FUNCTIONS MAXIMAL ALGEBRAS OF CONTINUOUS

MAXIMAL ALGEBRAS OF CONTINUOUS

B Y

K. H O F F M A N A N D I. M. S I N G E R ( 1 ) Massachusetts Institute of Technology

FUNCTIONS

1. Introduction

L e t X be a c o m p a c t H a u s d o r f f space a n d C (X) t h e a l g e b r a of all c o n t i n u o u s c o m p l e x - v a l u e d f u n c t i o n s on X . L e t A be a u n i f o r m l y closed c o m p l e x l i n e a r s u b a l g e b r a of C(X).

Our i n t e r e s t centers a b o u t such a l g e b r a s A w h i c h are m a x i m a l a m o n g all p r o p e r closed s u b M g e b r a s of C(X). I n t h i s p a p e r we g a t h e r t o g e t h e r m o s t of t h e k n o w n facts concerning m a x i m a l algebras, give some n e w results, a n d s o m e n e w p r o o f s of k n o w n t h e o r e m s .

A m a j o r m o t i v a t i o n for t h e s t u d y of m a x i m a l a l g e b r a s s t e m s f r o m a n a t t e m p t t o generalize t h e S t o n e - W e i e r s t r a s s a p p r o x i m a t i o n t h e o r e m t o n o n - s e l f - a d j o i n t algebras. T h i s t h e o r e m s t a t e s t h a t if A is a s e l f - a d j o i n t closed s u b a l g e b r a of C(X) (lEA i m p l i e s lEA), a n d if A s e p a r a t e s p o i n t s a n d c o n t a i n s t h e c o n s t a n t f u n c t i o n 1, t h e n A = C(X). See [11;

p. 8] for a proof. This r e s u l t can be r e s t a t e d as follows: (i) e v e r y p r o p e r s e l f - a d j o i n t closed a l g e b r a A is c o n t a i n e d in a s e l f - a d j o i n t m a x i m a l a l g e b r a a n d (ii) t h e s e l f - a d j o i n t m a x i m M algebras, B, are of t w o kinds; e i t h e r B = [/EC(X), /(Xo)=0] for a f i x e d xoEX, or B = [ / E C ( X ) ; /(xl) = / ( x 2 ) ] for f i x e d xl,x 2 EX. T h e c o n d i t i o n , A c o n t a i n s t h e f u n c t i o n 1, s a y s t h a t A is n o t in a m a x i m a l a l g e b r a of t h e first k i n d . T h e c o n d i t i o n , A s e p a r a t e s p o i n t s , s a y s t h a t A is n o t in a m a x i m a l a l g e b r a of t h e s e c o n d k i n d . T h u s A is n o t c o n t a i n e d in a n y s e l f - a d j o i n t m a x i m a l a l g e b r a a n d c o n s e q u e n t l y , f r o m (i), A is n o t a p r o p e r s u b a l g e b r a , i.e., A = C(X). A r e f i n e m e n t of t h e S t o n e - W e i e r s t r a s s t h e o r e m classifies all s e l f - a d j o i n t closed s u b M g e b r a s of C (X) a n d s a y s t h a t such a n a l g e b r a A is t h e a l g e b r a of all c o n t i n u o u s f u n c t i o n s on a n i d e n t i f i c a t i o n space of X , w i t h t h e c o m m o n zeros of t h e f u n c t i o n s in A d e l e t e d . This r e s u l t can be r e i n t e r p r e t e d as s a y i n g t h a t A is t h e i n t e r s e c t i o n of t h e self- a d j o i n t m a x i m M a l g e b r a s w h i c h c o n t a i n it.

L e t us d r o p t h e s e l f - a d j o i n t n e s s c o n d i t i o n on A . One m i g h t n o w h o p e t h a t t h e w a y t o generalize t h e S t o n e - W e i e r s t r a s s t h e o r e m w o u l d be t o show t h a t (i) holds (with self- (l) This research was supported in part by the United States Air Force under Contract Nos. AF 18 (603)-91 and A F 49 (638)-42, monitored by the Air Force Office of Scientific Research, Air Rese- arch and Development Command.

218 K . I t O t r F M A N A N D I . M . S I N G E R

adjointness deleted) and then to classify all maximal subalgebras of C (X). (To avoid trivia- lities we now make the assumption that all subalgebras under consideration separate points, contain 1~ and are closed). I t turns out, however, t h a t (i) fails; in section 7 we exhibit a proper algebra A not contained in a n y maximal algebra. The example is easy to describe, but the proof t h a t it is not contained in a maximal algebra depends on several results of earlier sections. Even if A is contained in a maximal algebra, it is not necessarily the intersection of the maximal algebras containing it. Specific examples are given in section 6.

Despite these negative results, the study of maximal algebras does give approximation theorems. I n particular, if A is a maximal subalgebra of C (X), then of course the algebra generated b y A and a n y / E C ( X ) - A is all of C(X). For example, [16] the fact t h a t the algebra of continuous functions on the circle which are boundary values of analytic func- tions on the disc is maxifilal, implies t h a t every continuous function on the circle can be approximated b y polynomials in z a n d / , where ] is not the boundary value of an analytic function on the disc. This is a generalization of Fejer's theorem (the case ] =5). For some special spaces X, one knows enough about maximal algebras so that if A lies in a restricted class of algebras (just as one restricts ones attention to self-adjoint algebras in the Stone- Weierstrass theorem), then the possible proper subalgebras B containing A can be classified.

If A contains functions not in these algebras B, A must be C (X). I n section 6, this situation is analyzed when X i s the circle and A contains a separating subalgebra of analytic functions.

Wermer's results [18; 19; 20] give enough information about maximal algebras to give a strong approximation theorem.

The study of maximal algebras has one natural reduction which we now discuss.

Suppose A is a maximal subalgebra of C(X) and suppose S is a closed subset of X. Let A s denote the closure of A restricted to S, and let A 0 = []EC(X); [] sEAs]. Then A 0 is closed and A c A o c C(X). Since A is maximal, either A 0 = C(X) so t h a t As = C(S) or else A = A o so that A is actually a maximal algebra on S extended continuously in all possible ways to X. Among the closed sets S such that A s + C ( S ) there exists a unique minimal one E =

N S which we call the essential set for A [3]. Thus A consists of a maximal algebra of As4=C(S)

C(E) extended in all possible ways to X in a continuous fashion. Furthermore, if S is a proper closed subset of E, then As = C(S). If E = X, then A is said to be an essential maxi- mal subalgebra of C(X). The study of maximal algebras of X is thus reduced to the study of essential maximal algebras of X and its closed subsets.

I n section 2, we list the known examples of essential maximal algebras. Some new ones are exhibited in section 4. One observes that these examples all stem from algebras of analytic functions. I n [10], Helson and Quigley show t h a t essential maximal algebras display a number of properties enjoyed b y analytic functions. To our mind, the reason for

M A X I M A L A L G E B R A S O F C O N T I N U O U S F U N C T I O N S 219 this is that an essential maximal algebra A is pervasive, t h a t is, As = C (S) for any proper closed subset S of X. I n sections 3 and 4, pervasive algebras and their properties are ana- lyzed. These results together with an idea of Rudin [14] show that a n y pervasive algebra on a disconnected space is contained in a maximal algebra (Section 4). This leads to some new essential maximal algebras on the circle and an example of an essential maximal algebra on the unit interval.

Section 5 contains a discussion of the representation of complex homomorphisms of an algebra b y positive measures on the ~ilov boundary, emphasizing the usefulness of such representations in studying maximal algebras. This measure representation is playing an important role in the study of function algebras. I t seems clear t h a t it will play an increasingly important role.

