Relatively maximal function algebras generated by polynomials on compact sets in the complex plane

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A R K I V F O R M A T E M A T I K B a n d 8 n r 3 1.68 10 2 Communicated 5 June 1968 by OTTO FROSTM~A~

Relatively maximal function algebras generated by polynomials on compact sets in the complex plane

By

JAN-ERIK BJORK

Introduction

Wermer's maximality theorem states t h a t if J is a Jordan curve in the complex plane the function algebra P ( J ) generated b y polynomials on J is a maximal closed subalgebra of C(J), the algebra of complex-valued continuous functions on J . Wermer's theorem can also he stated in the following form: I f g E G(J) is such t h a t the polynomials and g generate a proper function algebra of

C(J)

then g has an analytic extension to the interior of J . I n this paper we t r y to extend Wermer's maxi- reality theorem in the following way: Let J be a Jordan curve in the complex plane.

We denote b y

H(J)

the compact set bounded by J. Let F be a closed subset of -~/(J) containing J . Suppose g E C(F) is such t h a t g and the polynomials generate a proper function algebra of

C(F).

Now we wish to find out if g has an analytic extension from J into the interior of H(J), i.e. if there exists a function

GEC(H(J))

such t h a t

G=g

on J and G is analytic in the interior of

H(J).

Of course we need some conditions on F to obtain such results. We say t h a t F satisfies (C) if the following holds:

I. R(F)=C(F),

where

R(F)

is the function algebra on F generated b y rational functions with poles outside F. _ _

2. H(J)-.F

is connected and

( F - J ) N J # J .

We show in Theorem 1 t h a t if F satisfies (C) and

gEC(F)

is such t h a t g and the polynomials generate a proper function algebra of

C(F)

then there exists an analytic function G in

H ( J ) - F

such t h a t lim

G(z)=g(x)

as

z E H ( J ) - F

tend to

x6J.

I n the final part of this paper we apply theorems 1 and 2 to solve an approximation problem on the unit interval. Let /EC(I), where I is the unit interval. Assume

](¼) =/(~)

w h i l e / ( x ) # / ( y ) for all other pairs of distinct points x, y E I. I f

g E C(I)

is such t h a t g(¼)# g(~) we wish to find out if the function algebra on I generated b y /, g and the constant functions is

C(I).

This problem has been discussed in several papers, see for example [1, 2 and 4]. The best result is contained in [2] where it is shown t h a t we get

C(I)

if / and g are continuously differentiable. A famous example in [3] indicates t h a t some smoothness on / and g is necessary. The example consists of a J o r d a n arc K in C a such t h a t K is not polynomially convex. This J o r d a n arc is used to construct a proper function algebra of

C(I).

Let us now p u t J =

{](x) lz E I}.

We see t h a t J has one of the following three forms:

The case when J has the form (3) is easy, we get

C(I)

with no extra assumptions on / and g. Also case (2) can be easily reduced to case (1) so we only consider t h a t case. To prove t h a t we now get

C(I)

we need some conditions on J . Obviously J satisfies the condition (C) if

R(J)= C(J).

We do not know if this alone is sufficient

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J.'E. BJORK,

Relatively

maximal function a~gebras

Fig. 1 Fig. 2 Fig. 3

to g u a r a n t e e t h a t we get C(I). I n order to p r o v e t h a t we get C(I) we shall n e e d some s m o o t h n e s s of t h e t w o J o r d a n arcs a a n d b i n Fig. 1. I t is for e x a m p l e sufficient to h a v e a a n d b c o n t i n u o u s l y differentiable. Notice t h a t we n e e d n o e x t r a c o n d i t i o n o n g. W e shall l a t e r i n t r o d u c e a c o n d i t i o n o n a a n d b which g u a r a n t e e s t h a t we get C(I). This c o n d i t i o n is r e l a t e d t o difficult p r o b l e m s o n a n a l y t i c e x t e n s i o n s u s i n g reflection principles. 1

