mmm mmmmm

S U B A L G E B R A S O F L ~ C O N T A I N I N G H ~

B Y

DONALD E. MARSHALL University of California, Los Angeles, CA., USA

1. Introduction

L e t H ~176 be the algebra of bounded analytic functions on D = { z : Iz] <1} and let L ~176 be the Banach algebra of bounded measurable functions on T = {z: ]z] =1} with the uni- form norm. Then H ~176 can be regarded as a uniformly closed subalgebra of L ~176 b y identifying each /E H ~176 with its b o u n d a r y function.

I f A is a closed subalgebra of L r176 let ~ [ A ] denote its maximal-ideal space. K. Hoff- m a n [13] has shown t h a t each TE 7~t~[H ~176 has a unique norm-preserving extension to a bounded linear functional on L % For example, if z E D t h e n evaluation a t z is an element of 7tl[H ~176 and its extension is given simply b y the Poisson kernel. Now if A is a closed subalgebra of L ~ containing H ~176 t h e n the usual Gelfand topology on 7iliA] agrees with the weak-* topology t h a t ~ [ A ] inherits as a compact subset of the dual space of L % Consequently, each ] EL ~176 is continuous on ~ [ H ~176 a n d harmonic on D. Moreover, if A and B are closed algebras such t h a t H ~ 1 7 6 ~176 t h e n ~ [ H ~ ] ~ [ A ] ~ ~ [ B ] ~

~[L~176 Our m a i n result is the following theorem:

TI~EOREM 1. Let A be a closed subalgebra o/JL ~176 containing H~ Let A t be the closed subalgebra o / A generated by H ~~ and { l - l e A : / e H ~ } . Then ~ [ A z ] = ~ [ A ] .

When combined with a recent result of S. u Chang [7], Theorem 1 proves a conjec- ture of R. Douglas [9]. To state Douglas' conjecture, we let Q be a subset of L ~ and write [H ~ Q] for the uniformly closed subalgebra of L ~176 generated b y H ~176 and Q. An algebra of the form [H ~176 Q], where Q ~ {u: [ul = 1 a.e. on T and ~ e H ~176 is called a Douglas algebra.

Since each positive function in (L~) -1 is the modulus of a function in (He~ -1, we see t h a t Az is a Douglas algebra whenever H ~ 1 7 6 ~176 and we see t h a t if A is a Douglas algebra, t h e n A =Az. Douglas' conjecture was t h a t every uniformly closed subalgebra A of L ~~

containing H ~ is a Douglas algebra, or, equivalently, t h a t every such algebra A satisfies A = A t. N o w S. Y. Chang has proved t h a t if A is a closed algebra lying between H ~176 a n d

92 D O N A L D E. M A R S H A L L

L ~ and if B is a Douglas algebra with ~?/[B] = 7//[A], t h e n A = B. I n light of this result, Theorem 1 has the following consequence.

T a E O R E M 2. Every closed algebra A , such that H ~ 1 7 6 A c L r176 is generated by H ~176 and { a e A : u is an i n n e r / u n c t i o n in H~176

Douglas' conjecture arose from the s t u d y of operator algebras generated b y Toeplitz operators. I t has been discussed b y several authors: S. Axler [1], [2], S. u Chang [6], A. M. Davie, T. W. Gamelin, a n d J. Garnett [8], R. Douglas [9], R. Douglas a n d W. l~u- din [10], D. Sarason [15], [16], [17], T. Weight [18]. I would like to t h a n k J. Garnett f o r invaluable discussions.

2. Interpolating Blaschke products

Z oo

We call a sequence { n}n=l in D an interpolating sequence if for every bounded sequence {wn}~=l, there is an / in H ~ such that/(zn) =wn for all n. E v e r y interpolating sequence m u s t satisfy Z ( 1 - [ z ~ [ ) < 0 % and so is the zero sequence of a Blaschke product. W e call a Blaschke product whose zeros form an interpolating sequence, an interpolating Blaschke produc~.