2. Examples

We shall list the examples of essential maximal algebras known to us.

1. (Wermer [16]). Let X be the unit circle in the complex plane and let A be the algebra of continuous functions on X which can be analytically continued to the interior of the unit disc. That is, ] is in A if and only if

2n

fe~e/(e~~ n = 1 , 2 , 3 . . . .

0

2. (Wermer [21]). If F is a Riemann surface and X is an analytic curve on F which bounds a compact subset K of F, let A be the algebra of continuous functions on X which can be analytically continued to K - X.

3. (Bishop [6]). Let X be the topological boundary of any simply connected plane domain and let A be the algebra of functions on X which are uniform limits of polynomials:

4. (Hoffman and Singer [8]). Let X be a compact abelian group whose character group X is a subgroup of the additive group of real numbers. Let A be the algebra of all continuous functions on X whose Fourier transforms vanish on the negative half of the group 2~.

5. Rudin [14], has proved the existence of essential maximal subalgebras of C(X) where X is a certain totally disconnected set in the complex plane. This is described in section 4.

As mentioned in the introduction, we shall add to this list in section 4.

3. The essential set

Suppose t h a t A is a subalgebra of C (X), which we remind tile reader means A se- parates the points of X, contains the constant functions, and is closed. We consider those

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220 K . H O F F M A N A N D I . l~. S I N G E S

closed subsets K of X such t h a t A contains every continuous function on X which van- ishes on K. Among such sets K there is a unique minimal one E, which we call the essential set for A (relative to X). The algebra AE, obtained by restricting A to the set E, is a closed subalgebra of C(E) and A consists of the algebra of all functions which are continuous extensions to X of functions in A E. The minimality of E is characterized by saying that the algebra A E contains no non-zero ideal of C(E). If E = X, we say t h a t E is an essential subalgebra of C (X). The terminology here is due to Bear [3].

The study of function algebras A is "reduced" to the study of essential algebras, and the purely topological problem of describing closed subsets of X. We should point out t h a t when A is a maximal subalgebra of C(X) this reduction agrees with that carried out in section 1; t h a t is, a maximal algebra is essential if and only if it is pervasive (see in- troduction).

I n [10], Helson and Quigley proved t h a t every essential maximal algebra is anti- symmetric, i.e., contains no non-constant real-valued functions, and is analytic, i.e., any function in the algebra which vanishes on a non-empty open subset of X is identically zero. They were motivated of course b y an interest in proving t h a t a n y maximal (essential) algebra has m a n y properties in common with the algebra of analytic functions on the unit circle (example 1, section 2). They did not specifically mention the pervasive property which such algebras share with the analytic functions. W h a t we should like to point out in this section is t h a t the pervasive property seems to be the fundamental one. B y this we mean t h a t a n y proper pervasive subalgebra of C(X) is analytic and antisymmetric.

THEOREM 3.1. A proper pervasive subalgebra o] C ( X ) is analytic.

Pro@ Let / be a function in A which vanishes on a non-empty open set U in X.

Choose a non-empty open set V such t h a t V ~ U. The assumption t h a t A is pervasive tells us that if g EC(X) then there is a sequence [/n] of functions in A such t h a t / n converges to g uniformly on the complement of V. Then the sequence []]n] converges uniformly to /g on all of X. T h u s / g is in A for each g, o r / . C ( X ) is contained in A. So A contains the closed ideal in C(X) generated by ], i.e., A contains every continuous function on X which vanishes on the null set K of ]. This means t h a t when we restrict A to K we get a closed subalgebra of C (K). Clearly then / must vanish on all of X; for, if K were a proper closed subset of X the restriction of A to K would be at once dense in C (K) and closed and A would contain all of C (X).

T ~ o ~ 3.2. Let A be a closed subalgebra o / C ( X ) . (i) I / A is analytic, A is an integral domain.

(if) I / A is an integral domain, A is antisymmetric.

(iii) I / A is antisymmetric, A is an essential subalgebra o / C ( X ) .

M A X I M A L A L G E B R A S O F C O N T I N U O U S F U N C T I O N S 221 Proo/. (i) is obvious. (ii) Let R be the self-adjoint part of A, i.e., the set of all functions / in A whose complex conjugate is also in A. Then R is a closed subalgebra of A, and since R is self-adjoint there is a compact Hausdorff space Y such t h a t R is isometrically isomorphic to C(Y). Since A is an integral domain, so is C ( Y ) . Clearly then Y consists of a single point and R contains only the constant functions.

(iii) I t is clear t h a t if the essential set for A is a proper closed subset of X then A contains a non-constant real-valued function.

An immediate corollary of these two theorems is t h a t an essential maximal subalgebra of C (X), being a proper pervasive subalgebra, is analytic, hence an integral domain; hence antisymmetric. I n fact we see t h a t for a maximal subalgebra of C (X) the properties of being essential, pervasive, analytic, an integral domain, antisymmetric, are all equivalent.

4. Pervasive algebras

I n section 3 we saw t h a t some of the known special properties of an essential max- imal algebra are possessed b y every proper pervasive subalgebra of C(X). There are pervasive algebras which are not maximal, a simple example being the uniformly closed algebra on the unit circle generated b y 1, z 2, z 3, z 4, . . . . Having observed this, we felt it interesting to inquire whether it is true t h a t every proper pervasive subalgebra of C (X) is contained in a maximal subalgebra of C(X). Motivated b y a result of Rudin [14], we did prove the somewhat strange fact that, when the underlying space X is not connected, this is true. This then is a mild existence theorem for maximal algebras. I t can be used to construct a new class of essential maximal algebras, and in particular to construct a new essential maximal algebra on the unit circle.

Let us first observe the following.

LEM~A 4.1. Let A be a subalgebra o[ C(X) such that ]or each [ E A the real part o / ] has connected range. I / X is not connected, then A is contained in a subalgebra o/ C (X) which is maximal with this property.

Proo/. The proof is essentially that of Rudin []4; theorem 2]. Let F be the class of all proper closed subalgebras B of C (X) which contain A and are such t h a t for each /E B the real part of / maps X onto a connected set. If IBm] is a linearly ordered subset of F, the closure of the union of the B~ contains only functions whose real part has connected range, and this closure is a proper subalgebra of C (X) since X is not connected. B y Zorn's lemma, F contains a maximal element.

2 2 2 K . H O F F M A N A N D I . M. S I N G E R

LEMMA 4.2. If A is an antisymmetric subalgebra o / C (X), then/or each /E A the real part o/ / has connected range.

Proo/. Again see R u d i n [14]. L e t / E A a n d suppose t h e real p a r t of / does n o t h a v e connected range. T h e n the range of / is t h e union of two n o n - e m p t y c o m p a c t sets K 0 a n d K 1 which are s e p a r a t e d b y a vertical line. W e can find a sequence of polynomials p~ (in one complex variable) which converges u n i f o r m l y to 0 on K 0 a n d to 1 on K 1. T h e n p~ (/) is a sequence of elements in A which converges to a non-trivial i d e m p o t e n t function in A, i.e., a n o n - c o n s t a n t real-valued function in A.

I t is clear f r o m t h e above a r g u m e n t t h a t t h e s t a t e m e n t t h a t t h e real p a r t of each / in A m a p s X o n t o a connected set is equivalent to the s t a t e m e n t t h a t A contains no non- trivial i d e m p o t e n t functions. This in t u r n is equivalent (by a t h e o r e m of ~ilov) to t h e maxi- mal ideal space of A being connected.

