132 Proc. Japan Acad., 53 (1977) [Vol. 53,
36. On
the Existence o f Invariant Functions for
Representations o f Amenable Semigroups
Markov
By Kokichi SAKAI
College of Liberal Arts, Kagoshima University (Communicated by Kosaku Y0sIDA, M. J. A., March 12, 1977)
§ 0. Introduction. Let (X, ~, m) be a c-finite measure space and S be a left amenable semigroup. By Ll and L°° we denote the usual Banach spaces L1(X, ~', m) and L°°(X, ~, m) respectively. Let T
_ {TS ; s e S} be a representation of S by positive linear contractions on Ll. For the sake of brevity such T is called a Markov representa-tion of S on Ll. By co(T) we denote the convex hull of {T8; s e S} and by O(T) the closure of co(T) with respect to the operator norm topology. For this T we consider the following conditions :
(A) There exists a strictly positive function f in Ll such that T S f =f for all seS.
(B) Every operator in d(T) is conservative. (C) TS is conservative for every s e S.
Then it is obvious that the condition (A) implies (B) and (C). In this paper we shall prove the next theorems.
Theorem 1. For any Markov representation T = {T,; s e S} o f a left amenable semigroup S on L', the conditions (A) and (B) are mutu-ally equivalent.
Theorem 2. Let S be an extremely left amenable semigroup. Then for any Markov representation T = {T; S s e S} o f S on Ll with the following property :
(1) Ts (gh) = Ts (g)T *(h) for any g, h e L°° and s e S, the conditions (A) and (C) are mutually equivalent.
Theorem 1 is proved by Brunel [1] for the case when S is the ad-ditive semigroup of positive integers, and by Horowitz [3] for the case when S is commutative. In the author's paper [4] we shall show that the main theorem in [3] is also valid for the case of left amenable semi-groups of Markov operators.
§ 1. Proof of Theorem 1. Let S and T= {T3; s e S} be as in Theorem 1. By L(T) we denote the closed linear subspace of L°° gen-erated by {Ts h-h; s e S, h e L~} and put L+(T)={h e L(T); h>0}. Then the next lemma is well-known (e.g., see Granirer [2, Theorem 5]). Lemma 3. For any f e L°° the following equality holds:
(2) inf {If-hI1; ~~ h e L(T)}=inf {~f Q* f H~ ; Q e co(T)}. Especially i f S is extremely left amenable, then
No. 3]
Markov
Representations
of Amenable
Semigroups
133
( 3)
inf {IIf-hM; h e L(T )} = inf {I
I T s f I. ; s e S}.
Combining with Lemma 3 and Theorem 1(5) in Takahashi [5], we
have
Lemma 4. For any Markov representation T = {T. ; s e S} o f S on
L', the condition (A) holds i f and only i f L+(T) _ {0}.
The next lemma is essential for us to prove Theorem 1.
Lemma 5. I f h e L(T), then there exists a V e i(T) such that
limn~.IIV*nhII..=0.
Proof. Let {an ; n=1, 2, ... } be a sequence of positive numbers
with
ai =1, and put pn = ~i=1 ai and jn =1- j3,. Moreover we
choose an increasing sequence {Tn}
of positive integers satisfying
limRn
=0. We can take a sequence {Q}
nin co(T) such that
(4)
II
Qn hn ll~
< 1
for n=1, 2,
n where h1=h, hn=h+~2; ~k2 o V*k h for n>2, and V1= j3' 1 =1 a~Q3.
Indeed, since hn e L+(T) for all n> 1, we have inf {I Q*hn I ; Q e co(T)} = 0 by (2). So the desired sequence {Q,j can be taken inductively.
N N
We now put V==1 1 c Q2 and Vn=fin 1 ~i n+1 aZQ2. Then V e co(T), N ~
V = j3nV n + pnV n, and from (4) we have
(5) II VnVnkhll~< 1 for all n>1 and 0<k< n.
n+1 r
For any given E > 0 we can find a positive integer n such that j9 h II. <~/2 and (n+1)-1<e/2. Putting N=rn, by (5) we have
1 V*h =1( nVn +~nVn)hll~<~3n 1 VnNh I~+ 1
n 1 (1-i) <s. ~ + So (I V*khI <E for all k>N. Hence this V has our desired property. q. e. d.
Using Lemma 5, we can prove the following lemma by the same method as in Theorem 1 in [1].
Lemma 6. For any h e L+(T) there exists a U e co(T) such that ~k=o U*kh e L~.
From Lemma 6 it follows that if L+(T) contains a non-zero func-tion, then in i(T) there exists at least one operator which is not con-servative. Hence if the condition (B) holds, then L+(T) _ {0}. Owing to Lemma 4, we can conclude that the condition (B) implies (A) for any Markov representation of S on L'. Thus Theorem 1 is proved completely.
§ 2. Proof of Theorem 2. Let S and T = {T3; s e S} be as in Theorem 2. Suppose now that L+(T) contains a non-zero function. Then there exists an A e ~', m(A) > 0 such that the indicator function h=IA of A belongs to L(T). By (3) we can take an element s e S satisfying (I Ts hll~ <1. Since h= h2, we have llT?hl.=II (T*h)2I(.<II T? h.
134 K. SAKAI [Vol. 53, So I T s h I I~ = 0. This means that T S is not conservative. Hence recall-ing Lemma 4, we conclude that the condition (C) implies (A) for any Markov representation of S on Ll with (1). Thus Theorem 2 is proved completely. [1] [2] [3] [4] [5] References
A. Brunel: New conditions for existence of invariant measures in ergodic theory. Springer-Verlag, Lecture notes in Math., 160, 7-17 (1970). E. Granirer : Functional analytic properties of extremely amenable
groups. Trans. Amer. Math. Soc., 137, 53-75 (1969).
S. Horowitz : On finite invariant measures for sets of Markov operators. Proc. Amer. Math. Soc., 34, 110-114 (1972).
K. Sakai: On finite invariant measures for Markov representations of amenable semigroups (to appear in Sci. Rep. Kagoshima Univ., 26
(1977)).
W. Takahashi: Invariant functions for amenable semigroups of positive contractions on L1. Kodai Math. Sem. Rep., 23, 131-143 (1971).