Before we s t a t e t h e following r e s u l t s we m a k e t h e following useful r e m a r k : L e t E be a c o m p a c t set s a t i s f y i n g (C). N o w we c a n use a c o n f o r m a l m a p of H(J) o n t o t h e u n i t disc. l=[ence we m a y a s s u m e t h a t J is t h e u n i t circle T . W e p o i n t o u t t h a t all e x t r a c o n d i t i o n s we m a k e are i n v a r i a n t u n d e r t h i s c o u f o r m a l m a p . So w h e n we n o w s a y t h a t a c o m p a c t set satisfies (G) i t is u n d e r s t o o d t h a t t h e J o r d a n c u r v e J is T.

T h e o r e m 1. Let K be a compact set o/ the unit disc D containing T and satis/ying ( C).

Let g E C(K) be such that g and the polynomials, generate a proper/unction algebra B of C(K). I / n o w m is a non zero measure on K annihilating B then

f dm(x)

j x - - z / j x - - z is a bounded analytic/unction in D - K with

sup {la(~)llzeD- K} <<. sup {lg(x)ll~eg}.

Also lim G(z)=g(e ~) exists uni/ormly as z e D - K tend to e~e T.

Before we p r o v e T h e o r e m 1 we wish to s t a t e T h e o r e m 2. L e t G(z) be as i n T h e o r e m 1, h e n c e G(z) is a n a n a l y t i c f u n c t i o n i n D - K . I f zoEK is s u c h t h a t t h e r e exists a J o r d a n arc J c D w i t h J N K = {zo} a n d lira G(z) = a exists as z E J t e n d to z 0, t h e n we s a y t h a t G h a s t h e a s y m p t o t i c v a l u e a a t z 0. I f zoEK is such t h a t lira G(z)=a exists as z E D - K t e n d t o z0, t h e n we s a y t h a t G h a s t h e u n r e s t r i c t e d l i m i t v a l u e a a t %.

T h e o r e m 2. I / G has an asymptotic value at some point ^{z o E}K which is di//erent /rom g(zo) then there exists an open neighborhood V o/ z o such that the restriction o/ B to K - V generates a proper/unction algebra o/ C ( K - V). There exists a smallest closed subset F o / K ~ c h that the restriction o / B to F generates a proper/unction algebra o/

C( F). T h e / u n c t i o n G is analytic in D - F and i / G has an asymptotic value at some point z o E F then G has an unrestricted limit value at z o which equals g(zo).

Since t h e proofs are n o t v e r y s h o r t we shall first give some p r e l i m i n a r y results.

L e t M s be t h e m a x i m a l ideal space of B. As u s u a l we i d e n t i f y K with a closed sub- i In a forthcoming paper by H. S. Shapiro and Shields it is shown that we get C(I) without any extra conditions on ] and g. We also remark here that Mergelyan's Theorem shows that R(J) = C(J) is always verified. In a forthcoming paper "Analyticity in the maximal ideal space of a function algebra" by the present author essentlell improvements have been obtained which indicate that f and g generate C(I) in the case where J is only assumed to be a curve with fi nitely many self-intersections.

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ARKIW FOR MATEMATIK. B d 8 n r 3 set of M s a n d t h e n K c o n t a i n s t h e S h i l o v b o u n d a r y of B. I f

x ~ M s

t h e r e e x i s t s a p o i n t

:~(x) ED

s u c h t h a t

P(x)=P(u(x))

for e v e r y p o l y n o m i a l P . W e s a y t h a t x lies a b o v e ~(x) a n d t h a t ~(x) lies b e l o w x. I f V is a s u b s e t of D we p u t u - l ( V ) = {x E M s I

~(x) E V}. T h e set ~ - I ( V ) is called t h e fiber of V in M s . T h e c o r r e s p o n d e n c e b e t w e e n p o i n t s z E D a n d t h e f i b e r s u - l ( z ) is c o n t i n u o u s in t h e following w a y : L e t W b e a n o p e n n e i g h b o r h o o d of ~ - l ( z ) in M s , t h e n t h e r e e x i s t s a n o p e n n e i g h b o r h o o d V of z in D s u c h t h a t u - x ( V ) is c o n t a i n e d in W. Since

