(s]q)/(s/q) is the Helson constant of an infinite Kq-set, then A ( ~ Kq) is a tensor algebra.

Representations of tensor algebras as quotients of group algebras

STEN KAIJSER

lnstit,ut Mittag-Leffler, Djursholm, Sweden

w O. Introduction

1. Tensor algebras, or to be precise, projective tensor products of C(K)-spaces have impo~%ant relations both with I~ilbe~ space and with 11. The relation with Hilbert space was discovered b y Grothendieck, and was called b y him >>the funda- mental theorem on the metric theory of tensor products>>. I t certainly is the deepest result in this metric theory [see e.g. 6]. The relations with 11 were discovered b y Varopoulos, and their importance lies in the fact t h a t t h e y relate the algebra structure of tensor algebras to the algebra structure of group algebras [8]. These relations are two-fold, in the first place, a group algebra can in a canonical w a y be embedded as a closed subalgebra of a tensor algebra. Through this embedding, information on tensor algebras can be obtained from information on group algebras.

In the second place a tensor algebra can be represented as a quotient of a group algebra, so that information on tensor algebras can be transferred to group algebras.

The main result in the second connection is the following; if {K~}~= 1 are disjoint compact subsets of a compact abelian group, and if O K~ is a Kronecker set (or a K f s e t ) , then A ( ~ K~) is a tensor algebra. In this paper we shall consider to what extent the ]~Lronecker condition in the theorem of Varopoulos can be replaced b y I-Ielson conditions on the sets. Our main results are the following.

T~EO~E~ A. To every natural number n ~ 2, there corresponds a real number

~ , such that i f {K~}~=I are disjoint compact subsets of a compact abelian group, and i f U K~ is a H e l s o n - ( 1 - ~) set, or < o;n, then A ( ~ KI) is a tensor algebra.

THEOI~nM B. To every natural number q > 2, and every natural number n > 2, there corresponds a real number aq, n such that i f {K~}~I are disjoint compact subsets

1 0 8 STEiW K A I J S E I ~

of the group Dq, and O Ki is a Helson-(yq- a) set, where ~ < ~q,, and

rq -- sin

(s]q)/(s/q) is the Helson constant of an infinite Kq-set, then A ( ~ Kq) is a tensor algebra.

The proofs of these theorems are based on the metric theory of tensor algebras.

The main tool in the proof of Theorem A is a rather elementary measure inequality with which we can s t u d y the dual space of a tensor algebra.

To prove Theorem B we shall combine the methods used to prove Theorem A, with a theorem ef Bohnenblust and Karlin on the geometry cf Banach algebras.

However, besides this we shall also have to s t u d y various possible definitions of gelson-eonstants for subsets of Dq.

Loosely speaking, the proof of Theorem A is based on a notion of weak approxi- mation, while the proof of Theorem B is based on uniform approximation.

w 1. Tensor algebras in general groups

1. We shall start b y some standard definitions and notations. Let G be a compact abelian group and let E be a closed subset of G. We shall denote b y

A(E)

the restriction of

A(G)

to

E. A(E)

can also be represented as the quotient

A(G)/I(E),

where

I(E)

is the closed ideal of all functions f in

A(G)

with f-l(0) D E. I t follows from this representation t h a t

A(E)

is a Banach algebra with maximal ideal space E, and t h a t every element f in A (E), has an expansion

f(x) ~- ~ axz(x),

Z e G, ~ lax] < IlfllA(E) + ~, X e E . (1.1.1) The dual space of

A(E)

will be denoted

PM(E),

and its elements will be called pseudomeasu_res.

Let X be a compact space (all topological spaces considered in this paper will be assumed to be H a u s d o r f f spaces). We shall denote b y

(i) l x the identity element of

C(X)

(ii)

C(X)I

the unit ball of

C(X)

(iii)

S(X)

the group of all functions

f e C(X)

with If[ = 1 (iv)

S~(X)

the group of all functions

f e C(X)

with f~ : 1.

With the uniform topology,

S(X)

is a topological (abelian) group under point- wise multiplication of functions, and

Sq(X)

is a discrete subgroup of

S(X),

which separates the points of X if and only if X is totally disconnected.

Let E be a closed subset of a compact abelian group G. We shall call E

a) a Kronecker set,

if GIE is dense in

S(E).

b) a

Kq-set, if G[E Sq(E).

c) a

Helson-(a) set,

if

A(E) = C(E),

and if for every f in

C(E)

[If tiC(E) ~

aIIfI1A(E),

I:~I~3pI:~ESENTATIONS Oli' T E N S O R A L G E B R A S A S Q U O T I E N T S O F G R O U P A L G E B R A S 1 0 9

or e q u i v a l e n t l y b y d u a l i t y , if for e v e r y measure /x w i t h support on E ,

2 E G

I n dealing w i t h Helson-(~) sets, ~ < 1, we shall in this paper often m a k e t h e technical assumption, t h a t for e v e r y ~ E M ( E ) , /x # O, we can in fact f i n d Z r ~, such t h a t I~(X) l > aII/zftM(s) - N o t a t i o n a l l y this a s s m n p t i o n m e a n s t h a t we can avoid certain e-quantities, a n d conceptually it means, a t most, t h a t we consider a Helson-(~) set as a Helson-(a') set, for some ~' < ~.

L e t {X~}f~z be c o m p a c t spaces. We shall d e n o t e t h e i r cartesian p r o d u c t b y X a n d t h e i r disjoint union b y X ' . L e t fltrther fi E C ( X , ) , we shall define functions f E C ( X ) a n d f ' E C I X ' ) b y rasp.

I(x) = f(x x, x., . . . . , Xn) = fl(zl)f2(x2) . . . f,,(X,) (I.1.2) a n d

f ' l x , = f f . (1.1.3)

W e shall denote b y

(i) ] - [ (X) t h e set o f all functions fi E C(Xi)x for all i.

f E C(X) d e f i n e d b y (1.1.2), such t h a t (ii) g ( X ) t h e projective tensor p r o d u c t C(X1) ~ C(Xe) ~) . . . ~) C(X,,) [S].

V(X) is a semi-simple B a n a c h algebra w i t h m a x i m a l ideal space X, a n d every e l e m e n t F E V(X), has a representation

~ i

F@) = ~ a~t~(z), t~ e

]-[ (X), Ia~l < llFllv~x~ + ~.

k = l k = l

I t is well-known t h a t convex linear combinations of S ( X ) are dense in C(X)I, so t h a t in t h e above representation we m a y in fact assume fn E ] - [ (X), Ifif ---- 1.

A n algebra isomorphic to Some V(X) will be called a tensor algebra.

The dual space of V(X) will be d e n o t e d B M ( X ) , a n d is canonically identified

= X

w i t h t h e space o f continuous e-linear forms on ]-[i=~ C ( i ) . An element of B M ( X ) will he called a m u l t i m e a s u r e of order n, a m u l t i m e a s u r e o f order 2 is called a bi- measure. F o r A E B M ( X ) , we have IIAIIBM = s u p { [ A ( / ) [ t / E ~ - / ( X ) } , a n d we should like t o p o i n t out, t h a t in c o n t r a s t to the similar case for measures, we can n o t t u r n t h e sup i n t o a m a x bY going over to Borel functions.

L e t A E B M ( X ) , a n d let f j E C ( X j ) , 1 < j < _ < n , j : / : i . The functional on C(X~), t a k i ~ g ~ E C(X~) into (f~ | f~ | . . . . | fi_~ Q q~ | fi+~ | .... | f,,), is t h e n linear a n d b o u n d e d a n d is b y t h e Riesz representation t h e o r e m given b y a measure on Xi which w e s h a l l denote b y A~(fl | 9 9 9 ~ _ j | fi+i | 9 9 9 | f~). I t is clear t h a t

liA,<fx | . . . . | f , - , | f,+~ | . . - | f=)[[MtX,) ~ [tA[IB~" [Ifx[loo.-. [[f,,[[~ -

110 STEN ~:AIJSEI~

We have thus interpreted A as an ( n - 1)-linear operator into M(X~). I n the same w a y we can consider A as a multilinear operator from the products of a n y of the C(X)'s into the multimeasure space over the rest. I t is clear t h a t all these operators have the same norm, n a m e l y

IIAIIBM.

