A CHARACTERIZATION OF DOUGLAS SUBALGEBRAS

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A CHARACTERIZATION OF DOUGLAS SUBALGEBRAS

BY

SUN-YUNG A. CHANG

University of California at Los Angeles. L.A. Ca. 90024. USA (x)

1. Introduction

Let L ~ be the complex Banach algebra of bounded Lebesgue measurable functions on the unit circle ~D in the complex plane. The norm in L ~176 is the essential supremum over aD. Via radial limits, the algebra H ~176 of bounded analytic functions on the unit disc D forms a closed subalgebra of L ~. This paper studies the closed subalgebras B of L ~ properly containing H ~176 For such an algebra B we let B x denote the closed algebra generated b y H ~ and the complex conjugates of those inner functions which are invertible in the algebra B. (An inner function is an H ~ function unimodular on OD). I t is clear t h a t B z c B.

1%. G. Douglas [4] has conjectured t h a t B = B I for all B, and consequently algebras of the form B I are called Douglas algebras.

A discussion of the Douglas problem and a survey of related work can be found in [11]. I n particular, it is noted in [11] t h a t the maximal ideal space ~ ( B ) of B can be identi- fied with a closed subset of ~Vn(H~), and when B is a Douglas algebra, ~ ( B ) completely determines B. This means t h a t if the Douglas question has an affirmative answer then distinct algebras B has distinct maximal ideal spaces. T h a t the latter assertion is true when one of the algebras is a Douglas algebra is the main result of this paper. We prove t h a t if B and B 1 are closed subalgebras of L ~ containing H `~, if ~ ( B ) = ~ ( B 1 ) and if B is a Doug- las algebra, then B = B r Using this theorem, D. E. Marshall [9] has answered the Douglas question affirmatively.

Using functions of bounded mean oscillation, D. Sarason [13] had proved the theorem above in the special case when B is generated b y H ~ and the space of continuous functions on OD. B y similar means, S. Axler [1], T. Weight [15] and the author [3] had verified the theorem for some other specific Douglas algebras.

Section 2 contains some preliminary definitions and lemmas. The more technical aspects of the proof are in section 3 and the main theorem is proved in section 4. Some (~) Research supported in part by National Science Foundation.

6 - 762909 Acta mathematica 137. Imprim6 ]e 22 Septembrc 1976

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8 2 S U N - Y U N G A. C H A N G

readers m a y perfer reading section 4 before sections 2 a n d 3. I n section 5 we describe t h e largest C*-Mgebra c o n t a i n e d in a Douglas algebra.

T h e proof of o u r m a i n result follows a p a t t e r n f r o m Sarason's paper [12]. T h e proof of T h e o r e m 6 b e l o w uses techniques f r o m C. F e f f e r m a n a n d E. M. Stein [6]. I would like to express m y w a r m t h a n k s t o Professor D. Sarason for giving invaluable aid, a n d to Professors R. G. Douglas a n d A. Shields for v e r y helpful discussions. I a m also grateful to Professor J. G a r n e t t for re-organizing t h e paper, i m p r o v i n g t h e English a n d giving a simplified proof of L e m m a 2 below.

2.

Preliminaries

F o r an integrable f u n c t i o n

](t)

on 0D, d e n o t e t h e h a r m o n i c extension of ] to D b y

1 l "

](re ~~ = - ~ J_ P(r, 0 - t) ](t) dt

where

P(r, t) =

(1 -

r~)/(1

- 2r cos t + r 2) is t h e Poisson kernel. L e t V/(re ~~

= (~]/~x(reO),*

~//ay(re'O) ),

a n d

I V/(re'O)]~ = ] e/ /ex(re'O) [ 2 + ] ~]/~y(re~O) ] 2.

Our first l e m m a is a Littlewood- P a l e y identity.

LE•MA 1. I ] / , gEL s and at least one o//(0)

and g(0)

vanishes, then

l f~_ /(e~t)g(e")dt=I fD

2-~ ~ V](re~~ Vg(re ~0) r

log

drdO.

