Iterates of a convolution on a non abelian group

A NNALES DE L ’I. H. P., SECTION B

S. R. F ^OGUEL

Iterates of a convolution on a non abelian group

Annales de l’I. H. P., section B, tome 11, n

^o

2 (1975), p. 199-202

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Iterates of

a

convolution

on a non

abelian _group

S. R. FOGUEL Hebrew University of Jerusalem

Ann. Inst. Henri Poincaré, ^{Section B :}

Calcul des Probabilités et Statistique.

ABSTRACT. - Sufficient conditions are studied

for v

^*

[[ --~

⁰

when Il

is a

probability

^{measure and v is}

absolutely

continuous with

respect

^{to the} left Haar measure.

RESUME. On etudie des conditions suflisantes pour que

~~

^{v *}

~u" ~~

^{-~ 0}

lorsque

^{est une}

probabilité

^et

lorsque

^{v est une mesure} absolument conti-

nue par

rapport

^{à la mesure de Haar à}

gauche.

NOTATION. - We shall use the notation of

[2] :

^{let G be a}

locally compact

group, which is ^not

necessarily

^Abelian. Let ~, be its left Haar measure and

Ma

^{the two sided ideal of bounded measures which are}

absolutely

^conti-

nuous with

respect

^{to ~, : see}

[2],

Theorem 19.18. Note that

Ma

^{is identified}

with

by

^{the Radon}

Nikodym

^Theorem.

The convolution of two measures, ⁱ and

0,

^is

given by

and

if f is

^{a bounded} measurable function then

Let _,u be a fixed

probability

^{measure. If}

vEMa

^{then v *} ~c ^E

Ma

^and

Annales de l’Institut Henri Poincaré - Section B - Vol. XI, n° 2 - 1975.

Vol. XI, ^n° 2, 1975, p. 199-202.

200 S. R. FOGUEL

~c ^*

vEMa. Thus ~

^{acts as an}

operator

^on

Ma

^{and on =}

Ma .

^Note

that if 03C4 is a non

negative

^{measure then}

~

ⁱ

~ -

^03C4,

1 >.

^Put

ⁿ

^{= * ... *}

(n times), ,u°

^{- b the}

identity.

THEOREM 1. Let

and ~

^be

probability

^{measures with} ,u ^* ~ ⁼ ~ ^* . Assume there is an

integer

^{h and a non}

negative

^{measure v,}

v(G)

^>

0,

^such that >- v and

,uh

^{>_ v *} r~, then

Proof

^{- Put}

where v

and 03B81

^{are non}

negative.

^Thus

Continue the same

computation

^{to obtain}

Now

Let us use

(1)

^{now to obtain}

and

by

^{an induction}

argument

Annales de l’Institut Henri Poincaré - Section B

ITERATES OF ^A CONVOLUTION ON A NON ABELIAN GROUP 201

Thus

Now the first sum is smaller then

ý;;

^see

^{[3], p. 1632,}

^and

^{1 )j~0j~~.}

The

following Corollary

^{is included in}

[I],

^{Theorem V.I. We}

present

it here since the

proof

^{is very} different. Let H be the center of G :

if y

^E

H,

x E G then _xy ⁼ _yx. Denote

by bx

^{the Dirac measure at x. Thus ~}

= ~e

where ex ⁼ xe ⁼ x for all x E G. Let

Ho

^{be the set of}

points

^{in H such that}

~cn

^and

~cn

^*

bx

^{are not}

mutually singular

^{for all n.}

If x E Ho

^then

eHo

^too.

COROLLARY 1. - Let the closed _group

generated by Ho

^{be H. Let v be a}

measure on H which is

absolutely

continuous with

respect

^{to the}

left

^Haar

measure on H. Then

~~

^{v *}

^~cn ~~

^{--~ 0}

provided v(H)

^{= 0.}

Proof By

^the Hahn Banach Theorem it is

enough

^{to show that}

if f

^is

a bounded measurable function ^on H such

that (

^v,

f ~ - 0

^whenever

~~

^{v *}

~~

^{- 0 then}

f

^{= Const. a. e. with}

respect

^{to the left Haar measure}

on H.

Now if 0 is a measure on H and is

absolutely

continuous with

respect

to the left Haar measure on H

then, by

^Theorem

1,

^{if x E}

Ho

^then

Thus

Now,

^this

equality

^{remains true for the closed}

subgroup generated by Ho namely

^{all of H.}

Thus f

^{= Const. a. e. with}

respect

^{to the left Haar measure} on H.

Throughout

^{the rest of the} paper ^we shall assume.

ASSUMPTION 1. ^- The measures

,un

^{are not}

pairwise mutually singular.

Thus there exists two

integers n and k

^{and a non}

negative

^{measure v}

with

v(G)

^> 0 such that

,u" > v,

^v.

Therefore

and we may ^use Theorem 1 with _r~ ⁼

COROLLARY 2.

If

^{and are not}

mutually singular

^then

Vol. XI, ^n° 2 - 1975.

202 S. R. FOGUEL

Remark. If k ⁼ 1 it follows that either

~~ I~

^{= 2 for all n or}

(~

^{--~ p. This «} zero-two »

law,

^was

proved

ⁱⁿ

[3],

^Theorem

3.1,

for

general

^Markov

operators

^{that are}

ergodic

^and conservative. In our case

the measure ~ induces a random walk which need not be neither

ergodic

nor conservative.

Now if

~)

^2014~ 0 then the

spectrum of J1

intersects the circum- ference of the unit circle

only

^{at the}

point

^{1: Let} qJ be a

homomorphism

of the commutative Banach

Algebra generated by

p. Then

I

^{1 -}

]

^{- 0} hence either

I ]

^{1 or = 1.}

By

^Gelfand’s

Theory

^the

spectrum of Jl,

^{even in the smaller}

algebra,

^touches

the circumference of the unit circle

only

^{at 1. In}

particular : a f where a ~ ]

^{= 1 then a = 1.}

Let us conclude with :

Thus

if v

^{* 0}

and Ilk * , f

⁼

f then (

^v,

f ) =

^0.

Conversely, put

^{A =}

{

^{v :}

v E Ma

^and

~i

^{v *}

0 ~. If (

^v,

f ) =

⁰

for all v E A then

by Corollary 2 (

^{0 *}

(~ - J1k), f) =

^{0 for all 0 E}

^Ma.

Thus f = J1k * f a.

^{e. ~, and our result} follows from the Hahn Banach Theo-

rem.

Remark. ^- If k ⁼ 1 and the

Choquet-Deny

^{Theorem holds}

namely :

J1

* f

=

f a.

^{e. ~,}

implies f

^{= Const. a. e. A then v E A if and}

only

^if

v(G) =

^0.

BIBLIOGRAPHY

[1] ^Y. GUIVARC’H, Croissance polynomiale ^et périodes ^{des fonctions} harmoniques.

Bull. Soc. Math. France, vol. 101, 1973, p. 333-379.

[2] ^{E. HEWITT and K. A.} Ross, Abstract harmonic analysis, Springer-Verlag, Berlin, ^1963.

[3] ^D. ORNSTEIN and L. SUCHESTON, ^An operator ^{theorem on} L1 convergence to ^zero with applications ^to Markov kernels. Annals Math. Stat., ^vol. 41, 1970, ^{p. 1631-} 1639.

(Manuscrit ^t reçu le 7 février 1975).

Annales de l’Institut Henri Poincaré - Section B