Kawamura, Shinzō

Takemoto, Hideo

Tomiyama, Jun

English

Kawamura Shinzō

Takemoto Hideo

Tomiyama Jun

University of Szeged

Acta Sci. Math., 54 (1990), 191—137 State extensions in transformation group C*-algebras SHINZÓ KAWAMURA, HIDEO TAKEMOTO and JUN TOMIYAMA Introduction. Let X be a compact (Hausdorff) space and G a discrete group acting on X as homeomorphisms: x-+t(x), (x£X, t£G). Throughout this paper we denote by A the C*-crossed product associated with the topological dynamics (G, X). Our purpose is to study state extensions in A. Since G is discrete, the algebra C(X) of all continuous functions on Xis regarded as a C*-subalgebra of A and the restric-tion of a state of A is a state of C(X) again. We are interested in the correspondence of the family of states of C(X) with that of A. So we study how to extend a state /i ofC(A-) to a state or a tracial state of A. Of course n is identified with a probability measure (throughout this paper, a measure means a Borel measure, which is always regular on the compact space X). Ultimately we get an equivalent condition for a probability measure and a G-invariant probability measure on X to be uniquely ex-tended to a state and a tracial state of A respectively. In Section 1, we prove that a probability measure n on X has a unique state extension if and only if the measure fi(t( •)) is singular with repect to /i for all t in G except t=e. In Section 2, we prove that a G-invariant probability measure fi on X has a unique tracial state extension if and only if n(X')=0 for all t in G except t=e, where X' is the set of fixed points of X for t. In the theory of C*-algebras, the unique tracial state plays an important role (cf. [6], [7]). Hence it seems to be inter-esting to consider the condition on (G • X) under which A has a unique tracial state. Those conditions are given as an application of our second result. Notation. For a topological dynamics (G,X), we use s,t,u,m,n,e (= the identity) and x, y as elements of G and X respectively. We denote by Gx the isotropy group for x and X' the set of fixed points for t, i.e., Gx={t£G: /(*)=*} and X'= ?(*)=*}. The algebra C(X) is the abelian C*-algebra with supremum norm and *-operation: f*(x) =f(x), where the bar means complex conjugate. We denote by a, the canonical *-automorphim of C(X) induced by the action of t in G, i.e., Received December 10, 1986. 192 S. Kawamura,.H. Takemoto, J. Tomiyaraa a,if)(x)=f{t~1(x)) f o r / i n C(X). Let K(G, C(X)) be the set of those functions of G into C(X) which vanish outside finitely many elements. For t in G and / in C(X), Jbt means the function in K(G,C(X)) defined by ( /5 , ) ( / )=/ and (fS,)(s)=0 for s ^ t . Then every function <P in K(G, C(X)) is of the form: <P = 2ft&n where F is a finite subset of G. We consider K(G, C(X)) as a dense *-suba!gebra of A by defining ^-operation and multiplication as follows; ( 2 /,5,)*= 2 a(-«(/)^<-i and (Z fA)( 2 &«.)= 2 2 Mgs)s,s-For a measure no n X the topological support S(fi) of n means the smallest closed subset such that n ( f ) =0 f o r / i n C(X) with supp (f)aX-S(ji). Given a family e c measures, we denote by iA=© jU, the linear functional on K{G, C(X)) defined by H2fA)= 2 Mf,)-tiF r € F Since G is discrete, if i¡/ is positive definite then it is transform bounded on K(G, C(X)) in the sense of [3]. Hence ip can be extended to a state of A, which is denoted by \j/ again. The action of G on X determines, addition to {a,}, a canonical transformation group on the state space of C(X). Those are denoted by for a state fi on C(X), i.e., P,(fi)(f)=fi(<xt-i(f)) f o r / i n C(X), which is regarded as a measure on Xdefined by P,(fi)(E)=fi(t-i(E)) for each Borel set E in X. 1. State extensions. Let ip be a