This l e m m a follows f r o m t h e Parseval i d e n t i t y after expressing t h e gradients in polar co- ordinates. T h e corresponding result for t h e u p p e r half plane is in [14, p. 83].

T h e second l e m m a can be p r o v e d using m e t h o d s in [6] b u t it also follows f r o m an in- v a r i a n t f o r m u l a t i o n of L e m m a 1. F o r z o = r 0 e ~~176 E D, let

(S(Oo, ro) = {re

% ] 0 - 0 o ] ~< 4(1 - r0) , r o < r < l }.

L~MMA 2. Let

s > O ,

I~ol =ro>~1/2. I ! / e i ~ , II/ll~<l and I/(~o)1 > l -~, then

~/S(Oo. ro)(1

^{- -}

^{r) lv112r} dr dO < C~

s(1 -- r0)

where C 1 is independent o] e and r o.

Proo/.

L e t

w = ( Z - Z o ) / ( 1 - 2 o Z ) = s e i~.

On

s = l , zo=roe ~~176 dqJ=P(ro, O-Oo)dO.

L e t

/(z) = F(w)

= ( 2 ~ ) - ~ _ ,

P(s, t-q~) F(t)dt.

T h e n

(2~)-~S ] F(e'~) - F(0) ] 2d~ = (2~)-~ S

P(r o, 0 -00)

]/(e '~ ) -/(Zo)[~dO

= (2~)-~S

P(ro, 0 - 0 o ) ] / ( e ' a )

] ~

dO -

I/(~0) I ~ < 2~

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

83

H e n c e b y L e m m a 1,

l-- f lVF(w)plog~lSdSd~<2e.

g~

N o w 1 - I w l ~ = ( 1 - Iz012)(1- ] z ] ~ ) / I 1 - 5 0 z ] ~ a n d w h e n

zeS(Oo, to),

I 1 - 2 o z I ~<c~(1-Iz01) for some c o n s t a n t cl, for all z. T h u s for

re~~ ro)

we h a v e

11 _-rro

~< c2(1 - I w l 2) ~< c31og [-~1 for some c o n s t a n t s c~, ca.

Because

IVF(w) l~sdsdq~ = IV/(z) 12rdrdO,

we h a v e

2 1

ffs(o.,r.)(1-r)lVl(z)l~rdrdO<c3(1-ro)ffnlVF(w)l l o g ~ s d s d e f < ~ C l ( 1 - r o ) 8 .

W e t h u s complete t h e proof.

I f I is an arc on ~D with center e u a n d length

I

= 2~, we let

R(I)={re'~ IO-tl

A finite positive measure It on D is called a Carleson measure if there exists a c o n s t a n t c such t h a t

#(R(1)) <c[I[

for all subarcs I of ~D,

LEigi~IA 3. (Carleson [2]).

Let tt be a Carleson measure on D such that It(R(I))

< c [ I [

/or all subarc8 I o/~D. Then/or 1 <19 < co

< CA. II /I1 /or all IeL~(aD), where the constant A~ depends only on p.

Following an a r g u m e n t in [14, p. 236] one can easily prove L e m m a 3 using m a x i m a l functions.

F o r a n arc

I c e D ,

let 5 =

]I[-1S~/(t)dt

be t h e average of a function / over I. F o r

/ eLI(~D),

define

W e s a y l has b o u n d e d m e a n o s c i l l a t i o n , / e B M O , if

11111, <

F u n c t i o n s in

B M O

can be related t o Carleson measures b y t h e following

L E M M A 4. (Fefferman a n d Stein).

For / E L ~(aD), the following conditions are equivalent:

(i) I e

B M O

(ii)

I]

dit = (1 - r) I

V /(rei~ 12 r dr dO, then It is a Carleson measure.

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84 SUN*YU~TG A. CHANG

Furthermore i/

~(R(~))

e = s u p - - ,

I.r[<2. ] I I

then there is a constant .4 i such that

A-~ < II/ll~, < A l e .

