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(1)' tttttt tt tt ttttttl. Adjoint Ergodic Theorems for Amenable Semigroups of Operators By. t. Wataru TAKAHASHI Y6ichi KIJIMA and ''tt'(ReceivedMay31,1973) 1. Introduction. Let B be a Banach space and B* be the dual space of B, then we shall denote by <f; h> the value of fEB at hEB* and denote by L(B) [L(B")] the set of all bounded linear operators on B[B*], with operator norm topology. Particularly, we denote by L(B, B*) [L(B*, B)] the set L(B) [L(B*)] with the. weakoperatortopologydeterminedbyB*[B]. . ' In [4], Lloyd proved the following theorem:. Let {T(t): O<t<oo} be a semigrouP of operators on a Banach sPace B such that T(t)T(s) =T(t+s) and. ･, M=limIIT(t)1]<oo･ t-co. Then there exist oPerators PEL(B*) (not necessarily acijoints of oPerators in L(B)) in the closed convex hzall of {T(t)*: O<t<oo} in L(B*, B) zoith the ProPerty. T(t)*P=PT(t)*=P,. '. where T(t)* are acijoint oPerators of T(t). Such an oPerator is a Projection of. B*onto{hEB*:T(t)*h=h,O<t<oo}andhasthenormIIP]l:ilMl i･ ' This theorem is very usefui in the ergodic theory of Markov proces66s. In this paper, we shall extend Lloyd's theorern to the case of an amenable semigroup by considering that the semigroup {T(t): O<t< oo} is commutative,. The authors wish to express their hearty thanks to Professor Hisaharu Umegaki for many kind advices and suggestions in the course of preparing the present paper.. 2. Adjoint ergodic theorems. Let X be an abstract sernigroup and m(2) be the space of all bounded real valued functions of X with the supremum norm. An element ptEm(X)* (the dual space of m(X)) is mean on .S if ptt(e(t))=Il pt l[=1 where e(t)==1 for all tE2. A mean pt is left [right] z'nvariant'if pt,(A(t))= pt,(f<t)) [pt,(fS(t))=pt,(7C<t))].

(2) 2 Y. KiJiMA and W. TAKAHAsHi for all fEm(X) andsE27, where A(t) =f(st) [fS(t)==f(ts)] for all tE.S. A semi-. group that has a left [right] invariant mean is called left [right] amenable. A semigroup that has a left and a right invariant mean is called amenable; see [2].. The following Lemma was proved by Arens [1]: LEMMA. The closed unit ball of L(B*) is a comPact set in L(B*, B). By using this Lemma, at first we shall prove the adjoint ergodic theorem for a right amenable semigroup of operators on a Banach space.. b i ･pt 1･. `. THEOREM 1･ Let 2={T)t412-sei:frgguhpt"asmTelr::tlSle.semzgrouP zn L(B) such that. ST Then there ,exists an oPerator PEL(B*) in the closed convex hull of {T*: TE.X}. in L(B*, B) such thatT*P=Pfor all TE.X. Such an oPerator P is a Pro]'ection of B* onto the szabspace of vectors invariant under all T* in .X* and 1[Pil;SIML. PROOF. For an integer n, there exists S.E.X such that. tt. sup ll S.T II ;:$ M+ 1/n.. T. Denote by K. the convex hull of the set ,(ll).X*S,* and by iZr. ' the closure of. K. in L(B*, B) and let co K=A..K),. By Lemma, K is a nonempty compact convex. n= 1. subset of the multip!e M of the closed unit ball in L(B*, B). It is obviQus. co oo. that "X* U X*St*E U 2*Si Since each U* in X i･s linear and continuous with. i==n i=n B)-topology, k.. L(B*, is also invariant under U*E.X. Therefore we have i"KEK Now, by using Day's fixed point theorem [3], we obtain PEK such that T*P==P for all T in .X. Let P be such a point. Let {V.} be a generalized sequence in K. converging in L(B*, B) to PEK. That is,. ･s. I I. ". ;/. " --lim<f,V.h>==<f,Ph> ･' a. '. '. '. '. forfEBandhGB*. FromV.P=P,weobtainthat /, <f, PPh>==lim<f, V.Ph>. 'a. =lim<i Ph> a =<f, Ph>. Therefore, PP=P. Suppose that h, is an element such that T*ho==ho for T*EX* and <f, V..h>-><f, Ph> for fEB and hEB*. Then, from. ''' '. .<f,Pho>=lim<f,Vcrho> .. cr '' ' ' ' ' ,1...,.･ , , ==li.m<f,ho>, ,. ,... tt. '. ... .,.,. '.