THEOREM 4.3. Let A be a proper pervasive subalgebra o / C ( X ) , and suppose that X is not connected. Then A is .contained in an essential maximal subalgebra o / C (X).

Proo/. Since A is proper a n d pervasive, A is antisymmetric, so l e m m a s 4.1 a n d 4.2 tell us t h a t A is c o n t a i n e d in a subalgebra B which is m a x i m a l with t h e " c o n n e c t e d r a n g e "

p r o p e r t y . B u t t h e n B is a m a x i m a l subalgebra of C(X); for a n y proper subalgebra B 1 which contains B contains A a n d is therefore pervasive. So B~ is a n t i s y m m e t r i c , hence has the " c o n n e c t e d r a n g e " p r o p e r t y a n d m u s t be equal to B.

W e shall n o w combine t h e o r e m 4.3 with some work of W e r m e r [17] to prove t h e existence of n e w essential m a x i m a l algebras.

L e t X be a c o m p a c t set in t h e complex plane with these properties:

(i) X has no interior.

(ii) X does n o t separate the plane.

(iii) X is n o t connected.

(iv) X has positive Lebesque measure at each of its points, i.e., if x E X t h e n for a n y n e i g h b o r h o o d U of x t h e set U N X has positive plane measure.

L e t A x be t h e algebra of all continuous functions on t h e R i e m a n n sphere S which are analytic on S - X. T h e functions in A x separate the points of S [17].

E a c h function in A x assumes its m a x i m u m m o d u l u s on the set X, so t h a t we can identify A x with a proper closed subalgebra of C (X). These properties of A x require o n l y t h a t X h a v e no interior a n d positive measure.

We n o w observe t h a t properties (ii) a n d (iv) i m p l y t h a t A x is a pervasive subalgebra of C(X). L e t K be a proper closed subset of X. Choose a point x 0 E X - K. F o r simplicity let us assume t h a t x 0 = 0. Choose ~ > 0 such t h a t t h e disc [ [ z I ~< ~] does n o t intersect t h e set

M A X I M A L A L G E B R A S O F C O N T I N U O U S F U N C T I O N S 223 K. When 1In < (~, let An be the intersection of X and the open disc I z I < ! / n . B y condi- tion (iv), the measure of An, ]A~ I, is positive. Define

On (z)- 1 f ( dxdy l• x~-V~y: ~"

A n

The functions dp~ are (can be extended to) functions in A x. A routine verification shows t h a t On (z) converges to 1/z uniformly on the compact set K. Since X does not separate the plane (and has no interior) a theorem of Mergelyan [12] tells us t h a t polynomials in 1/z are dense in the continuous functions on K. Thus we see t h a t the restriction of A x to K is dense in C (K).

Using theorem 4.3 we then have

T~EO~EM 4.4. I f X is a compact set in the plane which satisfies conditions (i)-(iv) above, then the algebra A x is contained in an essential maximal subalgebra o / C ( X ) .

When X is totally disconnected, this result was obtained b y Rudin [14].

The following special case of theorem 4.4 is of particular interest. Suppose the set X consists of two disjoint homeomorphic images of the unit interval. (These two arcs can be so embedded as to satisfy condition (iv)). The algebra A x is then included in an essential maximal subalgebra B of C (X), where X consists of two disjoint copies of the unit interval.

F r o m B we wish to obtain an essential maximal subMgebra on the unit circle, by taking the subalgebra of functions which identify the respective ends of the two intervals. If we identify only one pair of endpoints we obtain an essential maximal Mgebra on the unit interval. We shall need the following lemma.

L ~ M M A 4.5. Let B a maximal subalgebra of C ( X ) and let x and y be two points in X . Let B o be the subalgebra o] B o/]unctions ] ]or which /(x) = ](y), and let Y be the compact space obtained from X be identi]ying x and y. Then B o is a maximal subalgebra o] C ( Y ) . Proo]. W h a t we must prove is this. If g is a continuous function on X such t h a t g (x) = g (y) and g ~ B 0 then the closed subalgebra of C (X) generated b y B 0 and g contains every 9 continuous function ] for which ](x) = ] (y). I t clearly will suffice to consider the ease in

which g (x) = g (y) = 0.

Since g (x) = g (y) and g ~ B0, g ~ B. Thus, the linear algebra [B, g] generated b y B and g is dense in C (X). This linear algebra consists of ull functions of the form

/o + /:g + " " + /.g"

224 K . H O F F M A N A N D I . M . S I I ~ G E R

w h e r e / 0 . . . /~ a r e in B. L e t I be t h e set of all f u n c t i o n s in [B, g] such t h a t / k (x) = / k (Y) = 0, k = 0, 1 . . . n. T h e n I is a n i d e a l in [B, g], so t h e closure of I is a closed i d e a l in t h e closure of [B, g] (which is C ( X ) ) . T h e i d e a l I c o n t a i n s e v e r y f u n c t i o n in B w h i c h is 0 a t b o t h x a n d y, a n d since B s e p a r a t e s p o i n t s on X t h e set of p o i n t s on w h i c h e v e r y f u n c t i o n in I v a n i s h e s consists of t h e t w o p o i n t s x a n d y. T h u s t h e closure of I m u s t c o n t a i n e v e r y con- t i n u o u s f u n c t i o n on X w h i c h v a n i s h e s a t b o t h x a n d y. B u t I is c o n t a i n e d in t h e a l g e b r a g e n e r a t e d b y B 0 a n d g. T h u s t h e closed a l g e b r a g e n e r a t e d b y B 0 a n d g c o n t a i n s e v e r y c o n t i n u o u s f u n c t i o n v a n i s h i n g a t x a n d y. Since B 0 c o n t a i n s t h e c o n s t a n t s , B 0 a n d ff g e n e r a t e all f u n c t i o n s w h i c h i d e n t i f y x a n d y.

N o w l e t us r e t u r n t o t h e a l g e b r a A x a b o v e w h e n X = 11 U 12 w h e r e 11 a n d I~ are d i s j o i n t h o m e o m o r p h i c i m a g e s of t h e u n i t i n t e r v a l . L e t B be a n e s s e n t i a l m a x i m a l s u b a l g e b r a of C (X), c o n t a i n i n g A z. I f x~ a n d u~ are t h e e n d p o i n t s of I~, i = l , 2, we consider t h e s u b a l g e b r a B 0 of B of f u n c t i o n s ] such t h a t ](xl) = / ( x 2 ) , / ( u l ) = / ( u 2 ) . B y t h e a b o v e l e m m a , B 0 is a n e s s e n t i a l m a x i m a l s u b a l g e b r a of a h o m e o m o r p h i c i m a g e of t h e u n i t circle.

W e wish t o s h o w t h a t t h e a l g e b r a B o is n o t i s o m o r p h i c t o a n y of t h e e x a m p l e s c i t e d in section 2. To do t h i s i t will suffice t o show t h e following. I f A~ is t h e s u b a l g e b r a of A x w h i c h identifies x 1 w i t h x 2 a n d u 1 w i t h us, t h e n A~ c a n n o t be i s o m o r p h i c t o a closed s u b a l g e b r a of t h e a l g e b r a of b o u n d a r y v a l u e s of a n a l y t i c f u n c t i o n s on a R i e m a n n surface (with b o u n d a r y ) .