R(K) = C(K)

a n d D - K is c o n n e c t e d we see t h a t if z 0 E D - K t h e n t h e e l e m e n t P = z - z o i n B c a n n o t b e i n v e r t i b l e . H e n c e t h e r e e x i s t s a p o i n t x E MB such t h a t

P(x) = z o

a n d i t follows t h a t x lies a b o v e z 0. W e h a v e n o w p r o v e d t h a t t h e fibers ~ - l ( z ) a r e n o t e m p t y w h e n z E D - K . W e shall l a t e r p r o v e t h a t t h e f i b e r ~ - l ( z ) is r e d u c e d to a single p o i n t w h e n

z E D - K .

I f

z E K

t h e f i b e r u-~(z) c o n t a i n s a t r i v i a l p o i n t , n a m e l y z itself. I f z E K a n d if ~ - l ( z ) o n l y consists of t h i s t r i v i a l p o i n t we s a y t h a t ~ - l ( z ) is a t r i v i a l fiber. I f z E T is is e a s i l y s e e n t h a t u - l ( z ) is a t r i v i a l fiber. F o r s u p p o s e t h a t x E~-l(z). N o w we c a n f i n d a p o s i t i v e m e a s - u r e v on K s u c h t h a t

g(x)= ~gdv

for all g G B. I n p a r t i c u l a r

P(z)=.~Pdv

for e v e r y p o l y n o m i a l . I t follows t h a t v is t h e u n i t p o i n t m a s s a t z a n d h e n c e

g(x) = g(z)

for all g E B w h i c h p r o v e s t h a t

x =z.

A s s u m e n o w t h a t we h a v e p r o v e d t h a t z - l ( z ) consists of one p o i n t

x(z)

w h e n

z e D - K .

T h e f u n c t i o n

G(z)=g(x(z))

is t h e n well d e f i n e d in D - K . W e shaft l a t e r p r o v e t h a t

d x - z I J x - z

if qn is a n a r b i t r a r y n o n zero m e a s u r e on K a n n i h i l a t i n g B. I t follows t h a t G is a n a - l y t i c i n D - K . Since K c o n t a i n s t h e Shflov b o u n d a r y of B in M s we h a v e

]g(x(z) I'<~

sup { I t ( x ) [

[ x E g } = IgIK

w h e n

z E D - K . If z E D - K

a n d l i m

z=e~aET

we see t h a t lira

x(z) =e ~

h o l d s in M s too, b e c a u s e ~-l(e(a) is a t r i v i a l fiber. H e n c e l i m

G(z)=lira g(x(z)) =g(e ~'~)

a s z E D - K t e n d t o e% I f z E K - T t h e f i b e r ~ - l ( z ) m a y b e n o n t r i v i a l a n d t h e n we g e t t r o u b l e s . A n i m p o r t a n t r e s u l t w h i c h we s h a l l p r o v e i s t h e following:

I f G h a s a n a s y m p t o t i c v a l u e a t

z o E K - T

w h i c h is d i f f e r e n t f r o m

g(zo)

t h e n rc-l(z0) c o n t a i n s e x a c t l y t w o p o i n t s . L e t x 1 b e t h e n o n t r i v i a l p o i n t in ~-l(z0). W e shall l a t e r p r o v e t h a t lira

G(z)=g(xl)

as

z E D - K

t e n d t o z o, hence G h a s a n u n r e s t r i c t e d l i m i t v a l u e a t z 0. W e c a n use t h i s t o p r o v e t h a t t h e p o i n t z 0 E MB h a s a n o p e n neigh- b o r h o o d W in M s such t h a t u ( W ) is c o n t a i n e d i n K . L e t t h e n V be a n o p e n neigh- b o r h o o d of z o in D s u c h t h a t ~ ( W ) is c o n t a i n e d in K N V. W e c a n use t h i s f a c t t o p r o v e t h a t t h e r e s t r i c t i o n of B t o F = K - V g e n e r a t e s a p r o p e r f u n c t i o n a l g e b r a of

C(F).