Let now A E B M ( X ) , let fi C C(Xd,, and define f E C(X) b y (1.1.2). We shall denote

2,(f) ---- A~(fl | .. 9 | fi_l @ f,+l | 9 .. | f.) 9 (1.1.4) Since we m a y consider each X~ as a subspaoe of X', we m a y also consider ~he

~,'s as (mutually singular) measures on X', and we can therefore define 2'(f) e M ( X ' ) b y

~.'(f) = 21(t) 4- ~,~(I) + . . . + 2 , , ( f ) . ( 1 . 1 . 5 ) VVe also observe t h a t since we have for each Xi II~i(f)llM(Xl)_< llAIIBM we have

IIZ(t)ilM<x,) < n llAli,mx).

I t is also obvious, b u t nevertheless important, t h a t BM(X) is a module over each C(X~) if we define the multiplication as follows: Let A E BM(X) a n d let ei E C(Xd, we define eiA, b y

(e~A)(fl | | J~ | . . , | f , ) = A ( f l | | e,f, | . . . | f~), A e O(Xj),

l < _ j < n . (1.1.6}

Clearly we have then Ile~AI[BM ~< [Ie~I[~ " [IAllBM-

The simplest example of a closed subset E of a group G, for which A ( E ) is a tensor algebra is the following:

L e t E i c Gi be Helson-(~i) sets, and let E = l - [ E l , then A(E) = A ( " ~ G~)/I(E), is a tensor algebra, and for every f in A ( E ) we have

tlflIv(E) ~ [ ~ ~ " HflIA(E) "

The above example works however only in product groups; and the importance o f the theorem of Varopoulos lies in the fact t h a t it does not, require the group to be a product. 'On the other h a n d it does require a ~>product seb> in a n arbitrary group, and such a thing is found in the following way.

L e t {K~}~ ~ be a set of closed subsets of a compact abelian group G, denoting their cartesian product b y K, we have K ---- ]-[ K~ c ]--[ G = G ~. L e t further s be the group addition map taking the point (gv g2 . . . g~) C G" to ~he point g l + g ~ + . . . - t - g n C G . The image of the set K under the. map s is usually called t h e s u m of the sets Kf, and we shall denote it b y ~ K ~ . The map s is a continuous group h o m o m o r p h i s m , and induces a map ~ from G into G ~ = G'.

B y restriction, the set of functions GlzK~ is mapped into a subse~ of the set o f

^ ,

funetioI~s G IK- Exte~ading ~ by lineari*y ~ e have a map, which w e shall also,

~ E P R E S E N T A T I O N S O ~ T E N S O ~ G E B R A S A S Q J ~ O T I N N T S O ~ G ~ O U r A L G E B R A S 111

denote by ~ of A ( ~ K ~ ) i n t o A(K). Identifying A ( ~ Kd with its image u n d e r in A(K), we m a y consider A ( ~ Ki) as the space Ao(K ) of all functions f in A(K), which have an expansion

f(ic) = ~ a,z,(ic), ~

ia, l < ~ , k = (ICl, I c e , - . . , Ic~), (1.1.7) and with )/i E s(G). ~ is the diagonal imbedding of G into G ~, and is of co~rse the set of all characters Z in G~, for which

z ( I c ) = z(Ic~ + Ic~ + - - 9 + Ic~), Ic = ( I c . Ic2,- 9 9 Ic~) 9

The natural embedding of A~(K) into A(K) is of course normdeereasing. We have further

A(K) : A ( K ~ • 2 1 5 2 1 5 =-

= A(K~) ~ A(K2) ~ . . . Q A ( K , ) c c(g~) ~ C(K2) Q . . . Q C(K~) = V(K) and .this gives a natural normdecreasing injection of A(K) into V(K). We shall denote b y T the composite of ~ and this natural i~jeetion. T is t h e n a norm- decreasing injective algebra homomorphism of A ( ~ Ki) into V(K), or equivalently of A~(K) into V(K).

2. To prove our main results we shall need the following two theorems from the mer t h e o r y of tensor algebras. These theorems will be proved in the next two sections.

THI~ORE~ 1.1. For every natural number n > 1, there exists a continuous real- valued function e,(x, y), 0 < x < 1, 0 < y ~ 2n, such that sn(O, O) = O, e,(x, y) is concave and increasing in each variable, and having also the following property:

Let {Xi}i~ 1 be compact spaces, and let X and X" be rasp. their cartesian product and disjoint union. Let A e BM(X), IIAll = 1, let f~, g~ ~ C(Xdl, be such that

B e { A ( f ) } > 1 - - x (1.2.1)

{ f

~ n - - y , (1.2.2)

X '

where f, g' and 2'(f) are defined by resp. (1,1.2), (1.1.3) and (1.1.5). Defining now also g by (1.1.2), we have

]1 - - A(g)] <

an(x, y).

(1.2.3)

T~EOnEM 1.2. For every natural number n > 2, there exists a continuous real- valued function %(y), 0 < y, with ~ ( 0 ) ~-- O, and having the following property:

X "

Let { ~}~=~ be compact spaces, let T be a locally compact space and let u be a positive measure on T with

]]Ul]M(T) __< 1 + y . (1.2.4)

112 S T E N K A I J S E I %

Let further f i ( x , t) E C(X, • T)I , resp. (1.1.2) and (1.1.3), and let

Writing now

we have F E V(X),

define f E C ( X •

f ^d r q x ' t)du(t) ^{\ ,} = l x ,

T

F ( x ) = i., [ f(x, t)du(t)

IIFll~ < 1 -~ y, and

1tl - - Fl[v <_ ~ ( y ) .

and f ' E C ( X ' • T ) by

(1.2.5)

(1.2.6)

3. Using T h e o r e m 1.1 we shall n o w s t a t e a n d p r o v e T h e o r e m A in a slightly m o r e precise f o r m as follows.

T~EOREM 1.3. Let (5, 0 < (5 < 1, be a real number. For every natural number n ~_ 2, there exists a real number fi, > O, such that i f {K~}i~l are disjoint compact subsets of a compact abelian group G, and i f U Ki is a Helson-(1 - - f i ) set, fl < fin, then the m a p

T : A ( ~ K , ) - * V(K) is a topological isomorphism, and lIT-l]] < (3-1.

Proof. W e f i r s t define fi= in t e r m s of t h e f u n c t i o n en(','), b y t h e r e l a t i o n 1 - - e,(O, nfln) = d. Since we f u r t h e r assume fl < ft,, we can t h e n choose e > O, such t h a t

1 - - en(e, n(fl ~- e)) = 5 . ^(1.3.1) L e t n o w A E B M ( K ) ,

[]A[]BM

^: 1. T o p r o v e t h e t h e o r e m it suffices b y well- k n o w n d u a l i t y a r g u m e n t s to p r o v e t h a t

t[T'(A)IIPM(ZKi) >-- 5 , a n d this m e a n s t h a t we m u s t f i n d a c h a r a c t e r Z in

[A(Z)I > a .

s(G), such t h a t

(1.3.2)

(1.3.3) L e t e > 0 be t h e e d e f i n e d b y (1.3.1). B y t h e d e f i n i t i o n of n o r m in B M ( K ) we can t h e n f i n d functions fi E C(K~), such t h a t

A ( t ) = ~ e { A ( t t } > 1 - ~ ,

I ~ E : P R E S E I g T A T I O N S O:F TENSOI% A L G E B R A S A S Q U O T I E N T S O F GI%OIJF A L G E B I ~ A S 1 1 3

where we h a v e d e f i n e d f b y (1.1.2). L e t n o w 2'(f) 6 M ( U Ki) = M(K') be t h e measure d e f i n e d b y (1.1.5), a n d let f ' 6 c ( u K d = C(K') be d e f i n e d b y (1.1.3).

Since we n o w huve b y definition

f f'dA'(f) = nA(f) > n(1 -- e), (1.3.5) uK~

a n d since [!f'll~ --< 1, We h a v e II),'([)IIM(UKI ) ~ n(1 -- e). B y the a s s u m p t i o n on the set U K , we can n o w f i n d a character Z 6 G, such t h a t

l f zdZ'(f) I > (l -- fi)Ill'(t)l[ > n -- n(fl + e) .

(1.3.6)

OK i

L e t e -~~ be t h e a r g u m e n t of t h e integral in (1.3.6), a n d p u t gl ~-e~OZ[K~.