state of A. For each element t in G, let fi, denote the bounded linear functional of C(X) defined by ,) f o r / i n C(X). Then it follows that ^ = © fit. t£G Propos i t ion 1.1. Let ij/= © p, be a state of A. Then has the following properties: (1) ¡xe is a probability measure on X, (2) n, is absolutely continuous with respect to pe and P,(fie), and S(n,)(zS(ne)r\ t(S(fie)), <3) l iMf )=H<HJ) ) -Proof . (1) is trivial. (2) By the Cauchy—Schwarz inequality, we have, for / in C(X), (a) l/U/)|2 = I WW* S iA(<5(-i<5,)iA(/<5e/<5J = ne(\f\2) and (b) k ( / ) l 2 = l>KA)l2 = ^(¿t«,-i(/)<5e)|2 S ^(a,-.(/M-i(/)<5e)«K<5e) = State extensions 193 By inequality (a) and the regularity of ¿u and n,, we have that fi, is absolutely conti-nuous with respect to fie and S(fx,)czS(ne). For / in C(X) with s u p p ( f ) < z X — -t(S(ne)), it follows that supp (a, . i(/)) = i-1(supp ( / ) )c : / - 1 ( ; r - r ( supp (A,e))= = X—supp (fte). By inequality (b), we have that fx, is absolutely continuous with respect to and S(p,)czt(S(fxc)). (3) Let <P= £ f,d, be in K(G,C(X)), where F={t1, Q. Then t £ F ijj(&*$)= 2 f*t-'s(at-l(ftfs))—Given each set of complex numbers {A,}"=1, t , s£F setting X,ft in place of ft., we have i,j = 1 This means that the nXn matrix (/¿,-t¡fat,-1 (Jt/t)))ij i s positive. Hence fit- ,s(ce,-i(I,/J)=(Is/,))- Putting and ft=1, fe=at{f), we have Let ¡J. be a probability measure on X and e the conditional expectation of A onto C(X). Let ju(=/ioe) denote the canonical state extension of ¡i. In order to find a condition under which p. is the unique state extension, we consider the possibility of existence of another extension of Given a measure ¡ion X and a characteristic function /_E of a Borel set E in X, we define a measure y_En on X by XEH(J) = f / d f i for / in C(X) (= n(y.Ef)).~ E Then we have xEpt(n)(f)= f f{t(x))dn (= M(z.-.(£>«t-iC/))), t-HE) and it is easy to see that S(xEP,(n))=Ef)t(S(n)), where E is the closure of ¿J in X. For / in G, let P,(p)=P,(iJ.)a+Pt(fi)s be the Lebesgue decomposition of the positive measure P,(p.) with respect to fi. Namely there exists a measurable subset C, of X satisfying the condition: n(X—C,)=0 and, for each Borel set E in X with EcC„ P,(li)(E) = P,(n).(E) = Jk,(x)dfi, E where k, is the Radon—Nikodym derivative of P ,((*)„ with respect to \i. Let £>,= = {xeC,: k,(x)>0}, E,={x£Ct\ k,(x)^ 1} and F,={x£C,\ Since A;(.1(A-)=l/fc((?(x)) for x in D„ it follows that t~1(Dt)=Dt.1, i " 1 (£ ' ( )=F t . l and t~1(F,)=Et.l. Using those facts we prove the following proposition, and applying it we show a characterization for fi to have a unique state extension. 13 194 S. Kawamura,.H. Takemoto, J. Tomiyaraa P r o p o s i t i o n 1.2. Let ¡i be apositive measure on X. For a fixed t in G, let {/¿3}j€G be the family of measures on X defined as follows: (1) In case let pe=n, (i,=XEtpfc and p,=0 for s${e,t, r1}, (2 ) /« case <=r\ let ne=n, ^ = + ^ for si {e, t}. Then (& is positive definite. a£G Proof . It is sufficient to prove the statement only in the case of (1). Let 4>= = 2 fmK be in K(G, C ( 4 Then we have mtF = 2Ai(2«--(/-/-))+ft( 2 *m-i(Lf„))+>!,A 2 «„-,(/„/„)). m€F m~1n=t m-1n=t~1 The second term of the right hand side = f 2 /m (>"(*))/• (>«(*)) dp. £t m->n=t The third term = 2 («*-.(/,,/,)) = = / 2 *,*m-ALfn)dii= f 2 <*n-i(Lfn)dn = 1) m - 1 " = < - 1 E t = f 2 fm{n{x))fn{n(x))^ = / 2 fn{m(x))fm{m(x))dp. E m-'n=t~I F m~'n—t E,  "=• E, If m~1n=t, since for JC in F t_,, we have f *n-i{JJn)dn ^ f «„-,(/Jn)kt-, dp = J «„-,(/„ftdp^ilt). = F,-. F,-i = /«*->(/,,/„)<%->(/*)= / «,(«n-*(Lfn))dp= f ztn-.