.1 applies and we have for y n k = 1, 2, ..., m, R A - R where k y(M) = (M( U ; L), M(u 2) M(u n)) , ". M £ R A k - 45 -For each M e R A , M i s of the form M = M where y(M ) = r E R n . At thi s juncture to overcome impending symbolic d i f f i c u l t i e s , we s h a l l agree to id e n t i f y R A^ with R n , that i s , M^ e R A^ with r e R n and each f* e M* (f e M) can be i d e n t i f i e d with f e C(R n) where f*(M r) = M r(f) = f ( r ) . Upon doing so we s h a l l see that u* i s i n fact the continuous i - t h projection on R . We have by ( i i ) u^(y) - x^ = g(y)f*(y) for some g E C(R n), f e M . Since u*(x) - x ± = g(x)f*(x) = g(x)-0 =0, we have u*(x) = x^ as required. Consider now f e A^ and r e R . Since M (f - f ( r ) ) = 0, f - f ( r ) e Mr = {g e : g(r) = 0} and by ( i i ) , n (2) f - f ( r ) = I g -(u - r.) j = l 2 f f om'- which- we-seeSfchat lim 1' ' i - 1 ' i ' i + 1 ' " * ' n y n x.^-r. x. - r. i x i i lim g i ( r 1 , . . . > r 1 _ 1 , x 1 > r i + 1 , . . . , r n ) = g ± ( r ) x.+r. x x so (3f/8x^)(r) exists for a l l r e R n . Moreover from (2) 9 k(f) = I 8 k(g.).(u.-r.) + g. x . L, X & J 1 1 & x - j = l J J J and evaluating, t h i s at r = (r^> r n)» w e obtain O k f ) ( r ) = g.(r) = (3f/ax ±)(r) - 46 -so 3^ = 3f/3x. . This shows that a l l the 3^ are p a r t i a l , d i f f e r e n t i a t i o n I X 1 operators. Since for f e A* , 3^(f) e C(R n), we see that A* C C 1^ 1 1) 2 and from the fact that 3^(f) e A* for f e A* we conclude that A? C C 2(R n) . Inductively we obtain A* C C m(R n). F i n a l l y we have that I m A i s isomorphic to a sub-ring of C m(R n) v i a the isomorphism a —-> a* . 5.7.2 Theorem : A = C m(R n) i f and only i f A i s a regular r i n g containing elements u^, u2> e A such that conditions ( i ) , ( i i ) , ( i i i ) , (iv) of Lemma 5.7.1 are s a t i s f i e d and (v) A has no ring extension which i s regular and s a t i s f i e s the same four conditions. Proof : (Necessity) I t i s clear that C m(R n) i s a regular r i n g from Corollary 5.2.2 and Theorem 5.5. Taking u^ to be the i - t h projection, we s h a l l prove that a l l the conditions are s a t i s f i e d . ( i ) C m(R n) has ring extensions C k ( R n ) , k = 0, 1, 2, m-1 which are also regular by the same reason as that for C (R ). ( i i ) By Proposition 2.3.1, for each k (1 < k < m), e R A xs fixed at some point r e Rn, i . e . = {f e C k(R n) : f ( r ) = 0} . Now i f f E Mj^, then - 47 -f (x^ » » • • J X ^ ) f (x^ jX^ , . . . ,x^_^ >x„) — f (^ -i » 1' 2' 3 , - " ? ^ n - l ' r„,r l>*2'*3*"" * n-1' n r ) = f ( x 1 , x 2 , x 3 , . . . > x n _ 1 , x n ) - f O ^ x ^ , . . . ^ ^ ) + f ( r 1 , x 2 , x 3 , . . . , x n _ 1 , x n ) - f ( r 1 . r 2 , x 3 , . . . , x n _ 1 , x n ) + f ( r 1 , r 2 , x 3 , . . . , x n _ 1 , x n ) - f C r ^ r ^ , . . . ^ ^ ) + f C r ^ r ^ , . . . , ^ ^ ) - f ( r r r 2 , r 3 , . . . , , ^ , 1 ^ ) and f (x 1 , X „ , • . . , X ) , v £ : f e i ' , * 2 - . v " : r ± T i , x i ' ; ; T.; xn?, f r i ' ? 2 ' i = l , r i , x i + 1 - , . . . . , x n ) x . - r. ( x . - r . ) where x ± ^ r ± ( i = 1, 2, ..., n). Now since f i s d i f f e r e n t i a b l e on R n , we in f e r that 8^ (x-^ »x2»• • • » x n^ f (r]_'r2 r i - l , x i ' * *' ' Xn^ ~ £ ^ r l ' r 2 >' " ' r i , x i + l ' * * * ' xn^ x. - r. l l , x . * r. 3f 3x. l (r) x . = r. i x i s continuous and we have f = Y g.