A finite measure ~ on the upper half-plane, H +, is called a Carleson measure if there is a constant C for which # ( S ) < C~, whenever S is a square of the form S = {x + iy: x o <~

x ~< x 0 + 6, 0 < y < ~}. The analogous definition is m a d e for D, where squares are replaced b y sectors of the form S = { r e % 1 - ( ~ < r < l and 00~<0<00+~ }. A n y rectifiable curve F in H + or D induces a measure on H+ or D, respectively, b y defining the measure of a Borel set S to be the length of F • S. The hyperbolic distance between two points is defined b y

z - w H+

V ~ ] on O(z, w) = Z--'W I

o n D .

Interpolating sequences can be characterized in the following w a y [12]. On H +, a sequence

Z r

{ ~} ~-1 is an interpolating sequence if and only if there exists an e > 0 such t h a t 0(z~, z~) ~>

for n=~m and the measure E ( I m z~)~z~ is a Carleson measure. Here ~ denotes the point mass at zn. On D, we replace the measure with Z(1 - I z n I)6z~.

3. Proof of Theorem 1

W e claim t h a t it suffices to prove the theorem for algebras of the f o r m A u = [H ~, u, ~]

where u is a unimodular function in L % To see this, note t h a t a n y algebra A containing

S U B A L G E B R A S O F L co C O N T A I N I N G H co 93 H ~ is generated b y its invertible elements. Now, if lEA -1, there exists g 6 (Ho~ -1 such t h a t Ill = Igl a.e. Then u=]g-land ~t=9] -1 are unimodular, and we see t h a t A is ge- nerated b y H ~ a n d G={uEA: u is unimodular and ~EA}. I t is easy to see t h a t ]/[[Au] = { ~ e ~ [ H ~ ] : I~(u)] = 1 } and ~ [ A ] = { q 0 e ~ [ H ~ ] : I~(u)] = 1 for all u in G}. Now (An)it A I c A for all u in G, so t h a t ~ [ A ] c ~ [ A ~ ] ~ N uEa ~[(Au)z]. I f ][/l[(Au)t] = :ff/[Au] for all u in G, we have ~ [ A ] ~ ~ [ A 1 ] c fl uEa]l'l[Au] = ~ [ A ] . Thus ~ [ A ] = ~ [ A I ] . This proves the claim.

F o r the remainder of this discussion, let u be a fixed, nonconstant, unimodular func- tion in L ~ and let Au = [H ~, u, ~]. F o r each ~, 0 < ~ < 1, we wish to find an interpolating Blaschke p r o d u c t B~, such t h a t

There exists ~ < 1 such t h a t if B~(z)= 0, then -<8. (3.1)

If < t h e n I < 10" 1 (3.2)

Assuming we can do this for the m o m e n t , we prove our theorem as follows. L e t B = [H% {/~} 0 < ~ < 1}. Suppose ~E ~[A=], and suppose q ( B ~ ) = 0 for some ~. K. H o f f m a n [13, p. 206] has shown t h a t since B~ is interpolating, ~v is in the closure of the zeros {z.}

of B~. B y (3.1) above, ] u(z~) ] ~<fl < 1, so t h a t IF(u) I ~<fl < 1, contradicting the assumption t h a t ~E ~[A~]. So q ( B ~ ) 4 0 for each qE 7~/[A=] a n d each B~. Thus each B~ is invertible in A=. We see now, t h a t B c (A=)z, so t h a t 711[B]~ ://I[(A=)i]~ 7/l[A=]. F o r the opposite in- clusions, note t h a t ~ [ B ] = { F E ~[H~~ I (B )I = 1 for each ~}. Suppose ~vE ~ [ B ] and B y the corona theorem [4], there exists a net {z~} in DI such t h a t zv con- verges to q0. There exists a $0, such t h a t if ? >Y0, t h e n I < ~. B y (3.2) above, I B~(%,)I <~

1/10 for y >~0, and we see t h a t I (B )I ~< 1/10. This contradiction implies t h a t I (u) l = l, so t h a t ~v E ~ [ A j . We have now shown t h a t ~ [ B ] = ~[(Au)z] = ://I[A=].