{Jn/„)dp = rt-1 f(F,-i) £, = J fn{m(x))fn(m(x))dp. E, E, Hence, the first term s 2 ( f«n,-^LfJdfi+ f <*„-,(Jnf„)dp) s E m~'n=t. State extensions 195 Therefore it follows that 2il/(4>*4>) ^ f % (fm{m(x))fm(m(x)) +fm(m(x))fn{m(x))+ B~ m-'n=i = f 2 (fJm(x))+fn(>n(x)Wm(™(x))+f„(m(x)))dti ^ 0. ¿j m~1n=t Theorem 1.3. Let p be a probability measure on X. Then p has a unique state extension if and only if P,(n) is singular with respect to p. for all t in G except t=e. Proof . Let \]/= © /1, be a state extension of fi. By (2) of Proposition 1.1, each <€G PI, is absolutely continuous with respect to FIE=FI and P,(N). Hence the assumption on {/?,(aO}Kc implies that fi,=0 for all t^e. Next suppose that /?,(/i) is not singular with respect to ft for some t^e. Set ij/=(\j/,+\l/t.l)/2, where ip, and ipt-i are the states constructed in the above propo-sition corresponding to / and i - 1 respectively. If n(E,)=0, then /¿(F,)>0, whereas we have /'№-.) = n(rl(F<)) = AG0.(>?) = f k,(x) dp > 0. Hence ^ is a state extension of fi, which is different from fi=pos. In the following, we give an example of a state of C(X) which has a unique state extension and whose topological support is the full space X. Example 1.4. Let Re be an irrational rotation on the unit circle [0,1). Let WnLi be ^ e set of all rational numbers in [0,1). We define a probability measure PQ on [0, 1) by HQ(E)= 2 1/2" for E<Z[0,1). Then {P„(FIQ)UZ are mutually singular and S(P„(fiQ))=[0,1) for all n in Z. Namely pQ has a unique state exten-sion but S(P„(fiQ)) is the full space. The theorem mentioned above gives a characterization for the pure state p{x) of C(X) to have a unique pure state extension. Namely we have the following. Coro l la ry 1.4. Let be the Dirac measure on a point x of X. Then ft{xj has a unique (pure) state extension if and only if Gx={e). Here we note that this result can be derived by Lemmas 4.19, 4.22 and 4.25 of [3]. (Though the second countability on G and X was assumed in [3], the proofs of these lemmas are still available here.) Moreover ANDERSON [1] has given an equivalent condition for a pure state to have a unique pure state extension in a more general case. He proved that for any C*-13* 196 S. Kawamura,.H. Takemoto, J. Tomiyaraa subalgebra D of a C*-algebra C, a pure state pofD has a unique pure state extension to C if and only if C is £>-compressible modulo n, i.e., M{ycd-e\\\ d£D,Ó S d^ 1, p(d) = 1, e£D) = 0 for each c in C. Of course, in our case, this condition on p(xi is equivalent to Gx={e). In the case of state extension, Example 1.4 shows that the condition mentioned above is merely a sufficient condition for p to have a unique state extension. In fact, since the identity is the only element in C(X) with pQ(d)=1, O s d s l , we have \\dcd—e\\ = = | | c - e | | s d i s t ( c , C ( * ) ) > 0 for c$C(X). As a matter of course, it is interesting to study representations of A associated with states extended from //{x}. Those are discussed in [4]. 2. Tracial state extensions. Let p be a G-invariant probability measure on X and [¡/ = 0 n¡ an extension of p.. We show a necessary and sufficient condition for p FGG to have a unique tracial state extension. First we consider the condition on 4>= (¡B p, tiG under which \J/ is a tracial state. P ropos i t i on 2.1. Let p, be a state extension of a probability measure T€G p on X. Then \J/ is a tracial state of A if and only if 6G satisfies the following two conditions: (1) S(,pt)aX' for all t in G, (2) f°r al1 s and t in G. Proof . Suppose that ^ is a tracial state. Then f o r / a n d g in C(X), we have (*) ^(fas(g)) = Hf«s(g)Sst) = = MgStf$s) = t(gxt(f)Sts) = pts(ga,(f)) Putting s=e in (*) , we have p,(fg)=pt(gat(f))- If x$Xt, then there exists a neighbourhood U of x such that UC\t~1(U)=Q. For any non-negative real-valued continuous function / on X with s u p p ( f ) a U , we have /* , ( / )= /* , ( ! / / / / )= =/¿ , ( | / /a ( ( / / ) )=() . Thus S(pt)czX'. Next, put t=us~1 and g= 1 in (* ) . Then we have /W>(/) = -•(/)) = ff(su~l(x)) dpu = Jf(s(x))dpu = xu x" = = &<>„)(/)• Conversely we suppose that {^(}l€G satisfies the conditions. Then we have = ¡}t->Mf*s(g)) = PtsW)«,Ag)) ="\ = ff{r1(x))g((tsri(x))d^= ff(t-1(x))g(x))dpts = pts(a,(f)g). X" State extensions- 197 This implies that \p (fSsgS,)=\j/ (gS,fSs). By the linearity and the continuity of \j/, <?)