C

L e m m a 4 is proved in [6] for the case of upper half spaces. The proof there can easily be a d a p t e d to the present case using L e m m a s 1 and 3.

3. A distance estimate

Throughout this section we fix an non-constant inner function

b(z)6H ~176

and we set, for 0 < ~ < 1 ,

G~ = {z6D:

Ib(~)l/>

1-(~}.

F o r convenience we assume G$ c {1/2 < [z l < 1}.

L ~ M ~ 5. Let 0<e, ~<1. I / / e L ~ ,

II/ll~<l a~d I/(~)l >~1 - ~ on G~, then the measure

tt defined by

dtt = Za~(z)(1

-r)[V/(z)12rdrdO satis/ies

sup

#(R(I)) <

where C1 is the constant in Lemma 2.

Proo/. Let I

be some arc on ~D. ]~y L e m m a 2 it suffices to find points rj e *~ in G, such t h a t

G, ~ R(I)= [JjS(Oj, rj)

and such t h a t ~(1 - r j ) < ] I [ .

F o r n = 0 , 1, 2 .... and 1 <k~<2 ~, let

{In.k}

be the partition of I into closed arcs of

length [In,k] =2-~[I[.

L e t

T(In.k)={zeR(I~.k); 1 - ] z ]

>~2-~-2]I]} be the top half of

R(I~.k).

We select a subfamily 3 of {In,k} b y the rule

IjeY

if l j is a m a x i m a l arc among those I~.k for which

T(I~.k)~ G,#O.

Then

G~ N R(I)= (J~R(Ij)

and the arcs in Y have pairwise disjoint interiors.

F o r I j 6 y choose

r f ~ 6 T(Ij) A G~

with smallest modulus rj. Then

G, (~ R(Ij)= S(Oj, rj)

and 1 - r j ~ < I I j [ . Hence

G~ ~ R(I)= [J j S(Oj, rj)

and X(1

-rj) < E l lj] <. [I[.

Now consider a function f ~vith the following property:

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A CHA/tACTERIZATION OF DOUGLAS SUBALGEBRAS 85

(P1)

/ E L ~176 a n d t h e r e exist e a n d ~, 0 < e , ~ < 1 , such t h a t t h e m e a s u r e /~, defined b y d#~ =Za~(z) (1 - r ) I VII

~rdrdO

satisfies sup1

#8(R(I))/I I l `<~.

F o r e x a m p l e , a f u n c t i o n satisfying t h e h y p o t h e s i s of L e m m a 5 has p r o p e r t y (P1).

THI~OREM 6.

There is a constant C such that i/] has property

(1)1)

then

lim sup

d(/b ~ , H ~) <~ C e 1/2 .

n--~*O0

Proo/.

Since

L~/H ~

is t h e dual of H 1 = {g ell1; g ( 0 ) = 0 } we h a v e

d(/b ~, H~176 = sup { ] l ; /(e'~ b ~( e~~ g(e'~ dOl : g E H~, H g Hl <<. l }.

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B y a d e n s i t y a r g u m e n t we can a s s u m e g E H % Moreover, if u is t h e Blaschke f a c t o r of g a n d

k=g/u,

t h e n

g=k

+ k ( u - 1 ) where n e i t h e r k n o r k ( u - 1 ) h a s zeros in D. T h u s in esti- m a t i n g

d(]b n, H ~)

using (1), we can a s s u m e

geH~

a n d

g=h 2,

h e l l ~176

HhH~`<I.

Finally, replacing / b y

a] +c

w i t h l al ~< 1 does n o t h a r m p r o p e r t y (P1), so t h a t we can a s s u m e

[[/[[~ ~< 1 a n d f(O) = O.

W i t h these a s s u m p t i o n s we h a v e b y L e m m a 1,

1 9 r log ! dr

2,.

1 = f for, ^V(bna) ^dO.