I f G h a s a n a s y m p t o t i c v a l u e a t

zoEK

w h i c h e q u a l s

g(zo)

we c a n p r o v e t h a t z-X(Zo) is t r i v i a l . I t follows t h a t if z E D - K t e n d t o z o in D t h e n

x(z)

t e n d t o z 0 in MB. H e n c e l i m

G(z)

= l i r a

g(x(z)) =g(zo),

i.e. G h a s a n u n r e s t r i c t e d l i m i t v a l u e a t z 0.

W e s h a l l f r e e l y use r e s u l t s a b o u t f u n c t i o n a l g e b r a s . W e refer t o [5] a n d [6] for a dis- cussion a b o u t these. H e r e we s t a t e some r e s u l t s w h i c h a r e u s e d in t h e following proofs.

L e t A b e a f u n c t i o n a l g e b r a w i t h t h e m a x i m a l i d e a l s p a c e MA a n d t h e S h i l o v b o u n d - a r y SA. T h e set

D.4 = M . 4 - S A

is c a l l e d t h e i n t e r i o r of M~. T h e L o c a l M a x i m u m P r i n c i p l e is h e r e u s e d in t h e following form: L e t W b e a s u b s e t of DA a n d l e t

bW

be t h e t o p o l o g i c a l b o u n d a r y of W in MA, t h e n [/(x)] < sup

{[](y)]yEbW}

for e v e r y x E W. I n p a r t i c u l a r t h e r e exists a p o s i t i v e m e a s u r e mx c a r r i e d on

bDA

for e a c h p o i n t x E D A such t h a t / ( x )

= ~/dmx

for all / e A . I t follows f r o m t h i s t h a t if D~ is n o t e m p t y t h e n t h e r e s t r i c t i o n of

A to bDA

g e n e r a t e s a p r o p e r f u n c t i o n a l g e b r a of

C(bDA). A

closed s u b s e t F of MA is A - c o n v e x if for e v e r y p o i n t

x E M A - F

t h e r e e x i s t s ] E A

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J.-E. BJ~RK, Relatively maximal function algebras

such t h a t / ( x ) > sup

{[/(y)lyeF}. If F

is a n A - c o n v e x subset of M a the function algebra A~ on F generated b y restricting A to F has F as its maximal ideal space.

Proo/ o/ Theorem 1. Let m

be a n o n zero measure on K annihilating B. L e t us p u t

W(z) =~g(x)dm(x)/x-z

^{a n d}

R(z)=~dm(x)/x-z.

Obviously W a n d / t are analytic functions in

D - K .

Because m annihilates B we get

~g(x)dm(x)/1-~x=O

for

z e D - K

and hence W ( z ) - - ~ ( 1 - I z ] 2 ) g ( x ) d m ( x ) / ( x - z ) ( 1 - 2 x ) when

z e D - K .

Let us p u t

K I = ( K - T ).

B y assumption there exists a closed arc

L={e~tla<~t<<.b }

such t h a t K1 • L is empty. Hence there exists r 0 < 1 such t h a t if r >/r 0 a n d a ~ t ~< b then

re ~t $K1.

F r o m now on we always assume t h a t

r>~r o.

We also put K - T = S.

Lemma

1. ~ f ~ I W(rdt)ldt< co

as r tends to 1.

Proo/. We have [W(re") dt< dt

+ Jr[g(x)[ [dm(x)[~

^{( 1 - r}

a) I/(x-re't[~dt =A(r) + B!r). Obviously

^lira

A(r~ =0 as r tends to 1 because K~ fl L is empty, also B(r) <~ 2~ f r[ g(x) [ [ dm(x) [ holds.