W e shall c o n t e n t o u r s e l v e s w i t h a s k e t c h of t h i s proof. L e t F be an a n a l y t i c circle on a R i e m a n n surface w h i c h b o u n d s a c o m p a c t piece K of t h e surface. L e t A be t h e a l g e b r a of all c o n t i n u o u s f u n c t i o n s o n K w h i c h a r e a n a l y t i c on K - F. S u p p o s e t h a t t h e a l g e b r a A~ is (isomorphic to) a s u b a l g e b r a of A . E a c h c o m p l e x h o m o m o r p h i s m h of t h e a l g e b r a A gives rise t o a c o m p l e x h o m o m o r p h i s m h 0 of A ~ b y r e s t r i c t i o n . T h e m a p p i n g 7~ : h - ~ h o is a c o n t i n u o u s m a p p i n g of K i n t o t h e s p a c e S o of c o m p l e x h o m o m o r p h i s m s of t h e a l g e b r a A ~ T h e s p a c e S o can be i d e n t i f i e d as t h e R i e m a n n s p h e r e S w i t h t h e p a i r s of p o i n t s (xl, x2) a n d (ul, u2) i d e n t i f i e d . T h i s follows f r o m a r e s u l t of A r e n s [1] t h a t t h e space of c o m p l e x h o m o m o r p h i s m s of A x is S. T h e m a p p i n g ~, w h e n r e s t r i c t e d to F , gives a h o m e o m o r p h i s m of F o n t o t h e circle on S o o b t a i n e d b y i d e n t i f y i n g t h e ends of t h e i n t e r v a l s 11 a n d I2.

I t is n o w r e ] a t i v e l y e a s y to a r g u e t h a t such a c o n t i n u o u s m a p p i n g ~ c a n n o t exist. F o r , let p be t h e p o i n t on F such t h a t q = z p is t h e p o i n t of S o w h i c h arises f r o m i d e n t i f y i n g x 1 a n d x~. A s u f f i c i e n t l y s m a l l n e i g h b o r h o o d F of t h e p o i n t q is h o m e o m o r p h i c t o t w o discs w i t h t h e i r c e n t e r s i d e n t i f i e d . T h u s V - [g] is h o m e o m o r p h i e t o t w o d i s j o i n t copies of t h e p u n c t u r e d o p e n disc. S u p p o s e we select a n e i g h b o r h o o d U of t h e p o i n t p w h i c h is c o n n e c t e d a n d for which g ( U ) c V. T h e n U - [p] is still c o n n e c t e d a n d m u s t b e m a p p e d b y ~ i n t o one of t h e t w o p u n c t u r e d discs c o m p r i s i n g V - [q]. B u t t h i s is i m p o s s i b l e , since t h e p a r t

M A X I M A L A L G E B R A S OF C O N T I N U O U S F U S T C T I O ~ S 225 of F which lies in the neighborhood U is mapped by 7e partly into one of the punctured discs and partly into the other.

We should perhaps comment t h a t in a rough sense the algebra A~c is not an algebra of analytic functions on a l~iemann surface with boundary, because the circle we have on the pinched sphere S O does not bound. We should also note t h a t it m a y well be t h a t Ax is alreudy a maximal subalgebra of C (X). One can prove t h a t if B is any proper subalgebra of C (X) which contains Ax, then every complex homomorphism of A z extends to a complex homomorphism of B. B u t whether Ax is actually maximal remains unknown.

5. Measures and the Silov boundary

Our discussion thus far of maximal subalgebras of U (X) has not involved a n y detailed information about the relation of the space X to the algebra A. Further discussion requires the introduction of the maximal ideal space and ~ilov boundary for A.

Let A be a closed subalgebra of C (X), as usual containing the constants and separating points. The space o/ maximal ideals (or complex homomorphisms) of A is the set S(A) of all non-zero complex linear functionals on A which are multiplicative. Each such multi- plicative functional is automatically of norm 1, and we give to S(A) the weak topology which it inherits as a subset of the unit sphere in the conjugate space of A. The space S(A) is the largest compact Hausdorff space on which the algebra A can be realized as a separating algebra of continuous functions. I n S (A) there is ~ unique minimal closed subset F(A) on which every function in A assumes its maximum modulus. We call F(A) the

~ilov boundary for A [7].

For each point x EX we have a complex homomorphism hx of A defined by

hx (/) =/(x).

Since A separates points on X, the mapping x--->h z is a homeomorphism of X onto a closed subset of S(A). The image of X under this mapping includes F (A) because each function in A certainly assumes its maximum on X.

Since each function in A assumes its maximum on F(A) we m a y (if we wish) regard A as a subalgebra of C(F). The minimality of F shows that F is the smallest compact Hausdorff space on which A can be realized as a closed separating algebra of continuous functions.

If p ES(A), there is a (not necessarily unique) positive Balre measure # , on F such that

/(p) =f /dl~,

F

226 K . H O F F M A I ~ A N D I . M . S I N G E R

for every / in A [see 2]. We say that/~p "represents" p. This representation results from the fact t h a t a n y continuous linear functional on C(F) which has n o r m 1 and is 1 at the identity is positive. We are particularly interested here in the role of these measures in the study of maximal algebras.

Let us first make some simple observations. We have been discussing special types of subalgebras of C(X): antisymmetric, pervasive, etc. The representation of homomorphisms by positive measures makes it clear t h a t A is antisymmetric if and only if A is an anti- symmetric subalgebra of C(F). I n other words, a n t i s y m m e t r y is independent of which space X we represent A on, because X always contains F. Likewise the p r o p e r t y of being an essential subalgebra of C(X) is independent of X. However, certain properties we have discussed do depend upon the underlying space X. For examplel the exact description of the essential set depends heavily on X. Also the p r o p e r t y of being pervasive depends on X. To rule out discrepancies, let us make the following conventions. The essential set for A will be the essential set for A relative to F. We shall call A pervasive if A is a pervasive subalgebra of C(F). (It follows t h a t if A is a pervasive subalgebra of C(X) t h e n X = F ; b u t A m a y be pervasive on F and not on X.)

We begin our consideration of measures with two facts which were proved for essential maximal algebras b y Bear [4].

THEOREM 5.1. Let A be a pervasive subalgebra o/ C(F), let p E S ( A ) - F and let #v be any positive measure on F which represents p. Then the closed support o//~v is all o / F .

Proo/. L e t K be the closed support of ~up. Suppose K is a proper closed subset of F.

Since

l(p) = f ld~v,

K

[/(p) l <s p ^[/[,

and since A is pervasive the measure/~p defines a multiplicative linear functional on C (K).

T h u s / % m u s t be a point mass, which is absurd since p ~F.

COROLLARY: Let A be a pervasive subalgebra o/ C(F) and let / be a /unction in A which has norm I. I / t h e r e is a point p E S ( A ) - F such that I/(P)[ = 1, then / is constant.

Proof. Choose a measure #p representing p. Since #~ has mass 1, I/] ~ 1, a n d

l=l/(P)l=lf/d l,

F

it is clear t h a t / (x) = / (p) for all x in F.

Of course Theorem 5.1 and its corollary hold for essential m a x i m a l algebras. We have

M A X I M A L A L G E B R A S O F C O N T I N U O U S F U N C T I O N S 227 s t a t e d t h e m for pervasive algebras to emphasize once again t h a t t h e pervasive p r o p e r t y of m a x i m a l algebras is t h e f u n d a m e n t a l one (among t h e k n o w n special properties).

W e shall later need t h e following.

T ~ O R ~ M 5.2. Let / be a/unction in A which has norm 1, and let K be the subset o/

S ( A ) on which ] = 1 (assume K is not empty). Let AK be the algebra obtained by restricting A to the set K. Then A K is closed and

(i) S (A~) = K.

(ii) 1 ~ (AK) c p N K.

(iii) I / p EK, then any measure #~ on F which which represents p (as a homomorphism o/

A) is supported on K N F.