⁽²⁾

Since b n a n d g are a n a l y t i c functions, we h a v e

(b~g)(z)=b~(z)g(z)

so t h a t V ( b n g ) =

bnVg+

gVb ~

on D.

W e n o w e s t i m a t e as follows:

l f fDV/" (b'Vg)rlog~drdO I

.<1 I'I" Ib~l

_{~ J J D}

Ivll Ivglrlog~ardO

_ 1~

ff ]b~l

I V / 1 2 i h l

Ih'lrlog~drdO

xl J J D

(l fro

¹ \ 1 / 2 / 4 l'l" ~ )I/2,

`<V~ Ib~l Iv/Plhprlog-drdOI l- / / Ih'prlog drdO r / \2"gj

JD

B y L e m m a 1 t h e second f a c t o r is

4 n "~112

('~r~f~rtlh-h(O)12dO)

~ '8 [I,II1) 1/2 9 (3) T o e s t i m a t e t h e first f a c t o r write

lffo'b =llV/l lhl rlog!araO=L+fo\o=S,+S,.

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86 SUN-YUNG A. C H A N G

Since

G~ c {]z I/> 1/2}

we have log

1/r ~< c(1 - r )

on G~. Using (1)1) and L e m m a 3 we then have

$1 < cA~llhH~ < ~A~lfg]ll. (4)

Also

$2< ( 1 - d )

2nl ffD IV/l~lhl2r log!drdO.

Since ]]/]]. < 2 H/I] ~ < 2, Lemmas 3 and 4 give

S 2 < (1

-cS)2n8A1A2[]gHI .

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Combining (3), (4) and (5) gives

lffDV].bn~grlog~drdO]<C(8112+(1-c~)n)]lg][ 1

(6) for a universal constant C.

We now estimate

~" ffV/'gVb"rlog!drdO=~+ ~\a =S3+S..

7g

Write

1 1 \~zu/1 1 \~/2

]Sa[< (~ffa~

[V/]Z[h]~r

l~ (~ ffa~ IVbnlzlhl2rl~ "

Since IIb-II, 42, the~e two factors can be bounded as were Sx and Sg so t h a t

]Sal <~ 4Axe~/2Az

I]g]ll. (7)

7~

For S a we again use the Schwartz inequality to get

iS41<(ff D iV/12[hlerlogldrdO~l/2((( \o~ r / \aj.\o~lVb"l~lhl~rl~

~ 1 ~ \112

"

As with the estimate for $2, the first factor is dominated b y

(8A~A2[[gU~)a/2,

and since

IW"l < ~(~ - ~)"~ IWl on D\G~, the ~econd factor does not e~ceed ~(X - ~)" 1(8A~ As IIgH~)i'~.

Combining our bound for S 4 with (7) gives

f f V/.gVb"rlog~drdO < C3(sa/2 +n(1-~)"-')[[gH1.

for a universal constant C 3.

With (6) and (2) this inequality implies

1 f/(e,O ) bn(e,O ) g(e,O ) dO < G(s ''2 +

n(1 ^-^- 0) n-x)

Ilalh

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A C H A R A C T E R I Z A T I O N OF D O U G L A S S U B A L G E B R A S 87 w h e n e v e r g ~ H ~ h a s no zeros. B y (1) a n d our r e m a r k s a b o u t g i m m e d i a t e l y following (1) we h a v e

d(/b n, H ~176 <. 3C(e ~/~ + n ( 1 - 6)~,1), a n d this p r o v e s t h e t h e o r e m .