=,/,(«?.<*>) for every $ and !P in A. Given a state \p= © p„ in many cases it is easy to check whether the family <ec . {//,},eG satisfies or not the conditions of the above proposition. However, given a family {/i,},ec of measures on X, it is not easy to see whether </ '=©// , is positive (gG definite or not. Here we give a systematic construction of a (tracial) state extension. We denote by H(G) the family of subgroups of G. Let J be a map of X into H(G). We put X'j = {x£X: 7(x)—/} and denote by yt the characteristic function of Xj. When X'j is a measurable set, is the measure on on X defined by y,n(f) = = J fdn= J fXtdn for / in C(X). X'j X Propos i t i on 2.2. Let J be a map of X into H(G) with the properties: (1) XJ is a Borel set for all t in G, and (2) J{x)czGx for all x in X. Let fi be a probability measure on X and //,=/,/! for each t in G. Then i]/= © p, is positive definite. In tiG addition, if p is G-invariant and (3) J(t(x))=tJ(x)t~1 for all t in G, then \]/ is a tra-cial state. Proof . Let <£= 2 f A be in K(G, C(X)), where F is a finite subset of G. tiF In case t~*s is in J(x)czGx, s(x)=t(x). Thus we have = *l>(2 <5,-//A) = H 2 = t,s(F t,s£F = 2 fMi(x))fs(t(x))x,-H(x)dfi = UsiF £ = f 2 f,(t(x))f(s(x))z,-is(x)dn- > X t,s£F For x in X, let F=F 1 U. . .UF„ be the disjoint partition of Fcorresponding to the equivalent relation determined by the subgroup J(x) of G, i.e., t and s belong to the same F, if and only if t~1s£j(x). Then we have 2 f,{t(x))fs(s(x))x,-.(*) = 2 2 /<(<(*))fs{Hx)) = s,(€F i=l s,t(zFt i = l I £Ft Hence is the integral of the non-negative function on X, so it follows that \1/(<P*<P)^ 0. 198 S. Kawamura,.H. Takemoto, J. Tomiyaraa Next we assume the additional condition. Then we have s(XJ)=X}"~' for all j and / in G. By the G-invariance of n, we get the following; f o r / i n C(X), P.QhKf) = /i,(«s-(/)) = f/(s(x)) dp = f f ( y ) dn = XJ XS}'-1 Since S(p,)czX' by Condition (2), Proposition 2.1 implies that \J/ is a tracial state. In the following, we show several examples of J, ¡i and ^ treated in Proposition 2.2. Example 2.3. Let J(x)={e) for all x in X. Then XJ=X and XJ=0 for aU t^e. Hence i]/=ji for each positive measure n on X. Example 2.4. Let x be a point of Xaadp=p{xi. Let J(x)=Gx and J(y) = {e} for y^x. Then * j={x} for t(^e)£Gx and XJ=0 for t$Gx. Then ij/= © fi, t€Gx is a pure state extension of (cf. Section 1). Example 2.5. Let X consist of a single point {x} and / ( x ) = / / for a (resp. normal) subgroup H of G. Then X*=X for t£H and * J = 0 for t^H. Since A is regarded as the group C*-algebra C*(G), i¡/ becomes a (resp. tracial) state of C*(G) with the property iA($)= 2 f o r ^l1(G)cC*(G). In the case <€H H={e), i¡/ is the conditional expectation e. On the other hand, when H=G, \j/ is a multiplicative linear functional of A, i.e., it is the trivial representation of C*(G). Example 2.5 gives two typical tracial states of C*(G). However, in contrast to C*(G), the reduced C*-algebra C*(G) does not necessarily have two tracial states. In fact, Powers [10] has shown that the conditional expectation is the unique tracial state of C*(F2) by using his result that C*(F2) is simple. Example 2.6. Let J(x)=Gx for each x in X. Then we have XJ—X'. Thus, if p is a probability measure then © %xtp is a state