Using L e m m a 1 we can now choose two different rays {re e} a n d {re ~} where

a < c < d < b

such t h a t lira W(ref~),

limW(re~), lim R(e ~)

and lim

R(e ~)

all exist finitely as r tends to 1. W e shall need the following elementary result:

L e m m a

2. Let J be a Jordan curve in D such that J ~ T={e~t[e<~t<~d}=J~. A~o J approaches T along the two rays {re ~c} and {re~}, i.e. J contains the two sets {re e ] r~ <~r<l} and {re~lr~ <~r<l} /or some

r x < l .

Let z be a point in the interior o/J. Let v be the unique positive measure on J such that P(z)= ~Pdv/or every 1aolynomiaI P.

Then dv(e~)=h(e~')dx when e<x<d. Here dx is the Haar measure on T and h is bounded on

^{(c, d).}

Lemma 3.

With J and v as in Lemma 2 we have

lim

C(r)=lim ]~ ]R(re~t)g(e~) - W(re i~) [dv(e te) =0 as r tends to 1.

Proof.

We have

C(r) <~ ~ ]dv(e't)l f s lff(re'~)-g(x)l(1-r'~)ldm(x)Hx-re'tl[1-xre-~t[

~r Idm(z) ] ~ Ig(rett) - g(x) l ( 1 - r2) h(e~t)/Ix- re'tl~dt= A(r) + B(r).

+

As in Lemma 1 we see that

A(r)

tends to zero as r tends to 1 and

B(r)

tends to zero because

h(d t)

is bounded on

(c,d)

a n d 9 is a eontinuos function.

L e t M s be the maximal ideal space of B. Let

z e D - K

and choose

x(z)ez-l(z)

in M s . Now we have:

Lemma 4.

R(z)g(x(z)) = W(z).

Proo/.

Choose a J o r d a n curve J as in L e m m a 2 which contains z in its interior.

This is possible since D - K is connected. I f r < 1 is sufficiently close to 1 the functions

W~(z)= W(rz)

^{a n d}

R~(z)=R(rz)

are analytic in a neighborhood of the closed set

H(J)

bounded b y J. Hence Runge's theorem shows t h a t we can approximate Wr and R~ uniformly b y polynomials on

H(J).

Let us p u t

J=z-I(H(J)).

If x E ] then ~ ( x ) e l l ( J ) . We define

VV~(x)= W~(z(x))

and R~(x)=R~(g(x)) on 3. I f (P~} are

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/t2tKIV FOR MA.TEMATIK. Bd 8 n r 3 polynomials such that lira ] P ~ - W~ ]

~(,,

= 0 then we see that lira [ P . - W~[ ~ = 0.

Hence we can a p p r o x i m a t e W~ a n d k r uniformly on J b y functions f r o m B. Assume now t h a t the l e m m a is false. T h e n we can find d > O such t h a t

lim[R(rz)g(x(z))-

W(r=) 1/> d. ^LetM >/sup { I W~l~_~ + I R, I ~-~ I r0 <r < ~ }. We can find M here ^beeanse

lira W(re ~) ....

exist finitely. Choose now a polynomial Q such t h a t

Q(z)=1

while

[Q [~-r <

d/2( [ gIK + 1)M. Let us

consider z~-t(J). Obviously ;z-l(J) contains the tope- logical b o u n d a r y of J in Ma. Beeanse ] - ~ - ~ ( J ) lies off t h e Shilov b o u n d a r y the Local M a x i m u m Principle shows t h a t

[/(x(z))] <~

sup {]/(x)] [x e~r-~(J)} for all / E B.

Hence we can also find a positive measure ;t on ~ - l ( j ) such t h a t / ( x ( z ) )

=S/dR

for a l l / e B . I n particular

P(z) =~Pd2

for e v e r y polynomial P a n d since ;z-l(z) is trivial when z E T it follows t h a t the restriction of )~ to ~-~(J) A T is identical to the measure v considered in L e m m a 2. I t follows from L e m m a 3 t h a t

as r tends to 1. We also have

f=-l<j)_~ Q[lk, g- r~,ld~ < ^d/2.

N o w we obtain a contradiction since

]Q(Rrg - ~rr)(x(z))] >1 d.