(3) AdjointErgodicTheoremsforAmenableSemigroupsofOperators 3. ., =<f,ho>' it follows that P is a projection of B* onto the subspace of vectors invariant under all T" in ;*. '. Secondly, we shall prove the adjoint ergodic theoreM for a left･amenable semigroup of operators on a Banach space. q. THEOREM 2. Let X= {T} be a left amenable semigrouP in L(B) such that. ' ' S.T ', , ' M=infsup]ITS"<oo.. -･,. Then there exists an operator PEL(B*) in the closed conbex hull of {T*.:. T(Ei ･X}. in L{B*, B) such that PT*==Pforall TG.S. Such an oPerator P is.a Pr6vectzon. of B* and I[P[I5M. ''. '. pRooF･ For an integer snu.,ptih/ illle. ifllliiilti.iiJrnE."X S,UCh thal. t ttt. ' Denote by K. the convex hull of the set ee Si*2* and by iiih the closure of i =n. ooKh. - As in the proof of Theorem 1, we can obtain Kn in L(B*, B) and let K=A n=1 PEK such that PT*=P for all T in X. We' shall show that KKgK It is obvious that K.K.EK. for each integer n. If AEK},, there exists a net {V.}CK. such that V.-A in L(B", B). Since TV.->TA for all TEK. and TV.EK., we have TAEK}, andthence K.K},EK},. Similarly, if AEK., there exists a net {V.} gK. such that V.->A in L(B*, B). Since V.B-.AB for all BEi{i}, and hence IZ,]Z,g]Z,. Therefore we have KKgK Now, let us define i. . K,={PEK:PT*==PforallTEX}, ' then it is obvious that Ko is nonempty, closed and convex. Let P, QEKo, then. we have PQEKfrom KKgK. Since (PQ)T*==P(QT*) =PQ for all TGjX, We '. have PQEKo and hence KoK,gK,. Therefore, we can obtain PEK such that PP=P. In fact, using Zorn's lemma, we can obtain a minimal nonempty compact set MgK such that MMgM. If P,EM, MP, is a nonempty compact subset of M and MPoMPog.MP,. SinceM is minimal, we have MPo==M. Let. M6t{UGM: UPo=Po}, then Mo is a nonempty compact subset of M and MbMbgM6. Since M is minimal, we have M==Mb. Therefore, PoPo==Po･ Finally, we shall completely extend Lloyd's theorem, that is, we shall prove the adjoint ergodic theorem for an amenable semigroup of operators on a Banach space.. THEOREM 3. Let .S={T} be an amenable semigrouP in L(B) such that ･M=inf sup ]1 TST' II < oo.. S TT' '. 7'7zen there exist oPerators PGL(B*) in the closed convex hull of ,{T*: T(Ei.X}.

(4) ':4 ''" Y.KiJiMAandW.TAKAHAsHi in L(B*, B) such that PT* =T*P= P for all TGX. Such an oPerator P is a Pro]'ection of B* onto the subsPace ofvectors invariantunder all T* in .X* and ll P ll ;:;l M.. PRooF. For an integer n, there exists S.E.X such that sup ti TST' li SM+1/n. T, Tr Denote by K. the convex hull of the'set V X*Si*X* and i<r. the closure of i=n K. in L(B*, B) and let K=ooA 1?h. Then K is nonempty, compact and convex. ･ oo. e. . n=1. ,. and also we can prove X*Kg.K and KX*g.K By using Day's fixed point theorem [3], we obtain PEK such that T*P=P for all TE.X. Now let us define. K,={PEK:T*P=:P forall TE.X}, ' then Ko is a nonempty compact convex subset of K and KoX*-C- Ko. Therefore, by using Day's fixed point theorem again, we obtain PEKo such that PT* =P for all TEE. This completes ttie proof.. Since a commutative semigroup is amenable [2], Theorem 3 is true for the semigroup. The above Theorems are useful in the ergodic theory of -various field [5] [6] [7].. ' References. ' [1] ARENs, R., Operationsin function classes. Monat$heftefurMath.U.Physik.55. (1951), 1-19. ･'. [2] DAy, M.M., Amenable semigroup. Illinois J. Math.1 (1957),509-544. [3] DAy, M.M., Fixed point theorem for compact convex sets. Illinois:.J. Math. 5. (1961),585-490. ･'' ･ ',. s. [4] LLoyD,S.P., An adjoint theorem. Ergodictheory. Academicpress. NewYork. (1963), 195-201.. [5] TAKAHAsHi, W., Invariant ideals for amenable semigroqps of Markov operators. K6dai Math. Sem. Rep. 23 (1971), 121-126. '[6] TAKAHAsHi, W., Invariant functions for amenable semigroups of positiveJ,contractions on Li. K6dai Math. Sem. Rep. 23 (1971), 131-143.. [7] TAKAHAsHi, W., Ergodic theorems for amenable semigroups of positive contractions on L'. Sci. Rep. Yokohama National Univ. 19 (1972), 5-11.. DEPARTMENT OF MATHEMATICSi TOKYO INsTITUTE OF TECHNOLOGY AND DEPARTMENT OF MATHEMATICS, YOKOHAMA NATIONAL UNIVERSITY.. v.

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