•(u. - r . ) . Hence . , l l i i= l n Mfc C I C(R n)(u. - r.) . i = l The other inclusion i s t r i v i a l . - 48 -( i i i ) This can be shown i n exactly the same way as i n the proof of Theorem 5.6.2. (iv) 9^ , (1 <_ k <^ m) i s just the i - t h p a r t i a l d i f f e r e n t i a t i o n operator so (iv) i s s a t i s f i e d . Now i f B i s a regular r i n g which contains C m(R n) and s a t i s f i e s conditions ( i ) to (iv) , then by Lemma 5.7.1, B* i s isomorphic to a sub-ring of C m ( R n ) , so each b* efiB* has.continuous p a r t i a l derivatives of a l l orders up to m . As done i n Theorem 5.6.2, we define a monomorphism 4> : B —> C m(R n) by $(b) = b*<>u ^ where u : R —> R n i s a homeomorphism. B Moreover i f = c(>|cm(Rn) i s the r e s t r i c t i o n of <j> to C m(R n) then (ipf ) ( x ) = f*o 1 J" 1(x) = f*(M ) = M (f) = f ( x ) X X implying tyf = f for f e C m(R n). Now l e t b e B then <|>(b) = f for some f e C m(R n), but <j)(b) = f = i|i(f) = <l>(f), therefore b = f since f / i s a monomornhism. This shows that maps B i d e n t i c a l l y onto C m ( R n ) , i . e . B = C m(R n) as a ring. (Sufficiency) Suppose A has a l l the conditions stated, by Lemma 5.7.1, A i s isomorphic to a regular sub-ring of C m(R n). Since C m(R n) s a t i s f i e s ( i ) to ( i v ) , condition (v) asserts that A = C m(R n). 5.8 The conditions imposed on A i n order to characterise C m(R n) oo w i l l turn out to be less formidable i n the case of C (R ). I t works out that d i f f e r e n t i a t i o n i n this case i s characterised algebraically by means of derivation. - 49 -5.8.1 D e f i n i t i o n : For an arbitrary ring A (containing R), a derivation on A i s a mapping 3 : A —> A such that (i ) 3(ra + sb) = r9(a) + s9(b) ( i i ) 9(ab) = a3(b) + b9(a) where a, b e A, r, s e R . As a consequence of this d e f i n i t i o n , 9(r) = 0 for each r e R . Typical examples are given by p a r t i a l d i f f e r e n t i a t i o n operators 3/3x^ on Cc°(Rn), i = 1, 2, . . . , n . The following lemma shows that under certain conditions the d i f f e r e n t i a l operators are completely determined by derivations. 5.8.2 Lemma : Let A be a sub-ring of C(R n) containing the projections u^, u2» u n , and the constant functions. Suppose n M_ = { f e A : f ( r ) = 0} = £ A(u - r.) , i = l r = ( r ^ , r 2 , r ) e R n and there exist derivation 3^ i = 1, 2, n on A such that 3.(u.) = 0 i f i ± j and 3.(u.) = 1. Then A C C°°(Rn) and 3.f = 3f/3x.), f erA . x x ' Proof : Let f e A and r e R n then f - f ( r ) e M , so there e x i s t g ^ e A , i = 1, 2, n for which (3) f - f ( r ) = I g .(u, - r.) . j= l J - 50 -We consider the i - t h p a r t i a l d i f f e r e n t i a l quotient f (r"L>r2» • • • »rx-l,xi'ri+l'" * " ' rn^ ~ x. - r. 1 1 g i ( r l ' r 2 r i - l , x i » r i + l , , , , » r n ) ' x i * r i Since g e C(R n), we see that the i - t h p a r t i a l derivative of f exists and (9f/9x i)(r) = g i ( r ) . However from (3) n ).f = T O.g.. (u. - r.)} + g. i 1 J 3 3 B i and evaluating at r = ( r ^ , rn^ ' w e ° b t a i n (3^f)(r) = g^(r). So 9^f = 9f/9x^ . By d e f i n i t i o n of derivations a l l p a r t i a l derivatives of f 0 0 n 0 0 n of a l l orders e x i s t , implying f e C (R ). Hence A C C (R ) . This completes the proof. I t i s not d i f f i c u l t now to give an algebraic characterisation of 00 -ft C (R ), rel y i n g heavily on the methods of proof employed i n Theorems 5.6.2 and 5.7.2. 5.8.3 Theorem : A r i n g A i s isomorphic to C (R n) i f and only i f A i s regular and there exist u^, u^, e A such that n ( i ) M e R. implies M = \ A(u. - r.) for some x=l $ ' * * * > £ R • ( i i ) I f M e R^, a e A and M(a) # 0, then there exist a e R, b e A such that - 51 -I (u. - M(u.)) 2 + a 2 = a2 + b 2 . i= l ( I i i ) There exist derivations 3^, i = l , 2, ...,n, on A such that 3.(u.) = 0 i f i 4 j and 3.(u.) = 1 . i J J i v i ' (iv) A has no ring extension which i s regular and s a t i s f i e s conditions ( i ) to ( i i i ) above. 