L e m m a 4 shows t h a t if

z E D - K is

sueh t h a t /~(z)#O t h e n

g(x(z))=W(z)/R(z)

for all

x(z)

E~-I(z). I t follows t h a t ~-l(z) consists of one point denoted b y

x(z).

Since

g(x(z))

is bounded when

z E D - K

it follows t h a t t h e meromorphie function

G(z)=

W(z)/R(z)

is analytic in D - K. N o w it is also easy to prove t h a t even if z E D - K is such t h a t

R(z) =0

t h e n ~-l(z) consists of one point

x(z)

and

g(x(z)) =G(z).

Theorem 1 is proved.

Before we prove Theorem 2 we need the following lemma.

L e m m a 5.

Zet ~ be a compact subset o/ ( D - J )

U {0} where

J is a Jordan arc in D having 0 and 1 as endlgoints, l.t now v is a positive measure on F such that P(O)=

SPdv /or every polynomial, t,',m v is the unit point mass at O.

Proo/ o/ Theorem 2.

Suppose t h a t G has an ash-mptotie value at some point

zoEK,

t h e n we shall prove t h a t G has an unrestrictcd l!mit value at z 0. We m a y assume t h a t

z o E K - T

since if

zoET

we have already proved that, lim G(z)=lira

g(x(z))=g(z o)

as

z E D - K tends to z 0. B y assumption there exists a J o r d a n arc J such t h a t J f3 K = {z0} a n d lim

G(z)

exists as z E J , {%} tends to %. L e t us first assume t h a t the asymptotic value is different from

g(zo).

Because

g(x(z))=G(z)

when z E D - K we see t h a t lira

x(z) = x 1

exists in M s as

zEJ--{Zo}

tends to z o in D. L e t zo be the trivial point in xrl(z0). Now

x l # z o

because

g(xl)

is assumed to be different from

g(zo)

here. Sup- pose now t h a t

x2E~-l(Zo)

is such t h a t

x ~ # x 1

a n d z0. L e t us p u t F = ( Z - J ) U {z0} if Z is a closed disc around z 0 in D such t h a t J intersects the b o u n d a r y of Z. Xow we choose Z so small t h a t

Ig(x~)-g(x(z)) I

~>d > 0 when z E (Z (3 J ) - {z0}. 5Tow we choose a closed neighborhood W of x~ in M s such t h a t W lies off the Shilov b o u n d a r y a n d W is contained in 7 r l ( F ) . L e t

bW

be the topological b o u n d a r y of W in Ms. I t follows that/(x2)

= S/dv

for all / E B, where v is a positive measure on b W. I n particular

P(zo)=SPdv

for e v e r y polynomial. Because

bW

is contained in ~ - l ( F ) L e m m a 5

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J.'E. BJORK,

Rela~vel~

mammal

function algebras

shows t h a t t h e s u p p o r t of v is c o n t a i n e d in ~r-l(z0). I t follows t h a t x~ c a n n o t be a p e a k p o i n t of t h e f u n c t i o n algebra B(%) on ~r-l(z0) g e n e r a t e d b y the restriction of B to ~r-l(z0). H e n c e t h e Shilov b o u n d a r y of

B(zo)

only contains x 1 a n d z 0. I t follows t h a t ~r-l(zo) o n l y consists of x i a n d z o. We n o w investigate t h e n e i g h b o r h o o d s of x 1 in MB. L e t W be a closed B - c o n v e x n e i g h b o r h o o d of xl such t h a t W lies off t h e Shflov b o u n d a r y a n d z 0 ~ W. Suppose n o w t h a t there exist Ya 6 D - K such t h a t Ya t e n d t o z o in D while W A ~r-l(ya) are e m p t y . F o r e v e r y n we consider t h e f u n c t i o n /a = (z - y,) on W. W e see t h a t / . G

Bw,

where Bw is the function algebra on W g e n e r a t e d b y restricting B t o W. Because W is B - c o n v e x we k n o w t h a t W is the m a x i m a l ideal space of

B W.

: N o w / a is different f r o m zero on W a n d hence there exists gn 6 B such t h a t

limalga/,~--llw=O.