We have t h e n g~ 6 C(K d a n d we define g a n d g' b y (1.1.2) a n d (1.1.3). B u t this implies t h e n t h a t

g~{ f g'dX'(t) }= P~e{e'~ f zdx'(t)}----[f zdX'(t)i > n-- n(~ + ~ ) .

^(1.3.7)

B y (1.3.4), (I.3.7) a n d T h e o r e m 1.1 we h a v e t h e n I1 - - A ( g ) l < e.(e, n(/3 @ e)) = 1 - - d ,

a n d therefore IA(g)l > ~. B u t we have also IA(g)] = [ei"~ = IA(z)], a n d therefcre /A(z)/ >_ b, a n d this proves t h e theorem.

w 2. T w o m e t r i c l e m m a s

1. To prove Theorem 1.1 we shall need two lemmas. The first l e m m a is the k e y l e m m a of the proof. I t is an i n e q u a l i t y for bimeasures, which follows from a more general i n e q u a l i t y valid in complex Le-spaces, 1 < p < 2. The second l e m m a is simply a eonvenient q u a n t i t a t i v e version of t h e qualitative s t a t e m e n t , t h a t for every # in M(K), the function g in the u n i t ball of L~(I#]), for which

fgd~

^{= I1~11,} is unique as an element of L~(I#]).

LEMMA 2.1. Let X 1 and X 2 be compact Hausdorff spaces, let A 6 B M ( X I X X 2 ) , and let e~ 6 C(X1), i = 1, 2, be such that sup~ex,{[el(x)] + /2(x)[} _< 1. Let further eiA 6 BM(X), be the bimeasure defined by (eiA)(f @ g) ~ A(eif @ g), f 6 C(X1), g e C(X~).

Then

IIAII~M >--[lelA[k~.M + ~

4

Ile2AII~M

1 1 4 S ~ E N tg:AI J S E I ~

The proof of Lemma 2.1 is based on the following general inequality.

PROeOSlTIOZ~ 2.2.

Let i~ be a 10ositive measure on a locally compact Hausdorff space X , and let f and g belong to Lv(#), then the following inequalities hold."

if 1 0 ~ 1, then

2:e 2~

2-~ llf -4- e~~ gil:dO

>~ 2-~ [ I[f!l~ -4-

e~~ :dO

(2.1.1)

o 0

I f

1 < p < 2 ,

then

(1?

2~

[If + e~~ dO >-- Jlf]l~ + ~i )]glib.

0

(2.1.2)

Note

1. In the case

is ~rue, and is of course elementary.

_Note

2. For all 10>~1, we have

2,~

sup IIf @

e'~ _ > 2~ IIf -~ e~~ dO

0

p ~ 2, a stronger inequality with 4/.~2 replaced b y 1

(2.1.3)

In a pplieatioI1 we shall often need to combine (2.1.3) witll the inequMities (2.1.1) or (2.1.2) above.

Proof of the proposition.

We s~art by observing that for arbitrary complex numbers a, b, we always have

and

2:v 2~

2Z4 ~ Ilal + e~~

0 0

(2.1.4)

if

~2V

if

2~ [ [ a / + e ' ~ ~ [J~l+e~~

o 0

(2.1.~)

> _ - (lal+e'~lbl)dO

= I a l + ~ Ibl

7g 0

4

\:12

= : a l ' + ~: Ib:•) .

We now prove (2.1.1).

Usi~g the definitio:~ of lterm, Fubi;:i's ~,hcorem a:td (2.1.4) we have

R E P r E S E N T a T I O N S O E T E N S O ~ A L G E B R A S A S Q U O T I E N T S O F G R O U P A L G E B R A S 115

2xr 2 ~

if , f f

2~ IIf + e'~ dO = -~ If + e'~gldtt dO =

0 0 X

2 ~ 2 ~

I 1

X o X 0

Now using Fubini's theorem again, then the triangle inequality, and then again the definition of norm, we have

2 z 2~

f l f ^-~u tifl + d~ = -~x l f f llfl -I- d~ >_

X 0 0 X

2z. 2~v

if

dO

= ~ l]If]l~ +

e'~ dO.

O

l ! / :

--> ~ (If] +

e'~ *

This proves (2.1.1).

Before proving (2.1.2), we observe, t h a t if p > 1, then by tI61der's inequality, (2.1.4) and (2.1.5), we have

2 ~ 2:v

12~ [a + e%FdO_> ~ [a + e%IdO _> [al 2 + -~ Ibl S

(2,I.6)

0 0

We now prove (2.1.2). Using the definition of norm, and Fubini's theorem, we have

2~: 2 ~

9 ~ 2/e

0 0 X

2.~ (2.1.7)

1 ~2/p

X 0

Applying (2.1.6) and Minkowski's inequality (observe t h a t

p/2 <

1), we have

~:rg

(f f

X 0

~ 2 lp

If + d~ dO dlx)

X

>-

. Ifl ~ + ~ 5gl ~) ~ ) >

X

4 \2/p

X

which is by definition [[fll~ +

(4/~)llgll~,

and so (2.1.2) is proved.

116 S T E N K A I J S E R

Proof of Lemma

2.1. W e f i r s t observe t h a t it follows i m m e d i a t e l y f r o m P r o 13osition 2.2, t h a t if /~, v E M ( X ) , t h e n

m a x I1# +

e'%il ~

II#ll 2 + ~ II~ll 2 9

0 _ < o E 2 a

W e consider A as an o p e r a t o r f r o m

C(X~)

into

M(X2).

n o t a t i o n s we h a v e t h e n o b v i o u s l y

IIAIIBM =-

sup

llA2(f)liM(xO

IIe~AIIBM =

sup

][A~(e~f)llM(XO f e

C(X1)

tIe~AII~M

= sup

I]A2(Qf)I!M(X~)

N o w let s be a positive n u m b e r . W e can f i n d fl C C(X1),

][A2(e~fi)llM(XO >_ ]Ie~AI].M- e.

B y l i n e a r i t y , a n d b y (2.1.8) we h a v e n o w

m a x

I[A2(effl + io

e

e2f2)llM(X, ) ~ =

m a x

[[A2(effl ) 4- io e Ae(eJ,2)lIi(x~ ) > 2

0_<o_<2:r

4 4

> llA2(elfl)l] 2 4- ~ HA2(eJ2)H 2 > (I[e~AIIBM - s) 2 4- ~

(][e2A[IBM - - e) ~ 9

(2.].s)

Using our s t a n d a r d

(2.].9)

such t h a t

Since s is a r b i t r a r y , a n d since Ilelfl

@ e~%2fi.tl~ ~__ 1,

t h e l e m m a follows.

2. I n t h e p r o o f of t h e t h e o r e m , we shall also n e e d t h e following lemma.

L~MMA 2.3.

Let a, b be positive numbers, and let X be a compact Hausdorff space. Let further # E M ( X ) , f, g c L| be such that

I1~11~1, f i l l ~ l , []g/l~l

and geffd# fgd.

then

( f I f - - g ] d ] t t O e ~ f ]f -- gied]t t] ~

2 ( % f f a 4 - %ffb)e.

(2.2.1) (2.2.2)

#

Proof.

W r i t e # = F . j#], IF[ = 1 a.e. (]#l).

]~eplaei~g t h e n f a n d g b y J'~, g~, a n d # b y f#l, we see t h a t we m a y assume t o be positive.

W e n o w w r i t e

f --- A + if2, g = g14- ig2 ; fi, gi

real. (2.2.3) W e h a v e t h e n

I~EP:RESENTATIONS OF TENSOI~, ALGF.Btl, AS AS Q U O T I E N T S OF G R O U P A L G E B R A S 1 1 7

fLd.>l-a, fgld.>,--b (2.2.4)

a n d therefore

f

f ~ d # >

( f ) '

f , d # > _ ( l - - a ) ~_> 1 - - 2 a , b u t t h e n

f <,' + < f (, - :;> ~, <

1 -- (1 --

2a) = 2a,

N O } V

f g~ d~ > 1 -- 2b (2.2.5)

f g~ d# ~ 2b

(2.2.6)

_< f (, - ~ a - e,>+ = ~ .