0 0 n Proof : (Necessity) I t i s evident that C (R ) i s a regular ri n g (by Corollary 5.2.2 and Theorem 5.5). Taking u^ to be the i - t h projection and 3^ to be 3/3x^ , we see that ( i i i ) i s c l e a r l y s a t i s f i e d . A repe t i t i o n of the proof i n Theorem 5.6.2 w i l l prove ( i i ) . • , co n ^ For M e R „ , M = M = { f e C (R ) : f ( r ) = 0} , so C°°(Rn) r n n I C°°(Rn) (u - r.) £ M . I f f E M , then f = \ g . (u - r.) for i = l i = l some g^, g 2 > g n e C(R n) as i n the proof of Theorem 5.7.2. Now since CO co f e C (R ) i t can be shown that each g^ e C (R ), so n f e l C (R )(u. - r.) and (i) ois v e r i f i e d . i = l 1 1 CO Next, i f B i s a regular ri n g containing C (R ) and s a t i s f i e s ( i ) to ( i i i ) , then by a re p e t i t i o n of an argument i n Theorem 5.6.2, there oo j"l exists a monomorphism <J) : B —> C (R ) and one can s i m i l a r l y show that <j>f = f for every f e C°°(Rn). Therefore B = C (R n) as ;a"ring. (Sufficiency) ( i ) and ( i i ) and Lemma 5.6.1 show that A i s i s o -morphic to a point-determining sub-ring of C(R n). Lemma 5.8.2 ensures that CO JI A i s isomorphic to a sub-ring of C (R ). Condition (iv) says that A = C°°(Rn) . - 52 -Bibliography [1] Frank W. Anderson and Robert L. B l a i r , Characterisations of the algebra of a l l real-valued continuous functions on a completely regular space, I l l i n o i s J. Math. 3 (1959), pp 121-133. [2] B. Banaschewski, An algebraic characterisation of C°°(Rn), B u l l . Acad. Polon. S c i . Ser. S c i . Math. Astro., Phys., 16 (1968), pp 169-174. [3] H. Cartan, Elementary Theory of Analytic Functions of one or several complex variables, Addison-Wesley Publishing Co., Inc. 1963. [4] A.G. Fadell and K.D. M a g i l l , J r . , Automorphisms of semi-groups of polynomials, Compositio Mathematica, Vol. 21, Fasc. 3, 1969, pp 233-239. [5] L. Gillman and M. Jerison, Rings of continuous functions, Van Nostrand, Princeton, N.Y. 1960. [6] Kenneth D. M a g i l l , J r . , Automorphisms of the semi-group of a l l d i f f e r e n t i a b l e functions, Glasgow Math. J. 8 (1967), pp 63-66. [7] I.P. Natanson, Theory of functions of a real variable, Vol. I. Frederick Ungar Publishing Co. N.Y. 1955. [8] L.E. P u r s e l l , An algebraic characterisation of fixed ideals i n certain function rings, P a c i f i c J. Math. 5 (1955), pp 963-969. [9] L.E. P u r s e l l , The ring of r e a l polynomials, Amer. Math. Monthly (1969) 76, pp 509-514. [10] Boris M. Schein, Automorphisms of polynomial semi-groups, Semigroup Forum, Vol. 1 (1970), pp2279-281. [11] L.P. Su, Ph.D. thesis, The University of B r i t i s h Columbia, 1966.
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Sub-rings of C(Rⁿ) Gan, Cheong Kuoon 1974
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Title | Sub-rings of C(Rⁿ) |
Creator |
Gan, Cheong Kuoon |
Date Issued | 1974 |
Description | The content of this thesis contains a study of the rings C(Rⁿ), L[sub c](Rⁿ), C[sup m](Rⁿ), C[sup ∞](Rⁿ), А(Rⁿ) and P(Rⁿ). We obtain the result that no two of the rings above can be isomorphic : in fact we prove the following : if Φ : A → B is a ring homomorphism where A, B are any two of the rings and A ⊂ B, then Φ(f) = f(p) for some p εRⁿ. We also characterise C(Rⁿ), C[sup m](Rⁿ), and C[sup ∞](Rⁿ) as rings. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-01-29 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080129 |
URI | http://hdl.handle.net/2429/19392 |
Degree |
Master of Science - MSc |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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