UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Sub-rings of C(Rⁿ) Gan, Cheong Kuoon 1974

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1975_A6_7 G35.pdf [ 2.77MB ]
Metadata
JSON: 831-1.0080129.json
JSON-LD: 831-1.0080129-ld.json
RDF/XML (Pretty): 831-1.0080129-rdf.xml
RDF/JSON: 831-1.0080129-rdf.json
Turtle: 831-1.0080129-turtle.txt
N-Triples: 831-1.0080129-rdf-ntriples.txt
Original Record: 831-1.0080129-source.json
Full Text
831-1.0080129-fulltext.txt
Citation
831-1.0080129.ris

Full Text

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. 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0080129/manifest

Comment

Related Items