I t remains to find B~, given u and ~. We will surround the places where [u] < ~ b y contours F which induce a Carleson measure. This construction comes from the proof of the corona theorem [4], b y L. Carleson. We will then uniformly distribute, in the ~-metric, a sequence {z~} on the contours. Our interpolating Blaschke product will h a v e {z~} as its zeros. Our method is v e r y similar to S. Ziskind's [19], except t h a t we work with a bounded harmonic function with unimodular b o u n d a r y values and we give several technical simpli- fications.

4. Preliminaries to the construction

The construction is best explained in the upper half-plane H +. So suppose u is a bounded harmonic function on H + with unimodular b o u n d a r y values, a n d fix ~, 0 < ~ < 1.

94 D O N A L D E . M A R S H A L L

L ~ M A 1. There exists an ~ ' < 1 , such that i/infa lu(z)l < ~ / o r some rectangle o/the form R~-{x+iy: xo <~x<xo+(~, ~12 ~<y <(~}, then sup~ lu(z)l <o:'.

Proof. B y a translation and a dilation, we can assume x 0 -=0 and (~ = 1. The result now follows by a normal families argument.

L e t B be a square of the form {x+iy: xo<~x<<.xo+(~ , 0<y~<(~]. Find f l < l such t h a t (1 -fl)](1 - a ' ) < 1 0 -a. F o r a set U c R , let I U] denote its measure.

L E M M A 2. Suppose xo<~Rea<xo+(~ and ~ / 2 < I m a ~ < ~ and lu(a)[ >/9. Let E = { x + i y e S : ]u(x+iy)] <~'}. Let E* be the vertical projection o/ E on {x+iy: y = 0 } . Then

I E*I <a/2.

This lemma is essentially proved in [12] and in [4].

Proof. B y a translation and a dilation, again, we can assume x0=0 and (~ =zr/2. L e t $ be the strip {x + iy: 0 <y <xt} and let ~(z)= e z. Now r maps $ one-to-one and eonformally onto the upper half-plane H+. First we assume E is bounded b y a finite number of J o r d a n curves. Let m and to 1 be the harmonic measures for E relative to S \ E and ~(E) relative to H+k~(E), respectively. L e t w* and to~ be the harmonic measures for E* relative to 5 and

~(E)* relative to H +, respeotively. Here ~(E)* is the circular (clockwise) projection o f ~ ( E ) on {[m z=O}. Hall's lemma [11, p. 208] and an elementary estimate show that

w(a) = eox(~(a)) >~ 2/3(o*(~(5 +xi)) >~ 2 x 10-'[ E*[.

Notice t h a t on the boundary of $\E, lu(z)] ~<l-(o(z)+ako(z). B y the maximum principle, the inequality persists on S\E. So fl < 1 - o ~ ( a ) +a'eo(a) and we conclude t h a t

IE*I

Now for an arbitrary E, we can cover the compact set {zCE: [u(z)l ~ : r I m z >~ l/n} b y a finite number of halls contained in E. Let E~ be the complement of the unbounded component of the union of these balls. Apply the above reasoning to each E n and E*.

5. The construction of F

Let S(~ 0<y~<l}. Partition the bottom half of S (~ into two squares with sides of length 1/2. Partition the bottom half of each of these squares into two more squares with sides of length 1/4. Continue the process indefinitely. We wish to describe two procedures which we will apply to a subcollection of the squares in S (~ For S a square in S (~ let Tz be the top half of S.

S U B A L G E B R A S O F L ~176 C O N T A I N I N G H ~ 95

mmm mmmmm

I I I I ~ I I I I ~ I

I l n I I i l ~ I I I i I ~ I I ~ l

C a s e 1.

I f S is a square c o n t a i n e d in S a n d [u(z)[ < a for some z in T~, shade S unless it is a l r e a d y shaded. N o t e t h a t b y L e m m a 1, supr~ [u[ < a ' < f l a n d b y L e m m a 2,

:~ IOl~<~ls*l. t5.1)

shaded

Case 2. SUpTs ]U] </~.