(2.2.7)

Writing this in terms of real a n d i m a g i n a r y parts, we have

= f (, - ~<:,., +:,.,>>~. = ~ f

(I - - f l @" f l ( 1 - - ~'1) - - f 2 ~ 2 ) d / ~ -

Using (2.2.4) we h a v e n o w

A < 2 (a @ b + f f2g2 d~ ),

a n d finally b y Schwarz's i n e q u a l i t y a n d (2.2.6),

< 2(a + b + (2a. 2b) 1/2) = 2 ( V / a + V / b ) ~ . This proves the lemma.

w 3. Two metric theorems

1. We shall n o w first prove Theorem 1.1. To do this we shall first prove t h e special case n = 2, a n d we shall t h e n prove t h e general ease b y i n d u c t i o n on n.

Proof of special case.

We shall prove t h a t t h e function

~2(x, s) = VTv + ~(2. + v/T. + ~/Yv)l/2

satisfies the conditions of t h e theorem, a n d we s t a r t b y observing t h a t it is d e a r l y increasing a n d concave.

We n e x t observe t h a t since 1121(l)[IM(x 0 < 1, we clearly h a v e b o t h

t ~ e { / g l d & ( f ) } > l - - y

a n d

I~e{fgfl,~2(f)}>l-y.

^(3.1.1)

XI X2

118 STE?r KAIJSE~

B y linearity we have n o w

A(g) ---- A(g~ | g2) = A((f~ § g~ -- fl) | g2) = A(f~ | g2) -- A((f~ - - gl) | g2) (3.1.2) a n d therefore

ll -- A(g)I _< ]1 - - A(f~ | g2)] § i A ( ( f l - - gl) @

gu)/=

A @ B . (3.1.3) :By definition of n o r m we h a v e n o w [A(f~ | g2)l <-- 1, a n d b y (3.1.1) we h a v e I~e {A(fl | g2)} >~ 1 -- y, a n d this implies t h a t A < ~ y y .

I t remains to prove that B _< ~(2x -b % / 2 x + %/~y)~/2, a n d towards this we firs~ observe t h a t

~ e { f fldAl(I)} = R e { A ( f ) } ~ l - - (3.1.4) X~

:By (3.1.1), (3.1.4) a n d L e m m a 2.3 we h a v e ~herefore

f lA - g1Id[21(f)[ ~< ~ 2 x - ~ %/~-y. (3.1.5)

X~

L e t us write e 2 = If: - - g11/2, a n d e: = 1 -- % W e h a v e t h e n

{/ } 1/

g ~ { ( r | = g e (1 - - ~z)f~dXl(f) _> 1 - - x - - ~ Ill - - a l d l & ( t ) l >

X~ X~

1 (3.1.6)

1 - - x - - ~ ( V - ~ x q- @ T y ) > HelAI]BM.

B y L e m m a 2.1 we have f u r t h e r

IleaAIlsM <_ ~ 1 - - 1--

4

][qAII ~ + ~ lle2All 2 < 1, a n d therefore

~ (2X ~- ~-~X -]- V ~ y ) 1/2 Y~

F i n a l l y we have now, d e n o t i s g h = sign (fl -- gl),

B = IA((fl - - gl) @ g2)[ ⁼ [2A(e~ h | g2)[ = 2l(e2A)(h Q g2)[

a n d this proves t h e special case n = 2.

(3.1.7)

(3.1.8)

R E P R E S E N T A T I O N S O F T N N S O ~ A L G E B R A S A S Q ~ O T I N N T N OF G ~ O U P A L G E B R A S 119

2. T h e general case.

T h e p r o o f is b y i n d u c t i o n . W e d e f i n e el(x, y) = y, a n d we t a k e f o r e2(x, y) t h e f u n c t i o n d e f i n e d a b o v e . W e n o w a s s u m e sk(x, y), 1 < / c < n - - 1 c o n s t r u c t e d , a n d we shall c o n s t r u c t edx, y). L e t t h e r e f o r e A ~. B M ( X ) b e a m u l t i m e a s u r e of o r d e r n, a n d let f , ff ~ ] ~ ( X ) , s a t i s f y t h e a s s u m p t i o n s of t h e t h e o r e m . W e ~ i t e f = f l @ f2 | fa | 9 9 9 | f - , g = gl | g~ | ga | 9 9 9 | g~, a n d we use t h e f a c t

t h a t a m u l t i m e a s m ' e m a y b e c o n s i d e r e d as a n o p e r a t o r f r o m a p r o d u c t of s o m e o f t h e C ( X ) spaces i n t o t h e m u l t i m e a s ~ r e s p a c e o v e r t h e rest, w h i c h is a m u l t i m e a s u r e s p a c e of lower order. This m e a n s t h a t we m a y use t h e i n d u c t i o n h y p o t h e s i s t o r e p l a c e a n y k, k < n - - 1, o f t h e f ' s b y

i r e ( A ( f l | f~ | g~ | g4 @ 9 9 9 Re {Atfi | g~ | g3 | g4 | . . . l~e (A(g I | f2 | ga | g4 | 9 9 9

g's. D o i n g this we h a v e e.g.

@ g,0} > 1 - - e~_~(x, y ) .

(3.2.1)

B u t n o w A(~ @ ~ o @ g a @ g 4 | @ g , ) , q~EC(X~), ~ f E C ( X ~ ) , is a bi- m e a s t u ' e in B M ( X 1 X X2), a n d we use t h e special case p r o v e d a b o v e to conclude t h a t

LA(G)I > 1 - - e2(G_2(x , y), 2 G _ l ( x , y ) ) . (3.2.2) W e define ghus e~(x, y) = s2(e,_2(x, y), 2e,~_l(x, y)), a n d t h i s c o m p l e t e s t h e p r o o f o f T h e o r e m 1.1. I t is e a s y t o v e r i f y t h a t s~(x,y) is c o n c a v e a n d increasing.

3. I n t h e p r o o f of T h e o r e m 1.2 a n essential role is p l a y e d b y a r e s u l t o f B o h n e n b l n s t a n d K arlin ( P r o p o s i t i o n 3.1 below). T o s t a t e t h i s r e s u l t we shall n e e d s o m e n o t a t i o n s ,

L e t A be a B a n a c h a l g e b r a w i t h u n i t e l e m e n t 1, a n d d u a l s p a c e A ' . W e shall define a c o n v e x set D A C A ' , b y

a n d

D A = { f e A ' [ f ( i ) = l, HfllA'~ 1}. (3.3.1) L e t f u r t h e r a E A, we shall t h e n w r i t e

V(a) ~ {z e C l z ~ f(a), f e D~} ,

(3.3.2)

v(a) =: s u p Izl, z E V(a) . (3.3.3)

V(a) is called t h e n u m e r i c a l r a n g e a n d v(a) t h e n u m e r i c a l r a d i u s o f t h e e l e m e n t a E A. T h e r e s u l t t h a t we shall n e e d is t h e following

PROPOSlTrOX 3.1. [2] Let A be a B a n a c h algebra w i t h u n i t 1, and let a E A , then llaIIA ~ e . v(a).

1 2 0 s:rEI~ K A I J S E R

P r o o f of Theorem 1.2. To see t h a t F C V we first observe t h a t for all t C T we h a v e t r i v i a l l y f E V, ][f]]7 < 1. W e n e x t use t h e fact t h a t t h e f u n c t i o n t ---~-f(x, t) is a c o n t i n u o u s f u n c t i o n f r o m T t o V. T h e r e f o r e t h e integral is well-defined a n d t a k e s its v a l u e in V. F i n a l l y we h a v e

HFHv ~_ f [If(x, t)llTdu(t ) ~_ 1 -[- y . (3.3.4)

T

F o r t h e second p a r t of t h e t h e o r e m we shall p r o v e t h a t for a n y f u n c t i o n e~(x, y) satisfying t h e conditions o f T h e o r e m 1.1, t h e f u n c t i o n Vn(y), d e f i n e d b y

~(y) = e. y + (1 + y)~o o, ~ (3.3.5)

satisfies t h e conditions of T h e o r e m 2. T o do this it suffices b y p r o p o s i t i o n 3.1 t o p r o v e t h a t

v ( 1 - - F ) < y + ( 1 - ~ y ) e , O, , (3.3.6) w h e r e v ( i - F ) d e n o t e s t h e n u m e r i c a l radius.