L e t

bW

be t h e topological b o u n d a r y of W in M s . L e t

S=Tr-I(bW),

o b v i o u s l y S is a closed subset of D a n d since ~r-l(z0) o n l y consists of x 1 a n d z 0 we see t h a t Zoq$S. Because y , t e n d s t o z 0 in D we m a y assume t h a t l/a(x)l

>~inf{]Z-ya] ]z6Si>>-d>O

^{w h e n}

x6bW.

W e m a y also assume t h a t ] g , / a - 1 ] w < l for all n. I t follows t h a t

I galbw < 2/d

a n d hence also l ga ] w <

2/d

for e v e r y n. N o w we get a c o n t r a d i c t i o n since lira,/,(xl)ga(xl)-=0 follows while lira a

[/a(xl)ga(xl)-

1 1 = 0 also holds. This shows t h a t there exists a n e i g h b o r h o o d V of z 0 in D s u c h t h a t g - 1 ( V - K) is c o n t a i n e d in W. Since z~-l(z) contains only one p o i n t w h e n z E D - K it follows t h a t t h e trivial p o i n t z o of zt-l(zo) has a n e i g h b o r h o o d U in M s such t h a t

•(U) is c o n t a i n e d in K. I f n o w

y, G D - K

t e n d to z 0 in D it follows t h a t

x(ya)Ezr 1

(Ya) m u s t converge t o x i in M s . H e n c e lira G(y~)=lim

g(x(ya))=g(xl)

which proves t h a t G h a s an u n r e s t r i c t e d limit value at z 0. W e m u s t finally consider t h e case when G has a n a s y m p t o t i c value at z 0 which equals

g(zo).

This case is simpler t h a n t h e previous a n d we c a n p r o v e t h a t ~-l(z0) is trivial. I t follows as a b o v e t h a t G has a n u n r e s t r i c t e d limit value at z o in this case too. :Now we complete t h e proof of Theo- r e m 2. L e t S s be t h e Shilov b o u n d a r y of B. L e t us p u t

W 1 = (x E K - T I G

h a s a n analytic extension t o a n e i g h b o r h o o d of x a n d

G(x)~ g(x)}.

Clearly W1 is a relatively o p e n subset of K. I f z o G W1 we can choose a n e i g h b o r h o o d U of z 0 in D such t h a t U A K is c o n t a i n e d in W 1. N o w t h e previous results show t h a t x~(U A K) = U A K a n d since

R(K)= C(K)

it follows easily t h a t U A K lies in t h e interior of $8. T h e n t h e local m a x i m u m principle implies t h a t t h e restriction of B to t h e set K 1 = K - W1 generates a p r o p e r f u n c t i o n algebra B 1 of

C(K1).

F r o m n o w on we w o r k with B 1 instead of B. W e c a n define G with respect to B 1 a n d clearly G is t h e same function as t h a t defined w i t h respect to B, i.e. we c a n represent G with a n o n zero measure on K 1 which annihilates B r I f we n o w define W 1 - W I ( B s ) with respect to

B1,

i.e. we

put WI(B1)= {xEK l - T[ G

has a n a n a l y t i c extension to a n e i g h b o r h o o d of x a n d

G(x)#g(x)},

t h e n WI(B1) is e m p t y . So n o w we assume t h a t B a n d K are such t h a t W 1 is e m p t y . L e t us n o w p u t W ~ =

{xEK - T[ G

has a n analytic extension to a neigh- b o r h o o d of x}. Clearly xE W2 implies t h a t

G(x)=9(x)

(since W 1 is e m p t y ) a n d it follows easily t h a t

W 2 ASs

is e m p t y . I t follows t h a t the restriction of B to t h e set K S = K - W 2 g e n e r a t e s a p r o p e r f u n c t i o n algebra of

C(K~).

Since we can represent G with a n y non-zero measure on K annihilating B we see t h a t K~ is t h e smallest subset of K such t h a t t h e restriction of B to K,. generates a p r o p e r f u n c t i o n algebra of

C(K2).