I f ~ is a square c o n t a i n e d in S a n d suprz lul >fl, shade S unless it is a l r e a d y shaded.

L e t Rs = S \ U ~ shaded S a n d n o t e t h a t

I~R~l <6Is*t.

P r o c e e d as follows. A p p l y t h e a p p r o p r i a t e case to S (~ o b t a i n i n g s h a d e d squares S(~ 1), S(~ '), S(31) ... On each S~ '), a p p l y t h e a p p r o p r i a t e case, o b t a i n i n g d o u b l y s h a d e d squares S(12), -~2), S(e)8, .... R e p e a t this process indefinitely. Observe t h a t we a l t e r n a t e cases in passing f r o m one s h a d e d s q u a r e t o a s h a d e d d e s c e n d a n t . Define F as t h e union of t h e b o u n d a r i e s of t h e Rs o b t a i n e d f r o m applications of Case 2. T o see t h a t F induces a Carle- son measure, it suffices to check I F N S] ~< C IS* [ where S is a square in t h e grid on S (~

B y (5.1) a n d (5.2), we see t h a t

Ir n sl ~< ~ 6 • 2-~ls*l = 121s*l.

n = 0

96 D O N A L D E . M A R S H A L L

F

k\\xtx\\ ~\,b,\x~ \,,\~ _A ~,~Xl_ \ ~ ' ~ , \ ~Nxx~ x\x~ A X-x,xkx,\~

C a s e 2.

F

N o t e t h a t a n y p o i n t in S (~ for which [u(z) [ < e will be in some Rs. Also, [ OR s f) { I m z =0}1 = 0 . This follows since u has u n i m o d u l a r vertical limits a.e. a n d anyi'point in ORs ~ { I m z = 0 } is a point where lim~_~o sup [u(x + iy) l <<.fl < 1.

6. T h e c o n s t r u c t i o n 0[ B~

I n this section, we will first consider t h e behavior of a Blaschke p r o d u c t whose zeros are located on F ~ S (~ Choose e < l / 1 0 a n d place points an on P so t h a t each z in I" satisfies Q(z, an) < 2 e for some n a n d so t h a t ~(%, am) ~>e if n4=m. I t is shown in [19] t h a t {an} is a n i n t e r p o l a t i n g sequence. L e t B be t h e Blaschke p r o d u c t whose zero sequence is (an}. W e wish t o v e r i f y (3.1) a n d (3.2) on S (~ B y c o n s t r u c t i o n (3.1) holds. H z E S ~~ a n d [u(z)[ < ~ , t h e n z is in some R s. B u t [ B[ < s on 0 R s \ { I m z = 0} a n d OR s Cl { I m z = 0} has h a r m o n i c meas- ure zero as a subset of ORs, since it has length zero. W e conclude t h a t

IBI

W e n o w wish to c o n s t r u c t B~. L e t u be a u n i m o d u l a r f u n c t i o n in L~176 F o r k = 0, 1 . . . . , 7 , let

SUBALGEBRAS OF LeO CONTAINII~G H ~ 97 a n d let {an.k} be t h e zeros o b t a i n e d b y a p p l y i n g t h e p r o c e d u r e described a b o v e t o t h e f u n c t i o n u o v ~ 1. W e can find a subset {zm} of [Jk.n V~l(a~.~) such t h a t @(z m, zz) >~, for m # l , a n d such t h a t for each m, there is a n l for w h i c h O(zm, z~) <3e. L e t B a be t h e Blaschke pro- d u c t whose zero sequence is (zm}. T h e n B a is a n i n t e r p o l a t i n g Blaschke p r o d u c t , a n d we h a v e t h a t (3.1) a n d (3.2) h o l d i n Iz] >~1]2. I f ]u(z0) ] < g f o r some z 0 w i t h ]%] < 1 / 2 , increase t h e zero sequence of B a w i t h a finite n u m b e r of distinct zeros in a n e i g h b o r h o o d of z0, so t h a t l B~(z)[ < 1 / 1 0 for all