Pro@ A p o r t i o n of this t h e o r e m was p r o v e d b y Bear [5]. I t is no loss of generality to assume t h a t K = [1II = 1], for we m a y replace / b y 1(1 + / ) if this is n o t so. T h e n / = 1 on K a n d ]/] < 1 on S ( A ) - K . I f I is t h e closed ideal of functions which vanish on K t h e n AK is isomorphic to A / I a n d t h u s inherits t h e quotient n o r m

[[go[[ = i n f [ [ g + go [I-

Clearly Ilgoll Ilg0ll = sup I gol. B u t the opposite inequality also holds since

K

IlY g0 II = I1 g0 II .

Thus t h e quotient n o r m is t h e sup n o r m on K so t h a t AK m u s t be complete in t h e sup norm.

(i) S(A~:) = K is well known, because K is a hull, i.e., t h e set of zeros of t h e function (1 - / ) .

(ii) is clearly implied b y (iii).

(iii) L e t p 6 K a n d let #v be a positive measure on F which represents p. F o r the func- tion / we t h e n h a v e

1

Y

a n d since

I/I

~< 1 we m u s t h a v e / = 1 on t h e closed s u p p o r t of tt,.

W e should like to m a k e some c o m m e n t s which place t h e o r e m 5.1 in w h a t we believe is its proper setting. F o r an algebra A

(i) t h e interior of S ( A ) is t h e set I = S ( A ) - P . (ii) the accessible set is t h e set L = i - I .

(iii) t h e minimal support set is t h e subset S , of P o b t a i n e d b y intersecting the closed supports of all measures try which represent points p in I .

228 K . HOFFMAI~" A~ID I . M. S I N G E R

(iv) the maximal support set is t h e set S* w h i c h is t h e closure of t h e u n i o n of t h e closed s u p p o r t s of all m e a s u r e s / z p which r e p r e s e n t p o i n t s p in I .

Of course I is a n o p e n s u b s e t of S ( A ) , while L, S , , a n d S* a r e closed s u b s e t s of F . I f t h e i n t e r i o r I is n o n - e m p t y , t h e n S , c S*. I f A is a m a x i m a l s u b a l g e b r a of C ( F ) , or m o r e g e n e r a l l y , if t h e a l g e b r a AE o b t a i n e d b y r e s t r i c t i n g A t o its e s s e n t i a l set E is a p e r v a s i v e s u b a l g e b r a of C ( E ) , t h e n t h e o r e m 5.1 tells us t h a t E is c o n t a i n e d in t h e m i n i m a l s u p p o r t set S , .

TH~O~]~M 5.3. For any algebra A, the accessible set L is contained in the essential set E.

Proo/. L e t A 0 be t h e i d e a l in C ( F ) of f u n c t i o n s w h i c h v a n i s h on E. B y d e f i n i t i o n of t h e e s s e n t i a l set, A0 is c o n t a i n e d in A . As is w e l l - k n o w n , e v e r y n o n - z e r o c o m p l e x h o m o - m o r p h i s m of A 0 is e v a l u a t i o n a t a p o i n t of F - E . T h u s if qb is a c o m p l e x h o m o m o r p h i s m of A , t h e r e s t r i c t i o n of 9 t o A 0 is e i t h e r i d e n t i c a l l y 0 or is e v a l u a t i o n a t a p o i n t of F - E . I f t h e r e is a p o i n t y o E F - E s u c h t h a t dp(/) =/(Y0) for e v e r y / in A0, t h e n dp(/) =/(Yo) for all / in A , b e c a u s e a n o n - z e r o h o m o m o r p h i s m o n a n i d e a l h a s a u n i q u e e x t e n s i o n t o a h o m o - m o r p h i s m of t h e full a l g e b r a . T h u s we see t h a t if p is a p o i n t of S (A) w h i c h does n o t lie i n F - E, t h e n e v e r y f u n c t i o n in A 0 v a n i s h e s a t p. I n p a r t i c u l a r t h i s is t r u e for e a c h p o i n t p EL. Thus, no p o i n t of L lies in F - E , i.e., L c E , q.e.d.

T ~ n O R E M 5.4. For any algebra A, S* c E.

Proo/. L e t p be a p o i n t of I a n d l e t / ~ be a n y m e a s u r e on I ~ w h i c h r e p r e s e n t s p. W i t h t h e n o t a t i o n of 5.3 (and a p o r t i o n of t h e proof), for e a c h / in A 0 we h a v e

o=/(p)= f f

where U = F - E. B u t t h i s s a y s t h a t t h e r e s t r i c t i o n o f / z p to U sends e v e r y c o n t i n u o u s f u n c t i o n on U w h i c h v a n i s h e s a t i n f i n i t y i n t o 0. T h u s t h i s r e s t r i c t i o n is t h e 0 m e a s u r e , i.e., t ~ (U) = 0. T h i s p r o v e s t h a t t h e closed s u p p o r t of/z~ is c o n t a i n e d in E .

I f we p u t t h e s e r e s u l t s t o g e t h e r for m a x i m a l a l g e b r a s we h a v e t h e following. I f A is a m a x i m a l s u b a l g e b r a of C ( F ) a n d if P =~S(A), t h e n

L ~ E = S , = S * .

I t seems r e a s o n a b l e to us t o c o n j e c t u r e t h a t w h e n A is m a x i m a l a n d I is n o n - e m p t y t h e n L = E = S , = S*. I n fact, one m i g h t c o n j e c t u r e t h a t for a n y a l g e b r a A w i t h I n o n - e m p t y t h e i n c l u s i o n S , ~ L holds. T h e q u e s t i o n p o s e d b y t h i s c o n j e c t u r e h a s t h e following t w o e q u i v a l e n t f o r m u l a t i o n s :

(i) I f p E I does t h e r e e x i s t a m e a s u r e / z ~ r e p r e s e n t i n g p whose closed s u p p o r t is con- t a i n e d in t h e accessible p a r t of t h e ~ilov b o u n d a r y , L = i - P.

MAXIMAL ALGEBRAS OF CONTINUOUS FUNCTIONS 229 (ii) I f A 1 is t h e closure of the restriction of A to i , is t h e ~ilov b o u n d a r y for A 1 e x a c t l y L?

W h e n the question is stated in f o r m (ii), we see t h a t we are asking w h e t h e r a strength- ened m a x i m u m m o d u l u s principle holds for function algebras. This seems to be a v e r y interesting question. A n affirmative answer could h a v e i m p o r t a n t consequences.

6. Tests for m a x i m a l i t y

I n our c o m m e n t s a b o u t pervasive algebras, we pointed o u t t h a t if A is a pervasive subalgebra of C ( X ) t h e n P = X. F r o m this it is easy to see t h a t if A is a n y m a x i m a l sub- algebra of C ( X ) t h e n I ' = X. So, in inquiring w h e t h e r an algebra A is m a x i m a l we need only inquire whether A is a m a x i m a l subalgebra of C (P).

H o w does one tell if a given algebra A is a m a x i m a l subalgebra of C(I')? This is, of course, a difficult question. One can a t t e m p t to check whether it is t i u e t h a t on the essential set for A t h e algebra is pervasive, an integral domain, antisymmetrie, etc. I n section 7, we shall use the analytic p r o p e r t y of m a x i m a l algebras to give an example of an algebra A which is contained in no m a x i m a l subalgebra of C(I'). B u t there are algebras A which are pervasive subalgebras of C (F) w i t h o u t being maximal, as we h a v e seen. One is t h e n cer- t a i n l y led to a search for other tests for t h e m a x i m a l i t y of A. W e should like to outline n o w a n o t h e r technique which, a l t h o u g h simple, does seem to h a v e some interesting conse- quences.