T o w a r d s this, let A E D v a n d let t h e (probability) measures 2~ --~ 2~(1) a n d 2' be d e f i n e d b y (1.1.4) a n d (1.1.5). W e h a v e n o w

[A(i - - F)] z [1 - - A ( F ) I : ]-- y -~ (1 ~- y - - A ( F ) ) I (3.3.7)

I I I

= i - y + (i - A(f(~, t))du(t/ < y + I1 - A(f(x, t))Idu(t/.

I

T T

Comparing t h e final t e r m in (3.3.7) w i t h t h e r i g h t h a n d t e r m in (3.3.6) we see t h a t it suffices to p r o v e t h a t

f

1 -~ y ]1 - - A ( f ( x , t))ldu(t ) < ~ O, . (3.3.8)

T

Let, n o w t E T. W e shall write

n - - y ( t ) ~

~e[fr'(x't)d~(x')}~,

^{, , (3.3.9)}

X '

a n d we h a v e t h e n

i l - - A ( f ( x , t))[ _~ sn(0, y ( t ) ) . N o w b y (1.2.5) a n d F u b i n i ' s t h e o r e m we h a v e also

(3.3.10)

R E P R E S E N T A T I O N S O F T E N S O R A L G E B R A S AS Q U O T I E N T S O F G R O U P A L G E B R A S 121

fv(t)du(t)=fnd. (t,-- e{f

T T T X "

= n(1 4 - y ) - -

f,

^{= n + n y - n = n y .}

X "

(3.3.11)

Since the function e~(O, y) is concave we h a v e b y J e n s e n ' s inequality (turned upside down)

1 + y [1 -- A ( f ( x , t)lldu(t ) < 1 + y

T T

e~ O, l @ y y(t)du(t) = s, O, i ~ y '

T

(3.a.12)

and this proves the ~heorem.

Remark." I n the case n = 2, one can using Grothendieck's ~)fundamental theorem~y prove t h a t the function V2(Y) ---- 15y satisfies the conditions of the theorem.

As an i m m e d i a t e consequence of Theorem 2, we have the following

COROLLARY 3.1. Let {X~}~I be compact spaces and let X and X ' be as above.

Let for each i, gi E S ( X 0, i.e. gi E C ( X d and Ig~l = 1, and define the functions g C C(X) and g' E C ( X ' ) by (1.1.2) and (1.1.3). Let T be a locally compact space and let u be a positive measure on T with HuH <_ 1 @ y. Let further for each i, f i E C ( X i x T h and let f and f ' be as above, and let

f f ' ( x ' , t)du(t) = g'(x') .

J T

Writing now F(x) =

f , f(x, t)du(t)

we have F E V, llF]lT < 1 - t - y and ] [ g - - F H <_ ~,,(y).

Proof. W e shall define functions hi E C(X~ • T) b y hi(xl, t) = g(x~)fi(xi, t).

Defining t h e n h, h', and H b y resp. (1.1.2), (1.1.3) and (1.2.6) we h a v e b y T h e o r e m 1.2.

H E V, J]H][ v_< 1 q - y and I [ 1 - - H [ l _ < ~ , ( y ) . B u t we h a v e also g - - g ' l , F - - - - g . H , a n d IIg/]v<-- 1, IIF[] _< []g]I" !]HH --< 1 @ y and l[g - F]I --< ]]g[[" [[1 -- HI] < ~7,(Y).

(3.3.13) and therefore

122 S T E ~ KAIJSEI%

COI~OLLAI~Y 3.2. Let {Xi}~_ 1 be compact spaces, and let F E V(X) ll~ll. < 1 , I!1 - FII~ < y ,

then II1 -- l i l y <-- lO~?~(y) .

be such that (3.3.14)

w 4. Helson sets in Dq

1. The group Dq is the preduct, as a group and as a topological space, of a denumerably infinite number of compact abelian groups isomorphic to Z(q), the

/ x

cyclic group of order q. The dual group Dq, is the sum of the corresponding dual groups. Dq is compact metrizable and totally disconnected, and ~ is discrete and denumerably infinite [7].

I n w 1 a closed subset E of a compact abelian group G was called a K~-set if GIE = St(E). I t is easy to see t h a t the group Dq contains K~-sets, and it was observed b y Varopoulos, t h a t if {K~}'~=I are closed subsets of Dq, and if U K~

is a Kq-set, then A ( ~ Kd is a tensor algebra. OR the other trend the gro~Ip Dq does not contain tIelson-(~)-sets, for c~ arbitrarily close to 1, so Theorem 1.1 e a r not in general be used to provide weak conditions oR a set {K~}~=I of subsets, ensuring A ( ~ Ki) to be a tensor algebra. Nevertheless it seems reasonable, in the light of Theorem 1.1 to conjecture t h a t if U Kf is almost a Kq-set then A ( ~ Ki) is a tensor algebra. In the next paragraph we shall prove this in the ease n = 2, i.e. A ( K 1 + K J is a tensor algebra if K 1 U K2 is almost a K~-set. I n the proof of this result an important role is played b y functions with fq = 1, in fact we shall have to assume t h a t the characters are sufficiently dense in the appropriate metric sense in S~(K 1 U J~TJ. This assumption is however not used in the definition of the ttelson constant, and we shall therefore consider also another type of Kelson- constant, which is more suitable for our purposes, and is conceptually more natural for subsets of D~. We then s t u d y the relations, between the two concepts, and we shall prove t h a t t h e y are essentially equivalent.

2. We shall presently consider subsets E of the group Dq t h a t are either Kq-sets or ]-Ielson-sets. I n these considerations an essential role is played by the groups S~(E) and its subgroup #q]F. To see the nature of this role we shall however first consider a more general problem. We shall need some definitions and notations.

Let K be a compact metrizable Hausdorff space, and let P be a compact convex set, containing 0, in the complex plane. We shall denote by ~P(K) the set of all functions f in C(K) with f ( K ) C P, and by P'(K) the set of Borel functions with f ( K ) c:: P. P(K) is a bounded convex subset of C(K). To avoid triviMities we assume t h a t P contains some point outside 0, and it is easy to see t h a t then, every function in C(K) can be represented b y a finite linear combination of functions

R E P R E S E N T A T I O N S O F T E N S O R A L G E B R A S A S Q U O T I E N T S O F G R O U P A ~ G E B ~ A S 123

f r o m

P(K).

W e can t h e r e f o r e define an e q u i v a l e n t n o r m in

C(K),

which we shall call t h e

P-norm,

as follows

J[f[[~ = i n f ~ ]akl o v e r all r e p r e s e n t a t i o n s

f - - ~ akfi, f i e P(K).

(4.2.1) W e shall d e n o t e t h e space

C(K),

w h e n g i v e n t h e P - n o r m b y

Cp(K).

T h e n o r m - d u a l of

Cp(K)

will be d e n o t e d

Mp(K),

a n d is t h e u s u a l space

M(K),

w i t h n o r m given b y

II~!L. = sup [ f f ~l~ l , f c P(K ) .

(4.2.2) W e shall l a t e r consider m a i n l y the cases, w h e n P is a set Pq = t h e c o n v e x hull of t h e qth roots o f u n i t y , a n d w h e n P is t h e refit interval. W e also observe t h a t t h e P - n o r m obtained, w h e n P is t h e u n i t disc, is t h e usual sup n o r m in

C(K)

resp. t h e u s u a l t o t a l mass n o r m in

M(K).

F o r a c o n v e x set E o f c o m p l e x n u m b e r s we shall d e n o t e t h e radius of t h e set b y

r(E),

i.e.

r(E) =

sup Iz[, z e E, a n d we shall d e n o t e t h e p e r i m e t e r b y

p(E).

F o r a b o u n d e d c o n v e x set c o n t a i n i n g 0, we h a v e 2r(E) _~

p(E) ~_ 2~. r(E).

F o r n o r m a l i z a t i o n of t h e base set

P, we

shall assume t h a t 1 e P , a n d t h a t

r(P) ~ 1.

D e n o t i n g t h e u n i t i n t e r v a l b y I , a~ld t h e u n i t disc b y D, we assume t h u s I ~ P c D . T h e s e a s s u m p t i o n s i m p l y t h a t t h e P - n o r m of a m e a s u r e is a t m o s t t h e t o t a l mass, a n d t h a t t h e P - n o r m o f a positive m e a s u r e is t h e t o t a l mass.