We shall n o w discuss h o w T h e o r e m 1 can be applied to t h e a p p r o x i m a t i o n problem on t h e u n i t interval.

Definition.

A J o r d a n arc J in t h e complex plane satisfies the reflection principle if the following holds: I f z 0 E J there exists a n open disc Z a r o u n d z 0 s u c h t h a t if G is

(7)

&RKIV FOR MATEMATIK. Bd 8 nr 3 a n y bounded analytic function in Z - J with the p r o p e r t y t h a t G has a n unrestricted limit value a t a point

z E J N Z

when G has a n a s y m p t o t i c value a t z, t h e n it follows t h a t G has a n analytic continuation to Z.

De/inition.

A J o r d a n are is almost smooth if J satisfies the reflection principle and

if R(J)=

C(J), i.e. the rational functions with poles outside J generate

C(J).

We r e m a r k here t h a t every smooth J o r d a n arc is almost smooth. We do n o t know if the condition

R(J)= C(J)

implies t h a t J is almost smooth. L e t us now consider a f u n c t i o n / e C ( I ) such t h a t

J = ( / ( x ) I x e I )

is of the form in Fig. 1. L e t us assume t h a t the two J o r d a n arcs a and b in (1) are almost smooth. Now we can prove t h a t

if g e C(I)

is such t h a t g(~)# g(~) then the function algebra generated b y [, g a n d the constant functions is

C(I).

We m a y assume t h a t

/(~)=](~)=0

while g ( ~ ) = l and g ( ~ ) = - l . L e t us p u t / 1 = / , / 2 = g ~ a n d /~=/g. On J we define

]j(z)=/j(x)

where

x E I

is such t h a t / ( x )

=z.

Obviously ~ are well defined on J . Suppose t h a t the function algebra on J generated b y / 1 , ]~,/3 a n d t h e constant functions is different from

C(J).

Now our previous results show t h a t ]~ a n d / 3 have analytic extensions from

~ J into the interior of J . Here ~ J is ~he outer b o u n d a r y of J (see Fig. 1). Call these extensions H2 and H a. On ~ J we have the relation

z~H2 = H a

and it follows t h a t

H3/z

is a bounded analytic function in the interior of J . Now we can approach 0 along ~ J in two different ways. We get lim

Ha/z=g(¼ )

from one w a y a n d lim

H3/z

g(~) from the other way. Now Montel's theorem (see [7], p. 170) gives a contradiction. I t follows t h a t / 1 , ~ , / 3 a n d the constant functions generate

C(J)

a n d then it is clear t h a t f, g a n d the constant functions generate

C(I)

too.

A C K N O W L E D G E M E N T

I wish to express here m y gratitude to Professor H. S. Shapiro who posed the problem considered in this paper. I n t h e final p a r t of this paper the application to solve the approximation problem on the unit interval the arguments are entirely due to H. S. Shapiro.

R E F E R E N C E S

1. MARKUSEVIC, L. A., Approximation by polynomials of continuous functions on J o r d a n arcs in t h e space K n of n complex variables. Amer. Math. Soc. Transl. 57, ser. 2, 144-170.

2. STOLZmVg~gG, G., Uniform approximation on s m o o t h curves. Acta Math. 111, 185-198 {1966).

3. W~.gm:R, ft., Polynomial approximation on a n arc in C a. Ann. of Math. 62, 269-270 {1955).

4. WE~MEg, J . , Subalgebras of t h e algebra of all complex-valued continuous functions on t h e circle. Amer. J . Math 78, 225-242 {1956).

5. WmaMEt~, 5., Banach Algebras a n d Analytic Functions. Academic Press, 1961.

6. G~CKSBEXG, L, Maximal algebras and a theorem of Rado. Pacific J. Math. l g , 919-941

(1964).

7. TrrC~At~SE, The Theory of FunctlorLs. Oxford.

T r y c k t den 14 maj 1969

Uppsala 1969. Ahnqvist & ~Tiksells Boktryckeri AB