L E N M A 6.1. Let p be a point in S ( A ) - I " and let #~ be a representing measure ]or p.

Then there is a proper subalgebra B o / C (F) which contains A and is maximal with the property that #p de/ines a multiplicative linear/unctional on B.

Pro@ L e t F be t h e f a m i l y of all proper closed subalgebras of C(F) which contain A a n d on w h i c h / ~ defines a multiplicative linear functional. I f [B~] is a linearly ordered subset of F t h e n / % is multip]icative on t h e closure of the union of t h e B~, a n d this closure is n o t all of C(I') since p ~ S ( A ) - I ' . B y Zorn's lemma, F contains a m a x i m a l element B.

B y a similar a r g u m e n t , one can prove

LEMMA 6.2. I / P is a proper subset o/ S ( A ) then among all closed subalgebras B o/

C (S) such that (i) A c B (ii) F ( B ) = F ( A ) there is a maximal one.

230 K . I ~ I O F F M A N A N D I . M . S I N G E R

Of course, if A is a m a x i m a l subalgebra of C(F) t h e n A is already a m a x i m a l closed subalgebra of C (S) with t h e p r o p e r t y t h a t each function in the algebra assumes its m a x i m u m on F. However, t h e converse is n o t t r u e as is shown b y t h e following example, due largely t o H. Rossi [Thesis, M.I.T., 1959].

L e t A be the closed bicylinder Iz] ~< 1, Iwl ~< 1 i n complex two-space, a n d let A be t h e algebra of all continuous functions on A which are analytic in the interior of A.

This algebra A is simply t h e uniform closure on A of t h e algebra of all polynomials in two complex variables. The space S ( A ) is A, a n d F ( A ) is the torus F = [(z, w);

Izl

= Iwl = 1].

T h e topological b o u n d a r y T of A is larger t h a n t h e ~ilov b o u n d a r y for A.

T H E o RE M 6.3. Let B be a uni/ormly closed algebra o/continuous/unctions on the topo- logical boundary T such that each/unction in B assumes its m a x i m u m modulus on the set F.

I / B contains A, then B = A .

Proo/. L e t (%, w0) be a p o i n t in F, i.e., I%1 = Iwol = 1. T h e n t h e disc K = [(z, wo);

[z I ~< 1] is contained in t h e topological b o u n d a r y T. W e wish to show t h a t every function ] in the algebra B is an analytic function of z on t h e disc K.

L e t F ( z , w) = 89 (1 + ww~l). T h e n F e A , II Eli = 1 a n d K = [I F I = 1] = [2' = 1] (these sets relative t o A ). Then, as we observed in t h e o r e m 5.2 t h e algebra BK o b t a i n e d b y restrict- ing B to t h e disc K is a closed subalgebra of C(K) a n d F ( B K ) ~ F • K. B u t F ~ K is e x a c t l y the circumference of t h e disc K. Hence BK is a u n i f o r m l y closed algebra of continuous functions on K each of which assumes its m a x i m u m modulus on t h e circumference. Also BK ~ AK, a n d A K is simply t h e algebra of all continuous functions on K which are a n a l y t i c for Iz] < 1. B y t h e t h e o r e m of W e r m e r which we m e n t i o n e d in section 2, t h e algebra AK is a m a x i m a l subalgebra of t h e circumference of K. Thus, it m u s t be t h a t AK = B~, or t h a t every function in B is analytic in u on t h e interior of t h e disc K.

Similarly we can show t h a t each function in B is analytic in w on t h e disc [(%, w);

Iwl

Since this holds for e v e r y point (%, w0) EF, t h e Fourier coefficients ] (m, n) = / e -tin~ e-int/(e i~ e u) d 0 d t

I"

vanish outside t h e q u a d r a n t m / > 0, n ~> 0 for e v e r y / E B . Thus B = A , q.e.d.

Of course t h e bicylinder algebra A is n o t a m a x i m a l subalgebra of C (1~). A larger algebra is for example t h e algebra of all continuous functions / on F such t h a t

2~

f e~n~ t~ 1 ) d 0 ~ 0 , n = 1,2 . . .

0

m A X I M A L A L G E B R A S O F C O N T I N U O U S F U N C T I O : N S 231 I t is interesting to contrast this algebra of analytic functions in the bicylinder with with the corresponding algebra when A is the closed unit sphere in complex two-space.

The algebra A of all continuous functions on A = [(z, w); [zl ~< l, lwl ~< 1] which are analytic in the interior has A as its space of maximal ideals and the full topological bound- a r y of A as its ~ilov boundary. I n this case there are larger closed subalgebras of C ( A ) in which every function assumes its m a x i m u m on the unit sphere. One such example is given in [9].

Let us return now to the consideration of our general algebra A and look more closely at l e m m a 6.1. I n certain special cases it m a y h a p p e n t h a t there is a point p E S ( A ) - F with these properties:

(i) if A c B c C(F) and no measure representing p is multiplicative on B, then B = C(F).

(ii) if A c B c C(F) and a n y measure representing p is multiplicative on B t h e n B = A. I t is then clear t h a t A is a m a x i m a l subalgebra of C(F).

This is a v e r y special situation of course, b u t it does arise. I t was this idea which was exploited in [8] to give a very short proof of the Wermer theorem on the maximMity of the analytic functions on the circle (and some generalization thereof). Royden [13] and Bishop [6] have also exploited this idea. We should like now to indicate how a simple extension of this idea leads to some examples of the " p r i m a r y " algebras described in the introduction.

F o r each positive integer k, let Ak be the uniformly closed algebra of continuous func- tions on the unit circle which is generated by 1, z ~, z k+l . . . Then A is (isomorphic to) the algebra of continuous functions on [z [ ~< 1 which are analytic for [z ] < 1 and have derivatives of orders 1 . . . k - 1 which vanish at the origin. The space of m a x i m a l ideals of Ak is the closed disc [z[ ~< 1 a n d the ~ilov b o u n d a r y for Ak is the unit circle F. The algebra A x iS (of course) the uniform closure on F of the algebra of polynomials, t h a t is, A~ consists of all continuous functions / on F such t h a t

2zr

f/(e~~176 n = 1 , 2 , 3 . . .

0

THEOI~EM 6.4. Let B be a closed subalgebra o / C ( F ) which contains ~4k (/or some [ixed Ic). Then either B = C(F) or B is contained in A r

Proo/. The origin of the unit disc defines a complex homomorphism of the algebra Ak bY

2~

0

232 K . H O F F M A N A:ND I . M . S I N G E R

I f this h o m o m o r p h i s m h does n o t e x t e n d to B, i.e., if no measure representing the origin is multiplicative on B, t h e n t h e function

/(e i~ = e ~~

^{is in}

B,

since the function z k inAk vanishes on the disc only at the origin. B u t then it is easily seen t h a t B contains t h e functions e ~n~ for every integer n, so t h a t B = C (F).

Suppose t h e h o m o m o r p h i s m h does e x t e n d to B. T h e n

2 z

h(/)= f /(e~~ / e B

0

for some positive m e a s u r e / ~ on F. I n particular t h e n # m u s t evaluate every function in Ak at the origin, or

2n

f e~n~ n>~]c.

0

Since/~ is a real measure of mass 1 we m u s t t h e n h a v e

d # (O)=21--~ r (O)dO

where q~ is a trigonometric polynomial

k - 1

r (0) = 1 § ~ [ape w~ + 5~ e-~P~

p = l

N o w let ] be a n y function in t h e algebra B, a n d let n ~> 0. T h e n

2~

1 f e~nOei~O /

o = h (z ~§ h (/) = h (z n

§

= ~ (e ~~ ) r (0) d O.