A nat~tral s t a r t i n g - p o i n t for i n v e s t i g a t i o n s on P - n o r m s is t h e following fact, wcl]-kno~vn in t h e t h e o r y of c o n v e x sets.

L e t C a n d D be c o n v e x sets in t h e plane, a n d let E be t h e sum-set E = C + D . T h e n E is a c o n v e x set, a n d

p ( E ) : p ( C ) -~ p ( D ) .

(4.2.3)

T h e i d e n t i t y (4.2.3) is i m p l i c i t l y c o n t a i n e d in some m o r e general f o r m u l a s in e.g. B o n n e s e n - - F e n c h e l [1], a n d is obvious on i n s p e c t i o n w h e n C a n d D are polygons. F i n a l l y a n ~nalytic p r o o f of (4.2.3) can be b a s e d on a i b r m n l a of Cauchy, showing t h e p e r i m e t e r of a c o n v e x set t o be a linear f u n c t i o n of t h e w i d t h of t h e set.

This f o r m u l a can be w r i t t e n e.g. as follows [see 1, p. 48]

2 z

if

p(C)

~ ~- [ sup

(x. u) --

inf ( y . u) ]

dO,

(4.2.4)

~ E C y E C

0

u : (cos

O,

sin 0), ( x - u ) t h e usual i n n e r p r o d u c t in 1t ~.

F o r m u l a (4.2.4) clearly shows t h e a d d i t i v i t y o f t h e p e r i m e t e r . W e observe also t h a t t h e radius is s u b a d d i t i v e u n d e r a d d i t i o n of c o n v e x sets, so t h a t if E = C -~ D, t h e n

r(E) <_ r(C) + r(D) .

(4.2.5)

] 2 4 S T E N X ( A I J S ] ~ I ~

W e n o w i n t r o d u c e some notions, t h a t will be used t h r o u g h o u t this w L e t P be a c o m p a c t c o n v e x set c o n t a i n i n g 0 in t h e c o m p l e x plane. L e t a b3 a c o m p l e x n u m b e r , we shall w r i t e

a P = {w l w = az, z E P } . (4.2.6)

a '~

L e t n o w a --~ { ~}~=~ be a set of c o m p l e x n u m b e r s . T h e P - r a n g e o f a is t h e set Qp(a) = ~ (akP) = {z [ z = ~ akzk, zk e P } . (4.2.7)

k = l k ~ l

I t is obvious t h a t p ( a P ) : [a] . p ( P ) , and it t h e n follows f r o m (4.2.3), t h a t

n

p(Qp(a)) = ( ~ lakl) " p ( P ) . (4.2.8)

k = l

L e t n o w K be ~ c o m p a c t m e t r i z a b l e H a u s d o r f f space, a n d let # E M ( K ) . T h e P-range of # is t h e set

Qp(#) = r { z l or e q u i v a l e n t l y

~= f fd~, f eP(K)},

K

(4.2.9)

Q.(~)={z z--ffd~, fcP'(K)}. ^(4.2.9')

Qe(#) is t h e c o n t i n u o u s linear image u n d e r #, of t h e bou'nded c o n v e x seC P ( K ) , a n d is t h e r e f o r e b o u n d e d a n d c o n v e x . I t follows m o r e o v e r e.g. f r o m a p a r t i t i o n o f u n i t y a r g u m e n t , t h a t

P(@P(~)) : P ( P ) " [r~[l~(~) 9 (4.2.10)

B y d e f i n i t i o n we h a v e f u r t h e r JI/~Jtp : r(Qp(#)). T h e fact t h a t b o t h t h e P - n o r m a n d t h e usual n o r m can be r e a d off f r o m t h e set Qp(/~), indicates t h e i m p o r t a n c e o f this set for p r o b l e m s on relations b e t w e e n P - n o r m s a n d usual n o r m s .

N o w t h e p e r i m e t e r of a c o n v e x set is a m o n o t o n e f u n c t i o n on t h e set of all c o n v e x sets, so a set of p e r i m e t e r L c a n n o t be c o n t a i n e d in a circle of radius r if r < L/2n, a n d we heJve t h e r e f o r e

P ( P )

[l#I]P ~ ~ - ~ [[/~[[~t(rc)- (4.2.11)

D e n o t i n g b y I F t h e i d e n t i t y m a p of C(K), considered as a m a p f r o m C(K) i n t o Cp(K), we see t h u s t h a t [[Ie][ ~ 2z~/p(P).

L e t n o w P be one o f t h e se~s Pq (resp. t h e u n i t i n t e r v a l I), a n d let us s p e a k of q-norms i n s t e a d o f Pq-norms, a n d of spaces Cq(K), a n d Mq(K), a n d o f ~,he q-~ange Q~(#). W e h a v e in this case p(P~) = 2q. sin (u/q) (resp. 1o(1) == 2), a n d f r o m (4.2.11) we h a v e t h e n

R E P R E S E N T A T I O N S O F T E N S O R A L G E B R A S A S Q U O T I E N T S O F G R O U P A L G E B R A S 125

2q sin

(n/q)

i]#[!q ~> 27r H#HM(~) (4.2.12)

1

resp., ll#il

~ -I]~ll~(~>.

^(4.2.13)

2~

I f t h e set K is totMly disconnected, th~n convex combinations of functions f r o m S~(K) are dense in

Pq(K),

a n d therefore the set Qq(/~) is in fact the closed convex hull of t h e set of all z in C, such t h a t z =

f f d # ,

w i t h f in

Sq(K).

I n particular we have therefore II#]lq ~

r(Qq(#))

= sup

ff@], f e Sq(K).

Combining this w i t h t h e definition of a Kq-set, we see t h a t the Helson c o n s t a n t of a Kq-set is yq = sin

(~/q)/(r~/q).

W h e n P is t h e u n i t interval, we use t h e general principle, t h a t w h a t can almost be done b y continuous functions, can be done b y Borel functions, to conclude the existence of a Borel function ~, which we m a y in fact assume to be i d e m p o t e n t , w i t h I f ~ d#l >--/I#/I~(K)/~.

These results are all well-known, [3, p. 565; 5]. The s t a n d a r d proofs are different, b u t also the more n a t u r a l approach given here is known, see e.g. [5, p. 674].

3. B y v e r y general a r g u m e n t s we h a v e proved t h a t no infinite subset of Dq can h a v e a t t e l s o n c o n s t a n t greater t h a n yq, a n d t h a t on the other h a n d a Kq-set h a s t h e 1-[elson eonsta'nt yq. I f K is a Kq-sct we have in fact

A(K) ~- Cq(K)

canonically, which m e a n s t h a t for e v e r y function f in C(K) resp. e v e r y measure

# in

M(K),

we have

llfllA(~) = Ilfl/q --< 7~111fll~ (4.3.1)

L e t n o w E be a Helson set in D~. T h e I{~lson c o n s t a n t ~(E) of E is d e f i n e d b y comparing n o r m s in

A(E)

a n d n o r m s in

C(E).

B y t h e above we m a y write a(E) = yq(1 - - 7 ) , a n d we h a v e t h e n

Ilf[l~ >- yq(1 - - 7)

"l]fI[4(E),

all

f e C(E),

(4.3.2) resp. H/*I[eM(~) ~> yq(1 --V)[I#][M(E), all # e

M(E).

(4.a.2') I f V is small, t h e n condition (4.3.2') is, in t e r m s of a f i x e d /,, a strong condition if

II#llq/][#llM

is close to yq, while it requires considerably less if I[#][q/][/~/[M is close t o 1. I~ is therefore n a t u r a l to consider i n s t e a d of (4.3.2) a n d (4.3.2') conditions of t h e following t y p e

][fll, >- (1 - - d)ll/llA, all

f e C(E),

( 4 . 3 . 3 ) resp. II/~IIpM ~ (1 - d)l!#l/~, all /~ e

M(E) ,

(4.3.3') a n d we shall m a k e the following definition:

1 2 6 S T E N K A I J S E I ~

Definition 4.1. Let K

be a I t e l s o n set in

Dq.

W e shall s a y t h a t K is a

H(q, fi) set

if for e v e r y

f C C(K),

we h a v e

[IfIIA(K) <

fi-~llfllq

or e q u i v a l e n t l y , i f for e v e r y # C

M(K)

we h a v e

I t is clear t h a t a

H(q, fl)

set w i t h fi close to 1, is a H e l s o n set w i t h IIelson c o n s t a n t close to yq. W e shall h o w e v e r also p r o v e , a n d this is t h e m a i n r e s u l t o f this section, t h a t a t I e l s o n set, w i t h H e l s o n c o n s t a n t close t o 7q is a

H(q, fl)

set, w i t h fl close to 1.