extension of p. In addition, »€G if p. is G-invariant then Hence, by Proposition 2.1, ^ is a tracial state. We get the following theorem by Proposition 2.1 and Example 2.6. Theo rem 2.7. Let p be a G-invariant probability measure on X. Then n has a unique tracial state extension if and only if p(X')=0 for all t except t=e. Coro l l a ry 2.8. The C *-crossed product A has a unique tracial state if and only if there exists exactly one G-invariant probability measure on X and p(Xty—Q for all t except t=e. State extensions- 199 The standard theory of topological dynamics (cf. Chapter II of [2]) shows that the two conditions on (G, X) in Corollary 2.8 are independent. Now, in the theory of C*-algebras, faithful tracial states such as the unique tracial state of C*(F^ have played an especially important role. Thus we consider faithfulness of tracial state extensions. In general, = © in Proposition 2.2 is not necessarily faithful. In »ec fact the canonical homomorphism of C*(G) (cf. Example 2.5) is not faithful. Here let us assume that G is amenable. Let n be a CP-invariant faithful measure on X. Then the tracial state extension fi=(ioe is faithful because the GNS representation of A by p. is nothing but the C*-reduced crossed product on the Hilbert space /2(G)<g> L2(X,fi), which is isomorphic to A (Theorem 7.7.7 of [5]). For p, and tZG l>co>0, let {¡/a=ojp.+(l —co)\l/. Then i]/m is a tracial state extension of p and aji^ip,,. Therefore i¡/a is faithful on A. Then we get the following. Coro l l a ry 2.9. Suppose that G is amenable. Then A has a faithful unique tracial state if and only if there is exactly one G-invariant measure p on X, which satisfies the properties: (I) S(p)=X and (2) p(X')=0 for all t except t=e. If the support of the unique G-invariant measure is X, then (G, X) is minimal (cf. Chapter II (Exercise 7) of [2]). In addition, if G is abelian, X'=0 since X' is G-invariant. Then we have the following. Coro l l a ry 2.10. Suppose that G is abelian. Then A has a faithful unique tracial state if and only if there is exactly one G-invariant measure on X with S(p) = X. We note that the unique tracial state of the rotation C*-algebra is a prototype of Corollary 2.10 and a motivation of our discussion. References c [1] J. ANDERSON, Extensions, restrictions, and representations of states on C*-algebras, Trans. Amer. Math. Soc., 249 (1979), 303—329. [2] J. R. BROWN, Ergodic theory and topological dynamics, Pure and Applied Math., Academic Press (New York—San Francisco, 1976). [3] E. G. EFFOS and F. HAHN, Locally compact transformation groups and C*-algebras, Mem. Amer. Math. Soc., no. 75 (1967). [4] S. KAWAMURA and J. TOMIYAMA, Properties of topological systems and corresponding C*-algebras, Tokyo J. Math., 13 (1990), no. 2. [5] G. K. PEDERSEN, C-algebras and their automorphism groups, London Math. Soc. Monographs 14, Academic Press (London—New York, 1979). 20Q S. Kawamura, H. Takemoto, J. Tomiyama: State extensions [6] M. PIMSNER and D. VOICULESCU, Imbedding the irrational rotation O-algebra into an AF-algebra J. Operator Theory, 4 (1980), 201—210. [7] R. T. POWERS, Simplicity of the C-algebra associated with the free group on two generators Duke Math. J., 42 (1975), 151—156. (J. T.) DEPARTMENT OF MATHEMATICS TOKYO METROPOLITAN UNIVERSITY FUKAZAWA, SETAGAYA, TOKYO 158 JAPAN (S. K.) DEPARTMENT OF MATHEMATICS FACULTY OF SCIENCE YAMAGATA UNIVERSITY YAMAGATA 990, JAPAN <H. T.) DEPARTMENT OF MATHEMATICS COLLEGE OF GENERAL EDUCATION T0HOKU UNIVERSITY SENDAI 980, JAPAN 

State extensions in transformation group C*-algebras

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State extensions in transformation group C*-algebras

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