0

I f we let ~p (e i~ = e~k~ (0) t h e above equation says t h a t for e v e r y / E B, F / i s anaIytic, i.e.,

2n

f e*~~176176

= 0 , n = 1, 2 . . .

0

T h u s e v e r y / E B is (the b o u n d a r y function of) a m e r o m o r p h i c function: / = W//y~. So / is meromorphic, y~ is analytic, a n d for every positive integer n t h e function ~/~ is analytic.

Clearly / is itself analytic. T h u s B c A 1.

I n the case k = 1, t h e above t h e o r e m states t h a t A 1 is a m a x i m a l subalgebra of C(F).

F u r t h e r m o r e , the t h e o r e m implies t h a t for a n y k t h e algebra Ak is contained in precisely one m a x i m a l subalgebra of C(F), n a m e l y A 1. So, w h e n k > 1, t h e algebra Ae is c o n t a i n e d in a m a x i m a l algebra b u t is n o t the intersection of t h e m a x i m a l subalgebras of C(F) which contain it. T h u s we m i g h t s a y t h a t Ak is a p r i m a r y algebra. W e shall see shortly t h a t this p r o p e r t y of

Ak(k ~

2) is r a t h e r easily deduced f r o m t h e m a x i m a l i t y of A 1 a n d the fact t h a t Ak has finite codimension in A~. W e h a v e given t h e o r e m 6.4 separately to

MAXIMAL A L G E B R A S OF C O N T I N U O U S F U N C T I O N S 2 3 3

show that the proof of the primary nature of Ak is not appreciably more difficult than the short proof of the maximality of A 1 given in [8].

We proceed now to enlarge our class of primary algebras. We shall need the following lemmas.

LEMMA 6.5. Let A be a maximal subalgebra o/ C(F) and let B be a subalgebra o / C (F) which contains an ideal I o / A . I / B contains a ]unction / which is not in A, then B contains every continuous ]unction on F which vanishes on the hull o / I . I/, in addition, the hull o / I is /inite, then B = C (F).

Proof. B y the hull of I we mean the set of all points in F where every function in I vanishes. Since A is maximal and ] is not in A, the linear algebra generated b y A and ] is dense in C(F). This means each function h in C(F) can be uniformly approximated b y functions of the form

~ gk /k, gk E A .

k - O

If g is a n y function in the ideal I, then the function g gk /~

k 0

belongs to B, because ] is in B and each ggk is in I, hence in B. F r o m this it is clear t h a t if hEC(F) and g E I , then gh is in B. I n other words, J = I . C ( F ) is contained in B. Now J is an ideal in C (F) so t h a t the closure ] consists of all continuous functions vanishing on the hull of J. Clearly J and I have the same hull. This proves the first statement of the lemma.

If we now assume that the hull of I is a finite point set, then since B separates points of F and contains every continuous function vanishing on t h a t finite point set, it is clear that B = C[F).

LEMMA 6.6. Let A be a commutative linear algebra with identity and let A o be a subalgebra o / A which has finite codimension in A . Then A o contains an ideal 1 o/the algebra A such that I has ]inite codimension in A.

Proo/. Let I be the set of all elements a EA o such t h a t a A ~ A o. Then I is an ideal in A and I c A 0. For each element a EA 0 let La be the linear transformation of A into A which is right multiplication by a. Since A 0 is a subalgebra of A, the space A 0 is invariant under La. Thus L a induces a linear transformation L~ of the linear space A / A o into itself.

The mapping a-+L~ is an algebra homomorphism of A o into the algebra of linear trans- formations on the finite-dimensionM space A / A o. The kernel of this homomorphism is the ideal I. Thus A o / I is finite-dimensional, hence A / I is finite-dimensional.

234 K . H O F F M A N A N D I . M . S I : N G E R

THEOREM 6.7. Let A be a m a x i m a l subalgebra o / C ( F ) and let A o be a subalgebra o / A which has finite codimension in A . I] B is a subalgebra o/ C(F) which contains Ao, then either B = C (F) or B is contained in A .

Proo/. B y Lemma 6.6, A 0 contains an ideal I of the algebra A which has finite co- dimension in A. Thus the hull of I must be a finite point set. If B is not contained in A, then by lemma 6.5, B = C(F).

Under the hypotheses of Theorem 6.7, the algebra A 0 is contained in precisely one maximal subalgebra of C (F). This theorem applies (of course) to the algebras A~ of theorem 6.4. We should like to discuss now some further examples of this situation.

For the remainder of this section, let F denote the unit circle of the complex plane, and let A be the uniform closure on F of the algebra of polynomials. Suppose t h a t / 1 . . . ].

are functions, each analytic in a neighborhood of the unit disc, satisfying (i) /1 . . . /~ separate the points of F

(ii) at each point of F, one of the functions/j has a non-vanishing derivative.

Let A 0 be the subalgebra of C(F) generated b y / 1 . . . /~. Results of Wermer [20; Lemma 3.2] and [19; Theorem 1.2] imply that there is a function

g (Z) = (Z - - ~ 1 ) p . . . . (Z - - h ) P k

such t h a t A 0 contains the ideal I = g A of the algebra A. Thus A 0 is contained in precisely one maximal subalgebra of C (F), namely A. As an approximation theorem, this result states that if/1 . . . /~ are analytic in a neighborhood of the closed disc and satisfy conditions (i) and (ii) above, then given any continuous function h on F which is not in A, we can ap- proximate every continuous function on F by polynomials in ]1 . . . /~ and h.

There is a theorem similar to this where one assumes t h a t / 1 . . . ]~ are analytic in an annulus containing F and satisfy conditions (i) and (ii). I n this case the algebra A 0 generated b y / 1 . . . /~ need not be a subalgebra of A; indeed, this algebra A 0 m a y be all of C(F).

If A 0 =4= C (F), Wermer's results [19; 20; 21] state the following. There is a Riemann surface F and an analytic homeomorphism z---->z' of F onto an analytic circle F ' on F which bounds a domain D such t h a t D U F ' is compact. Each of the functions/1 . . . /n extends analyt- ically from F ' into a neighborhood of D U F'. If A ' denotes the algebra of all continuous functions on F which extend from F ' analytically to D then A ' is a maximal subalgebra of C(F), which of course contains A o. I t is once again true t h a t A o contains an ideal I of A" which has a finite hull. Thus, in this case, the algebra A 0 is contained in exactly one maximal subalgebra of C (F), namely A'. But the algebra A ' m a y consist not of the functions which extend from F analytically to the unit disc but rather those which extend analyt- ically into the domain D on the Riemann surface F.

M A X I M A L A L G E B R A S O F CO:NTI:NIYOUS F U : N C T I O N S 235 I f we wish to state this last result as an a p p r o x i m a t i o n theorem, it says t h a t if h E C (F) a n d

h~iA'

t h e n / 1

...

/~ a n d h generate C(F). Given t h e function h on F it might be ex- t r e m e l y difficult to determine w h e t h e r or n o t h is in A ' . B u t one can state a slightly weaker a p p r o x i m a t i o n t h e o r e m whose hypothesis is more easily verified in specific cases.

Because conditions (i) a n d (ii) are satisfied b y ]1 . . .

/n

there will be an annulus con- taining F such t h a t t h e analytic h o m e o m o r p h i s m

z--->z'

of F into the surface F can be extended to an analytic h o m e o m o r p h i s m of this annulus onto a n e i g h b o r h o o d of the curve F'. T h u s one sees t h a t if h is in A ' t h e n in the plane h m u s t be extendable f r o m F so as to be analytic in an annulus of one of the two types: [z; 1 < [z[ < p] or [z; p < ]z[ < 1]. T h a t is, /1 . . .