T h e m a i n step t o w a r d s this is to p r o v e t h e following t h e o r e m .

TI~EOI~EM 4.1.

Let ~ be a positive number, 0 < ~ <

rain (~2/8q2, 1/15),

and let K be a totally disconnected metrizable compact Hausdorff space. Let further ~ c Sg(K) be a family of functions with the following property:

Eor every measure # C M(K), there exists g E ~, with

I

t I sin (=/q) (1 4.3.4)

f g ~ > ~/q

i

I 1

Then: I f ~ = ] . I~1,

that

and f g

f a Borel function with f~ ~- 1, there exists g E ~, such

d# > (1 - 14 ~ ( /q)' 2/3 )II~!I~<K) if q _> 2 (4.3.5) d/, > (1 - - 5~ 2/3) " ]]/~]]M(K) / f q = 2 . (4.3.6)

[

-Remark

1. W i t h o u t loss o f g e n e r a l i t y we assume t h a t t h e f a m i l y ~ is closed u n d e r m u l t i p l i c a t i o n b y qth r o o t h s o f u n i t y (otherwise s i m p l y adjoin all multiples b y qe, r o o t s o f u n i t y ) , a n d we shall u n d e r this condition p r o v e t h a t we can in fact f i n d g C ~, such ~hat

Re ( f g d#) > (1--14(~/q)2/3)llttl1M(K) .

-Remark 2.

T h e o r e m 4.1 is s t a t e d a n d shall be p r o v e d w i t h o u t using t h e con- cepts of q-norms. I n t h e following proof, t h e n o r m o f a m e a s u r e will always be t h e t o t a l mass n o r m .

.Remark 3. I f ~ ~-- z2/Sq 2,

t h e n (1 - - ~)7q ~ (1 - - 14(~/q)2/3), a n d this shows t h a t t h e given r a n g e o f a's contains t h e r a n g e o f i n t e r e s t .

Remark

4. T h e o b s e r v a n t r e a d e r has a l r e a d y n o t i c e d t h a t we h a v e followed o u r own advice f r o m w 1, a n d used s t r i c t inequalities in b o t h a s s u m p t i o n s a n d con- clusions o f t h e t h e o r e m .

REPRESENTATIONS OF T E N S O ~ ALGEBRAS AS QUOTIENTS OF G~OUP ALGEBRAS 1 2 7

4. Outline of proof:

A n y f u n c t i o n g in

Pq(K)

w i t h (I.(g)[

(and w i t h - - (~/q) < a r g

(I,(g)) < (z~/q))

is close in Ll([/~i) to t h e f l m c t i o n f . T h e p r o o f consists t h e r e f o r e essentially in findir~g a f u n c t i o n g in ff sufficiently close t o f. T h i s we can n o t do o~fly in t e r m s of t h e m e a s u r e #, a n d we shall t h e r e f o r e c o n s t r u c t a n o t h e r m e a s u r e v, which is more a d e q u a t e for our purposes. I n t h e ease q = 2, t h e m e a s u r e # is a s s u m e d t o be real, a n d t h e p r o b l e m is t o f i n d a suitable i m a g i n a r y p a r t to a d d to #. T h e c o n s t r u c t i o n s in t h e p r o o f are m a i n l y geometric, a n d a cruciM step in t h e p r o o f is a p u r e l y geometric l e m m a on t h e p e r i - m e t e r of c e r t a i n c o n v e x sets.

Proof of theorem.

F r o m general facts a b o u t sets Qp (#), we k n o w t h a t Qq(#) is for e v e r y

# 6 M(K)

a c o m p a c t c o n v e x set c o n t a i n i n g 0. W e also k n o w t h a t

p(Qq(/.z)) -~- p(Pq)

9 I]/~[I = 2 ~ . [1~[[.

W e n e x t observe t h a t if

f E Pq(K)

t h e n

f . P q ( K ) c

Pq(K), a n d t h e r e f o r e

Q~(f'a) c Q~(/~).

I f we h a v e m o r e o v e r

fq

= 1, t h e n

f . P ~ ( K ) -- P~(K)

a n d hence

Qq(f.#)

= Q~(/~). (4.4.1)

I n fact, (4.4.1) holds also if

fq = 1, and

if f is a B o r e l f u n c t i o n . I a p a r t i c u l a r it follows tha'~

Qq(#)

is i n v a r i a n t u n d e r m u l t i p l i c a t i o n b y qth r o o t s of rarity. A n o t h e r consequence of (4.4.1) is t h a t if # ~ - f - I # ] , fq = 1 t h e n

Qq(/~) = I[/~[[ 9 P u . (4.4.2)

T o see this we assume, b y (4.4.1), t h a t # is positive. W e t a k e t h e n

g = 1

6

Pq(K),

a n d we see t h a t t h e c o m p l e x n u m b e r I]~]l belongs t o Qq(#). Since Qq(#) is c o n v e x a n d i n v a r i a n t u n d e r m u l t i p l i c a t i o n b y qth roots of u n i t y , we h a v e t h e n

I]~lI-Pu c Qq(~).

N o w we also k n o w t h a t t h e sets h a v e equal p e r i m e t e r s , a n d t h e r e f o r e t h e y are equal.

T o illustrate t h e g e o m e t r i c n o t i o n s involved, we shall c o n s t r u c t explicitly t h e set Qq(#), for a m e a s u r e /~ w i t h finite s u p p o r t . I n this ease we m a y also write Qq(a), for a c e r t a i n set a = {ak}~=~. B y (4.4.1) we m a y assume t h a t

-- (z/q) <

arg (%) <

(u/q),

a n d it is also clear t h a t we m a y assume t h a t arg (%) <_ arg ( ( ~ k + l ) ,

all k. W e n o w t a k e z 0 = ~ = 1 ak, a n d we see t h a t z0 is a b o u n d a r y p o i n t o f Qq(a).

I n fact z0 m a x i m i z e s t h e real p a r t in Qq(a), a n d m a x i m i z e s t h e i m a g i n a r y p a r t a m o r g all such points. W e n o w define c o n s e c u t i v e l y

Z k = Z k _ 1 -~- a k ( @ - - 1 ) , @ = e 2 m / q .

All t h e p o i n t s {zk}~=0 are t h e n e x t r e m e points of Q~(a), a n d we h a v e f u r t h e r z, = @'z0. D , f i n i r g n o w zk+, = @'zk, we get all t h e e x t r e m e points of Qq(a).

See Figures 1 a n d 2.

128 STEN KAIJSER

Fig. 1. Q2(a)

a = ( a l , a2, aa)

Z3

~2

Z~

Fig. 2. Q3(a)

/

a = ( a 1, a.~)

Z* -- oa 1 --~, a 2

.Z3 ~ Z o = a i + a~

Z 4

Conversely, we see t h a t if Q is a c o n v e x polygon, i n v a r i a n t u n d e r m u l t i p l i c a t i o n b y q,h roots of u n i t y , t h e n Q is a Qq(a), for some set of c o m p l e x n u m b e r s . T o c o n s t r u c t one such set, we assume t h a t Q has say q . n vertices. W e e n u m e r a t e t h e n t h e v e r t i c e s consecutively, st~rtil~g f r o m a n a r b i t r a r y v e r t e x b y %, z l , . . . , zn,...zq,. W e h a v e t h e n z~ = @z0, a n d we define a k ~ (% - - zk_~)/( @ -- 1), 1 </c: < n.

R E P R E S E N T A T I O N S OF T E N S O R A L G E B R A S AS Q U O T I E N T S OF G R O U P A L G E B R A S 129 W e shall n o w p r o v e t h a t i f t h e given m e a s u r e /x is c o n t i n u o u s , or m o r e precisely i f # can b e d e c o m p o s e d i n t o s u f f i c i e n t l y small m u t u a l l y singular p a r t s , t h e n t h e eonehision o f t h e t h e o r e m holds. T o w a r d s this we f i r s t define a real n u m b e r

a,

b y t h e r e l a t i o n

1 + 3q

(tana--a)

= ( 1 - - ~ ) - I (4.4.3)

47~

or e q u i v a l e n t l y ,

I 4~ c~

r a n t s - - ~ ~ - - 9 - -

3 I - - c r q-"

Using now the elementary estimate ~2 < I0, and the assumption ~ <

~218q~ <

zz~/32, we h a v e

47~ ~x

This implies t h e n

(,+)','

< \ - - i - ~ / < \ ~ / < q "

On t h e o t h e r h a n d , we h a v e also c~ < 1/15 a n d t h e r e f o r e a < ~14.