4. A characterization of Douglas algebras

Before p r o v i n g t h e m a i n t h e o r e m we m u s t m a k e some o b s e r v a t i o n s a b o u t m a x i m a l ideal spaces. F u r t h e r details are in [11]. Because H ~176 is a l o g m o d u l a r s u b a l g e b r a of L ~176 [8], each ~0 E ~ ( H ~ has a unique r e p r e s e n t i n g m e a s u r e m~ s u p p o r t e d on ~(L~176 F o r a n y / E L ~176 we can define f(~o)= S/dmr a n d b y t h e uniqueness of m~, f is continuous on ~ ( H r 1 7 6

Of course, if for all g E H ~176 q)(g)=g(z) w i t h z e D , t h e n f ( T ) = [ ( z ) f o r / e L ~176 I f H ~ 1 7 6 B c L ~176 t h e n ~ ( B ) = {~ E 7~I(H~176 f(~v) ~(~0) = (/g) ^ (q)) for all /, g E B}. I f / e (L~176 -1 (i.e. / is a n invertible element of L ~176 a n d if ]/] = 1 a.e., t h e n we d e n o t e / - 1 = [. I f B is a Douglas algebra, t h e n ~ ( B ) = { q ~ : Iq0(b)[ = 1 w h e n e v e r b is i n n e r a n d

SeB}

(c.f. [11], [4]).

T H E O R E M 7. I / B and B 1 are closed subalgebras o / L ~176 containing H ~176 i/ ~ ( B ) = 7In(B1) and i ] B is a Douglas algebra, then B = B r

Proof. T h a t B c B 1 is n o t difficult. I t reduces t o showing t h a t 5 E B 1, w h e n e v e r b is a n inner f u n c t i o n invertible in B. B u t since ~ ( B ) = ~ ( B 1 ) , b h a s no zeros on ~ ( B 1 ) a n d as b E H ~ 1 7 6 b is invertible in B 1. H e n c e b = b -1 is in B r

T o p r o v e B l c B suppose B is g e n e r a t e d b y H ~176 a n d a f a m i l y {bz} of conjugates of in- n e r functions. F o r a n y finite set F of t h e i n d e x set {2}, let b F = H F b~, a n d let BF be t h e algebra g e n e r a t e d b y H ~~ a n d Sv- Clearly b~ e B y if % e F . W r i t e G~(bF) = {z E D: I bF(z) I >~

1 - ~ } , 0 < ~ < 1 .

L e t g e B I. A d d i n g a constant, we can a s s u m e geB11. L e t hE(H~176 -1 satisfy Ihl = ]g[

a.e. a n d let ] = g h - l E B 1 . T h e n / e B ; ~ a n d [][ = 1 a.e. I t suffices t o p r o v e ] E B 1.

Since B is a Douglas algebra, ~ ( B ) - - N { ~ ( B v ) : F ~ {bx}, F finite}. Since If[ = 1 on

~ ( B 1 ) = ~ ( B ) , c o m p a c t n e s s implies t h a t for a n y e > 0 t h e r e is a finite set F c { $ x } such t h a t Ill > 1 - e / 2 on :~l(B~). This m e a n s ]](z)l > 1 - e on s o m e region Ge(bv), 6 > 0 . I n d e e d , if t h e r e were z~eal/~(bF) with [/(zn) [ ~ < l - e , t h e n a n y cluster p o i n t ~0 of {zn} in ~ t ( H ~176 would satisfy [~(bv)[ = 1 so t h a t ~ 0 e ~ ( B ~ ) . B u t since ] is continuous on ~ ( H ~ 1 7 6 W e would h a v e a contradiction. Decreasing ~, we can a s s u m e Ge(bv)c {Iz I > 1/2}. F r o m L e m m a 5 a n d T h e o r e m 6 we now h a v e

d(/, B) < d(/, By) < d(/, b~ H ~) = d(/b~, H ~ < Ce ~/~

for s u i t a b l y large n. Because B is closed this m e a n s / ~ B.