]n

determine a positive n u m b e r p, either less t h a n or greater t h a n 1, such t h a t each function in A ' is extendable to a continuous function on [z; 1 ~< ] z I ~< P] which is analytic on [z; 1 < [z] < p ] , or a continuous function on

[z;p<~lz I

~<1] analytic on [z;

p < [zl < 1]. We m i g h t describe this b y saying t h a t /1 . . . ]~ determine one side of the unit circle such tha~ each function in A ' is analytic in an a n n u l u s on t h a t side of t h e circle.

N o w one can certainly state t h e following. S u p p o s e / 1 . . . /~ are analytic in an annulus containing F a n d satisfy conditions (i) a n d (ii) above. If h EC(F) a n d h is n o t analytically extendable to an annulus on either side of F, t h e n e v e r y continuous function on F can be u n i f o r m l y a p p r o x i m a t e d b y polynomials i n / 1 . . . /~ a n d h.

7. Bounded analytic functions

L e t L ~ be t h e Mgebra of b o u n d e d measurable functions on t h e unit circle of t h e complex plane, identifying functions which are equal almost everywhere. T h e n L~r is a c o m m u t a t i v e B a n a c h algebra with the n o r m

II/11 = ess sup l i (e I.

0

L e t H ~ be the (closed) subalgebra of L~r consisting of t h e functions f such t h a t

2=

f e~O/(ei~ n =

1, 1, 2, 3 . . .

0

F o r each ] E H ~ the function

2z

] (z) = f / (e ~~ P~ (0) d 0, 0

where P~ is the Poisson kernel for z, is a b o u n d e d analytic f u n c t i o n in the open u n i t disc, a n d

I1 11= sup

z l < l 1 6 - 6 0 3 8 0 8 Acta mathematica. 1 0 3 . I m p r i m 6 le 2 1 j u i n 1 9 6 0

236 K . H O F F M A N A N D I . M. SII~'GER

Furthermore, a well-known theorem of F a t o u states t h a t every bounded analytic function in the disc arises ~n this way. Thus H ~ is isometrically isomorphic to the algebra of bounded analytic functions in the open disc.

The algebra L ~ is self-adjoint, i.e., closed under complex conjugation, and is therefore isometrically isomorphic to the algebra

C(X) of

all continuous functions on its space of maximal ideals. This follows from the commutative Gelfand-Neumark theorem, see [11;

p. 88]. The space X is also the Stone space for the Boolean algebra of measurable subsets of the circle, modulo sets of measure 0. Thus X is an extremally disconnected compact Hausdorff space. We shall need to know only t h a t X is totally disconnected and t h a t a basis for the topology of X is given by the open and closed sets

[x~x;

k~(x) = o]

where M is a measurable set on the circle and

kM

is its characteristic function.

The algebra H ~ is isometrically isomorphic to a closed subalgebra A of

C(X).

For the sake of simplicity we shall sometimes write A = H:r a n d / o r C (X) = L~.

We proceed now to record some facts about the space of maximal ideals of H ~ which are contained in an unpublished paper of I. J. Schark.

Each point of the open unit disc defines a complex homomorphism of H ~ by

l-~l(z)

and the open disc is thus "embedded" in the space S (H~r Also, there is a natural continuous mapping z~ of

S(H~)

onto the closed unit disc, as follows. The algebra H ~ contains as a closed subalgebra the algebra A 1 of continuous "analytic" functions on the circle. The mapping z mentioned above sends a homomorphism of H ~ into its restriction to the sub- algebra A], and this restriction corresponds to a point in the closed disc. The mapping is one-one over the interior of the disc; t h a t is, if (I) is a complex homomorphism of H ~ such that (I) evaluates each function in A 1 at some point z, I z I < 1, then 9 is simply evalua- tion at z on all of H ~ . This is evident from the fact i f / E H ~ t h e n / ( z ) = 0 if and only if

1(~)

g(~)- z-~"

is in H ~ . The mapping ~ certainly maps

S(H~)

onto the closed unit disc since ~(S) is a compact subset of the closed disc which contains each interior point. The mapping ~ is (as we shall see) not one-one over the circumference. Notice also t h a t - 1 actually defines a homeomorphism of [z] < 1 into S (H~) since the topology of the disc is in either case the weak topology defined by H ~ . Let us call A the image of the open disc in

S(Hr162

One unsolved problem in this context is whether A is dense in

S(Hr

Since H ~ is an

M A X I M A L A L G E B R A S O F C O N T I I ~ U O U S F U l q C T I O N S 237 algebra of continuous functions on A, it is at least clear t h a t t h e closure of A in S(H~) will contain t h e ~ilov b o u n d a r y F (Hoo).

We wish n o w t o show t h a t F ( H ~ ) is (homeomorphic to) S(Loo)= X. Since Ho~ is a subalgebra of L ~ there is a n a t u r a l continuous m a p p i n g ~1 of X into S ( H ~ ) , obtained b y restricting each h o m o m o r p h i s m of Loo to the subalgebra H ~ .

LEMMA 7.1. The mapping ~ is a homeomorphism o] X into S (H~) and xq (X) is exactly r (Hoo).

Pro@ W h a t we m u s t show is t h a t t h e functions in Hoo, w h e n regarded as elements of L~o, separate points on X a n d t h a t the ~ilov b o u n d a r y for H~o is M1 of X. W e shall first show t h a t the ~ilov b o u n d a r y condition is satisfied. F r o m this proof it will be clear t h a t H ~ separates points on X.

L e t x E X and let U be an open set in X containing x. W e m u s t show t h a t there is a function h EHoo whose m a x i m u m m o d u l u s on U is greaLer t h a n the modulus of h a n y w h e r e on X - U. I t will suffice to do this when U = [/CM = 1], M a measurable set on t h e u n i t circle. Define

2n

( Z ) = ~ / e~O'3rZ iO . ~ 0

T h e n !P is analytic for ] z [ < 1 a n d h = e ~ is a b o u n d e d analytic function in t h e u n i t disc.

F u r t h e r m o r e , on the u n i t circle, [ h i = exp kM almost everywhere. T h u s on X, ] h i = e where kM -- 1 a n d [ hi = 1 where kM = 0; so h is t h e desired function.

Let us n o w regard X as a closed subset of S(Hoo). We t h e n h a v e the following picture of S (H~). This space contains the open u n i t disc A. The closure of A contains besides A the ~ilov b o u n d a r y P (Hoo) which is t h e space X of m a x i m a l ideals of L ~ . I t is n o t k n o w n w h e t h e r t h e closure of A is all of S (Hoo), b u t it is easily seen t h a t there are points in A -- A which are n o t on t h e ~ilov b o u n d a r y . F o r example, t h e function

z + 1]

/ (z) = e x p

\~:-- l/

is in H:r has no zeros in A, b u t m u s t vanish somewhere on A since /(z) tends to 0 as z approaches 1 along the positive axis. N o such 0 of / can occur on t h e ~ilov b o u n d a r y X since t h e function / has modulus I on t h e unit circle a n d is therefore invertible in Loo.

This is the e x t e n t of t h e information we shall need f r o m t h e w o r k of I. J. Schark.

The space S(H~) - A a n d a fortiori the space X , is fibered b y a n a t u r a l continuous m a p p i n g 7r onto the u n i t circle, ~z being t h e m a p which sends each complex h o m o m o r p h i s m