lzl

7 . F u r t h e r r e s u l t s

I n view of S. Y. C h a n g ' s result, we h a v e shown t h a t e v e r y subalgebra of L ~176 contain- ing H ~ is g e n e r a t e d b y H ~ a n d ( / ~ E A : B is a n interpolating Blaschke p r o d u c t ) . W h i c h algebras are of t h e f o r m [H ~, B], where B is a n i n t e r p o l a t i n g Blaschke p r o d u c t ? I f / is a simple function, it is possible t o see t h a t [H ~ , / ] = [H ~, B] for some interpolating Blaschke p r o d u c t B. I f A i s g e n e r a t e d b y H ~ a n d a countable collection of L ~ functions, t h e n A = [H ~ UI2], where U a n d V are inner f u n c t i o n s a n d / ) V CA. Such an algebra is c o n t a i n e d in some algebra of t h e f o r m [H ~,/~], b u t it is n o t clear w h e t h e r A = [H ~, B] or not.

R e f e r e n c e s

[1]. ~ L E R , S., Algebras generated b y H ~ and characteristic functions, Freprint.

[2]. - - Some properties of H ~ + L E, Preprint.

[3]. CARLESON, L., A n interpolation problem for bounded analytic functions. Amer. J . _Math., 80 (1958), 921-930.

[4]. - - Interpolations b y bounded analytic functions and the corona problem. A n n . Math., 76 (1962), 547-559.

[5]. - - The corona theorem, Proceedings o / 1 5 t h Scandinavian Congress, Oslo, 1968. Lecture l~otes in Mathematics. 118, Springer-Verlag.

[6]. CHANG, S. Y., On the structure and characterization of some Douglas subalgebras. To appear in Amer. J . Math.

[7]. - - A characterization of Douglas subalgebras. Acta Math., 137 (1976), 81-89.

[8]. DAVIE, A. M., GAlVrELHq, W.W. & GARZVETT, J., Distance estimates and pointwise bounded density. Trans. Amer. Math. Soc., 175 (1973), 37-68.

[9]. DOUGLAS, R., On the spectrum of Tocplitz and Wiener-Hopf operators. Proe. Con]. Abs- tract Spaces and Approximation (Oberwol/ach, 1968), Birkh/~user, Basel, 1969, 53-66.

M R 41 //4274.

[10]. DOUGLAS, 1~. & I~UDIN, W., Approximation b y inner functions. Paci]ic J . Math., 31 (1969), 313-320. M R 41 #4275.

[11]. DuREI~, P., Theory of H ~ Spaces, Academic Press, New York and London, 1970.

[12]. GAR)~ETT, J., Interpolating sequences for boundedharmonic functions. Ind. Univ. Math. J., 21 (1971), 187-192.

[13]. HOFFMA~, K., Banach Spaces of Analytic Functions. Prentice-I-Iall, Englewood Cliffs, New Jersey, 1962.

[14]. OHTSUXA, M., Dirichlet Problem, Extremal Length, and Prime Ends. Van l~ostrand, New York, 1970.

7 - 762909 Acta mathematica 137. Tmprim6 le 22 Septembre 1976

98 DOEALD E. MAI~SHALL

[15J. S ~ A s o ~ , D., Algebras of functions o n the u n i t circle. Bull. Amer. Math. Soc., 79 (1973), 286-299.

[16]. - - Approximation of piecowise continuous functions b y quotients of b o u n d e d analytic functions. Canad. J . Math,, 24 (1972), 642-657.

[17]. - - F u n c t i o n s of vanishing m e a n oscillation. Trans. Amer. Math. Soc., 207 (1975), 391- 405.

[18]. WEIGHT, T., Some subalgebras of L ~ ( T ) determined b y their m a x i m a l ideal spaces. Bull.

Amer. Mash. Soe., 81 (1975), 192-194.

[19]. ZmKI~D, S., I n t e r p o l a t i n g sequences a n d the Shilov b o u n d a r y of H~(A). To appear i n J .

~unctionat Anal.

Received August 29, 1975