W e n e x t d e n o t e b y

Aq(a)

t h e set d e f i n e d as t h e c o n v e x hull o f t h e u n i o n o f t h e u n i t disc a n d t h e p o i n t s Qk(1/cos a), 1 < k < q. W e f u r t h e r d e f i n e a f u n c t i o n

_F(x)

(or m o r e p r o p e r l y

F~,.(x))

^{b y}

Z o gr

F i g . 3. A~(a)

Z~ Z~

Z1

4 Zn

X5

1 3 0 S T E I ~ l~ A / J S E / ~

for

F ( x ) -~ sup y o v e r all (x, y) E Aq(a ),

cos (2~/q) 1

~ x ~

c o s a - - - - c o s a

(4.4.5)

(Figures 3 a n d 4.)

W e h a v e , since a ~ ~/q, p(Aq(a)) = 2 z + 2q(tan a - - a), so i f A ' is a p o l y g o n w i t h s u f f i c i e n t l y small sides a n d w i t h v e r t i c e s on t h e b o u n d a r y o f A s ( a ), t h e n

~o(A') ~ 2~ + -~ 2q(tan a - - a). W e shall p r e s e n t l y c o n s t r u c t a v whose Qs(v) is such a n inscribed p o l y g o n . W e shall t h e n h a v e

[fvl[ = 19(Qs(v))/p(Ps) = :p(Qs(v))/2zrs ~ F~-~ 1 + ~ - (tan a - - a) , (4.4.6) a n d b y (4.4.3) a n d t h e a s s u m p t i o n s on ~, we can t h e n f i n d g E ~ w i t h ]I~(g)] ~ 1.

T o m a k e the c o n s t r u c t i o n , we f i r s t b r e a k # into small pieces. W e p a r t i t i o n K into a f i n i t e n u m b e r o f disjoint B o r e l sets {Ei}N=l, a n d we a s s u m e for s i m p l i c i t y t h a t I # t ( E 0 ~ - r i ~ 0, all i. W e d e n o t e / ~ - r ~ - 1 . # 1 ~ i, so t h a t we h a v e

~u----~rl/~. H o w , for all i we h a v e Qq(#~) = Ps, a n d this implies t h a t i f

/~(a)

-~ ~. a@, t h e n Qs(#(a)) - - Qq(a). T h e m e a s u r e v will be such a linear com- b i n a t i o n , a n d for q ~ 2 t h e c o n s t r u c t i o n is i l l u s t r a t e d g e o m e t r i c a l l y in F i g u r e 3. W e shall c a r r y o u t t h e c o n s t r u c t i o n f i r s t in t h e s o m e w h a t simpler case q ---- 2, a n d t h e n for a n a r b i t r a r y q.

T h e case q --~ 2. W e a s s u m e ]]#]] --~ (cos a) -1, a n d we h a v e t h e n positive n u m b e r s {rl}N=l, such t h a t ~ ri ---- (cos a) -1. W e d e n o t e n o w

R E P R E S E N T A T I O N S OF T E N S O R A L G E B R A S AS Q U O T I E N T S OF G R O U P A L G E B R A S 1 3 1

k

x k = - - ( c o s a ) - i d - 2 ~ r ~ , 0 < k < x Y (i.e. x 0 - ~ - ( c o s a ) - l ) ,

i ~ 1

a n d we d e n o t e , in t e r m s o f t h e F ( x ) d e f i n e d in (4.4.4) zk = xk -4- i F @ t ) , 0 < k < N . W e can n o w f i n a l l y d e f i n e

Zk - - Z k _ l

a k - - , l < k < N . 2

T h e set a = {ak}N=, is c o n s t r u c t e d in such a w a y t h a t Q~(a) is a p o l y g o n in- scribed in A ~ ( a ) , a n d we h a v e R e (ak) = rk. W e d e f i n e n o w v = ~ ak#k, a n d we h a v e t h e n /~ = - i ~ e (v). W e a s s u m e n o w t h a t all r~ are so small t h a t Ilvll > (~/2)(1 _ ~ ) - x , so t h a t t h e r e exists g E ~ w i t h 1I~(9)1 > 1. A s s u m i n g R e (I~(g)) >__ O, we h a v e R e ( I , ( g ) ) > cos a, a n d since g is real a n d since g = R e (v) we h a v e

I A s ) > cos a = cos~ a "lbll >- (1 - 5 . ~/~)ll~*ll

A r b i t r a r y q. W e a s s u m e a g a i n [(/~[[ = (cos

a) -1,

a n d we h a v e t h e n also in t h i s case positive n u m b e r s {r~}~=l, such t h a t ~ r~ = (cos a) -x. I n this case we shall h o w e v e r f i r s t pass o v e r t o t h e n u m b e r s {sl}~=l d e f i n e d b y

W e d e n o t e n o w

W e define n o w L

W e d e n o t e f u r t h e r

a n d

D e f i n i n g n o w zk d e f i n e d for all k,

s~ = 2 (1 - - cos a cos (~/q))ri .

k

x~-~--{- ( c o s a ) - i - ~ s i , 0 < k < N .

/ = 1

b y xL >_ cos ( z / q ) > xL+l, a n d Zk = Xk E i F @ k ) , O < k < L .

N

y k = ( c o s a ) - i - ~ s . L - 4 - 1 < k < N ,

/ = k + l

wk = yk - - i F ( y k ) , L - 4 - 1 < k < N .

for L ~ - l < k < N b y z i == ~ . wk, Q = e 2"~/q, 0 < k < N . W e can n o w d e f i n e

w e h a v e z k

Z k - - Z k _ l

a k - - , l < k < N .

~ - - 1

T h i s finishes t h e c o n s t r u c t i o n of t h e n u m b e r s ak. W e set v = ~ ak/xk, so t h a t Qq(v) is a p o l y g o n inscribed in A q ( a ) , a n d a s s u m i n g all rk s u f f i c i e n t l y small we h a v e t h e n ?4(1 ^-- a)llv[I > 1. B y a s s u m p t i o n , t h e r e t h e n exists g E such t h a t lI,(g)[ > 1. Assuming - - ( x / q ) < arg ( I , ( g ) ) <_ ( x / q ) , we h a v e , since

132 S T E I ~ K A I J S E R

I~(g) e Qq(v): c Aq(a), Re (I,(g))

> cos a. W e shall e s t i m a t e R e

(I,(g))

a n d t o - w a r d s t h i s w e f i r s t d e f i n e f g dttk = tk = 1 - - uk, a n d since

tk E Pq,

w e h a v e

- - (~/2 - -

x/q) <

a r g (uk) _< (u/2 - -

~r/q).

W e c a n n o w w r i t e

~e

(I,(g))

= R e ( E ak(1 - - uk))

a n d

l~,e (I (g)) -= ~e ( E

rk(1 - - us))

O u r t a s k is t h u s t o e s t i m a t e R e ( ~

rkuk)

-

1

~ ( X a ~ )

COS ~/,

1

~e ( X ~ 0 . (4.4.7)

COS

in t e r m s o f R e ( ~

akuk).

E x a m i n i n g c a r e f u l l y t h e c o n s t r u c t i o n o f t h e n u m b e r s ak, w e see t h a t - -

(u/q)

< a r g (ak) <

ztlq,

so t h a t R e

(akuk)

> 0, u n l e s s uk = 0. W e d e f i n e n o w f o r e v e r y k a p o s i t i v e n u m b e r bk, b y t h e c o n d i t i o n t h a t bk is t h e l a r g e s t r e a l n u m b e r , s u c h t h a t

Re (bku) < Re (aku),

f o r all u w i t h - - (x/2 - -

x/q) <_

a r g (uk) < (x/2 - -

~r/q),

(4.4.8)

Fig. 5.

9 I

{ ' ;

I

I

I I

I I

I I

I I

I i

I s~ I

r- 1,

t I

I I