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8 8 S U N - Y U ~ G A. CHANG

**5. A description of the largest C*-algebra contained in a subalgebra**

Suppose B is a closed subalgebra of L ~ p r o p e r l y c o n t a i n i n g H ~176 The largest C*-algebra c o n t a i n e d in B is t h e algebra B N/~ w h e r e / ] denotes t h e space of complex conjugates of functions in B. The proof of T h e o r e m 7 yields a description of t h e functions in B (I/~

when B is a Douglas algebra. I n view of t h e paper [9] this description of B f//~ is valid whenever H ~ c B c L ~

T~V~OREM 8. Suppose B is a Douglas algebra. L e t / E L ~. T h e n / E B N B i / a n d only i/

/ satis/ies

(P~) /or every s > 0 there is an inner/unction b e B -1 and there is (~, 0 < 8 < 1 such that the measure d/z =Zc~(b)(z)(1 - r) I V/12rdrdO satis/ies /z(R(I)) <~e I I I / o r all subarcs I o/~D.

Proo/. Suppose / satisfies (P~). T h e n for a n y s > 0 there is b E B -1 so t h a t b y T h e o r e m 6, d(/, bnH~176 1/2 w h e n n is large. H e n c e / E B . Since [ also satisfies ( P 2 ) , / E B N B.

On t h e other h a n d , if / E B N/~ a n d I / I = 1, t h e n the proof of T h e o r e m 7 shows t h a t / has (P~). Being a C* algebra, B N/~ is t h e closed linear span of t h e u n i m o d u l a r functions in B N/~. A n d b y L e m m a 4 a n d t h e inequality IigiI.~<21[gll~, t h e space of functions in L ~176 h a v i n g (P2) is u n i f o r m l y closed. Hence each / E B N/~ has (P2).

I n t h e special case b = z , t h e closed algebra g e n e r a t e d b y H ~ a n d ~ is actually t h e space H ~176 + C ([7], [11]). T h e o r e m 8 t h e n gives t h e description f r o m [12] of (H ~ + C) N (H~176 C) as V M O N L %

References

[1]. A~LER, S., On some properties of H ~ +L~. Proprint.

[2]. CARLESO~, L., Interpolations by bounded analytic functions and the corona problem.

Ann. o/Math., 76 (1962), 547-559.

[3]. CHANO, S. Y., On the structure and characterization of some Douglas subalgobras. To appear in Amer. J. Math.

[4]. DOUGLAS, R. G., On the Spectrum of Toeplitz and Wiener-Hopf Operators. Prec. Con- /erence on Abstract Spaces and Approximation, (Oberwolfach, 1968), I.S.N.M., 10 (birkhauser Vorlag, Basel, 1969), 53-66.

[5]. - - Banach Algebra Techniques in Operator Theory, Academic Press, New York 1972.

[6]. FEF~'E~MAN, C. & STEIN, E. iYI., H p spaces of several variables. Acta Math., 129 (1972), 137-193.

[7]. HELSON, H. & SARASON, D., Past and Future. Math. Scand., 21 (1962), 5-16.

[8]. HOFFMAN, K., Banach space o/analytic /unctions. Prentice Hall, Englewood Cliffs, N.J., 1962.

[9]. M ~ S H ~ L , D., Subalgebras of Z ~176 containing H ~ Acta Math., 137 (1976), 91-98.

[10]. SARASON, D., An addendum to 'Past and Future'. Math. Scand., 30 (1972), 62-64.

[11]. - - Algebras of functions on the unit circle. Bull. Amer. Math. Soc., 79 (1973), 286-299.

[12]. Functions of vanishing mean oscillation. Trans. Amer. Math. Soc., 207 (1975), 391-405.

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A CHARACTERIZATION OF DOUGLAS SUBALGEBRAS 89 [13]. STEIN, E . M . & WEISS, G., Introduction to Fourier Analysis on Euclidean Spaces. Princeton

Math. Series, 1971.

[14]. STEIn, E. M., Singular Integrals and di]/erentiability Properties o] Functions. Princeton Math. Series 1970.

[15]. WEIGET, T., Some Subalgebras of L~(T) determined b y their m a x i m a l ideal spaces. Bull.

Amer. Math. Soc., 81 (1975), 192-194.

Received August 29, 1975

A CHARACTERIZATION OF DOUGLAS SUBALGEBRAS