Szűcs, József M.

English

Szűcs József M.

University of Szeged

Acta Sci. Math., 48 (1985), 477—476-On G-finite ^-algebras* JOSEPH M. SZUCS Dedicated to Professor Karoly Tandori on his 60th birthday Let M be a W *-algebra and G a group of ^-automorphisms of M. In [3] we have proved that if there exists a faithful G-invariant normal state cp on M, then for every t£M, the w*-closure of the convex hull of the orbit of t under G contains a unique G-invariant element t°. (In fact, we have proved this result under the more general assumption that the family of G-invariant normal states on M is faithful, i.e., M is G-finite. If M is <7-finite, for example, if M is an operator algebra in a separable Hilbert space, then this assumption obviously implies the existence of a faithful G-invariant normal state on M). In the present paper we shall prove that the assump-tion of normalcy of q> is superfluous in this theorem in case G contains all inner auto-morphisms of M. In fact, we shall prove the stronger result that the mapping t-~t° is normal, i.e., M is G-finite [3]. Under additional hypotheses, we shall also prove that (p is itself a normal state. At the end of the paper we shall make two comments on our paper [4]. P r o p o s i t i o n . Let M be an Abelian W*-algebra and G a group of *-automor-phisms of M. If there exists a faithful G-invariant (not necessarily normal) positive linear form <p on M, then M is G-finite. (For the notion of G-finiteness, cf. [3] J P r o o f . Let e be the least upper bound of the supports of all G-invariant normal positive linear forms on M. According to [3], we have to prove that e=l. Assume on the contrary that e^l. Since e is a G-invariant projection in M, the restrictions of the elements of G to the W*-algebra M{\ —e)=(l —e)M{\ —e) form a group Gx_ e of *-automorphisms of M(\ —e). By the definition of e, the only,Gx-e-invariant normal positive linear form on M( 1 —e) is the Zero functional. Consequently, to prove the theorem, i.e., to obtain a contradiction to the assumption e ^ l , it is suf-*) This work was supported in part by. organized research money granted by Texas A&M University at Galveston. Received July 23, 1984. 478 Joseph M. Szűcs ficient to show that there is a nonzero G1_(,-invariant normal positive linear form on M{\— e). Since the restriction of <p to M ( 1 —e) is a faithful Gx_e-invariant normal positive linear form on M{\— e), we may assume that e = 0 and thus M=M(l — e), <J1_e=G' and <pi-e=<p. In other words, we have to prove that under the hypotheses of the theorem, there exists a nonzero (/-invariant normal positive linear form on M. Let S denote the family of those projections p in M for which t-~<p(tp) = (p(ptp) is a normal positive linear form on M. We are going to show that (1) if p,q£S, then p\/q£S (2) sup S = l . To prove (1), let p, q£S. Then by the commutativity of M, we have p\/q~ —p+q —pq and thus the functional t — (p(t(p\Jq)) = (p(tp)+(p{t(q —pq)) is a normal positive linear form on M. (Because p, q£S and q—pq^q.) This proves (1). To prove (2), we have to show that every nonzero projection pin M majorizes a nonzero projection belonging to S. This can be shown by using arguments of J . DIXMIER [ 1 ] , which originate from Lebesgue's work. Let p be a nonzero projection in M. Consider a normal positive linear form p on M, such that p(p)^(p(p). Me are going to prove that there exists a nonzero projection q in M, such that (p(r)^p(r) for every pro-jection r in M, such that r^q. Then the spectral decomposition theorem will imply that (p(t)^p(t) for every t(LMq, ¿SO. Since every positive linear form majorized by a normal positive linear form is normal [1], this will prove (2). Now assume on the contrary that every nonzero projection q^p in M majorizes a nonzero projection r£M such that <p(r)>/z(r). By Zorn's lemma, there exists a maximal family C of mutually orthogonal nonzero projections J in M such that q>(s)>fi(s) for sdC. By the indirect hypothesis, 2 s € c s = p . Then (p(p)=(p{2sic ^ Z s e c V i t i ^ sic which contradicts the choice of p (i.e., that p.(p)^q>(p)). Consequently, there is a nonzero projection q^p in M, such that <p(r)^p(r) for every projection r£M majorized by q. Hence (2) is proved. Since S is an upward directed set, it may serve as an index set for generalized sequences. We shall prove that (*) <K0 = l™p6S<H'/>), t€M exists (and is finite). First let iSO. If p^q and p, q£S, then the equality tq = tp + t(q —p) shows that tq^tp. By the positivity of <p, the function p-*(p(tp) is a nondecreasing nonnegative numerical-valued function on S and <p(tp)^q>(t) for p£S. Consequently, the finite limit \\mpis<p(tp) exists and is equal to sup {<p(tp): p£S}: (**) \impiS(p(tp) = sup{<p(/p): peS}, / S O . The existence of Mmpisq>{tp) for all t£_M follows by linearity. On G-finite W*-algebras 479 It is clear that ip is a positive linear form on M. Moreover, i¡/ is normal. Indeed, it is an elementary observation that for p£S, the functional t-*(p(tp) is normal on M. The normalcy of ip follows from ( * * ) by using the elementary result that the supremum of an upward directed family of normal positive linear forms is normal [1]. Now we are going to prove that if/ is (/-invariant. First let us consider any ele-ment Pq of S, g0 of G and t0 of M. Since <p is G-invariant and p0£S, the linear form t->-(p(tg0(p0)) is normal. Consequently, g0(pa)£S. We have <Hgo(/o)go(Po)) = Em p 6 S (p(go(t0)g0(p0) p) = limpg9o(po)jpes<p(g0(4)go(po)i>) = = limpS3o(po),p€S <P(go(?o)go(/>o)) = <p(go('o)go(Po)) = <p(go(t0p0)) = 9(t0p0) = = limp§Po,P€S<P('o/>oP) = hmpZS <P(loPoP) = ^(toPo)-So we have shown that ^(g0(io)go(Po))=xl/(toPo)- By using property (2) of S, proved above, we can let pa w*-converge to 1 in this equality. Then relying on the normalcy of t/> and on the continuity properties of g„, we obtain that i//(g0(/0))=i/'(/0). Since g0£G and / 0 € M have been chosen arbitrary, we have proved that ip is G-invariant. Finally, \J/ is not identically zero. It is in fact faithful. Indeed, if t£M, t^O and t^0, then i>(T)=sup WUp)'- p£S}- By property (2) of 5", we have <p(tp)^Q for some p€_S. Therefore, >J/(t)>0 and the proof of our proposition is complete. T h e o r e m . Let M be a W*-algebra and G a group of %-automorphisms of M containing the inner automorphism group. If there exists a faithful G-invariant (not necessarily normal) positive linear form on M, then M is G-finite. P r o o f . Since G contains the inner automorphism group, q> is central [1]. Let teM be such that t*t= 1. Then < p ( l - t t * ) = <p(l)-(p(tt*) = (p{t*t)-(p(tt*)=Q. Since 1 — tt* is a projection, the faithfulness of <p implies that //* = 1, i.e., M is a finite W*-algebra. Let Z denote the center of M. It is easy to see that Z is invariant under the elements of G. Consequently, we can apply Proposition to Z, the restric-tion G z of G to Z and the restriction of <p to Z. We obtain that Z is Gz-finite. The G-finiteness of M follows from (simple) results of [3]. C o r o l l a r y . Suppose that under the hypotheses of the theorem, for every element t of the center Z of M, the uniformly closed convex hull of the orbit of t under G contains at least one G-invariant element. If the restriction of (p to the algebra Z° of G-invariant elements of Z is normal, then <p is normal on M. P r o o f . Let I be the inner automorphism group of M. We know [1] that for every t£ M, the norm closure of the convex hull of the orbit of t under I contains at least one element -f t of Z. Moreover, by the hypotheses of the corollary, the norrfi closure of the convex hull of the orbit of ft under G contains a G-invariant element tG. It is clear that tG is a G-invariant element in the norm closure of the convex hull of 480 Joseph M. Szűcs the orbit of t under G. Now let / c be the G-canonical mapping of M onto MG [3]. Since tG is the unique G-invariant element in the w*-closure of the convex hull of the orbit of t under tG, we have tc=tG and tG is in the norm closure of the convex hull of the orbit of t under G. Since (p is norm-continuous and G-invariant, (p(t) = cp(tG), i.e., cp is the compo-sition of the mappings t-+tG (t£M) and s-*(p(s) (s£Mg). We know [3] that t—tG is normal. If <p is normal on MG, then this composite mapping, i.e., (p, is also normal on M. R e m a r k 1. We do not have to use generalized sequences in the proof of the Proposition if G, as a subset of the space of linear self-mappings of M is separable in the topology of pointwise it>*-convergence. In this case we can choose a dense coun-table subgroup G 0 = { g i , g2, ...} of G, a non-zero projection q in M, such that (p is normal on Mq and take the ordinary limit \j/(t)=Y\m„^ca<p(t[g1 (</)V ••• (<?)])• It can be shown that \j/ is a non-zero G-invariant normal positive linear form on M. It is easy to see that G is separable if the predual of M is separable. This will always be the case if we only consider H^*-algebras M of operators in a separable Hilbert space. R e m a r k 2. The assumption of Proposition that <p is faithful is essential. Indeed, let G be an abstract infinite Abelian group. Then G acts naturally on M=l°°(G) as a group of ^-automorphisms. A G-invariant state on /" (G) is nothing else but an invariant mean on G. We know that there are infinitely many invariant means on G, none of which are normal. In this situation the entire proof of the theo-rem is valid except that i/' will be identically zero. Indeed, S will consist of the charac-teristic functions of finite subsets of G. It is an elementary fact that every invariant mean is zero on such functions. Finally, the author would like to make two comments on his paper [4]. The first comment is that in Proposition 2 and in its corollary the assumption that M is ff-finite should be replaced by the assumption that the predual of M is separable. The second comment is that all the results of the above mentioned paper remain valid if G is only assumed to be an amenable group (instead of an Abelian one). Indeed, if ( / , c C is a summing sequence [2], then it is easy to prove that under the hypotheses of Lemma 1, the sequence \UJ\~1 2gtv 8(0 ^-converges to tG for every t£B*. The remaining results of the paper can be extended to amenable groups G without altering the proofs. Added in proof: By adapting the method of proof of Proposition to the nonabelian case, Theorem can be broved withoit the assumption that G contains the inner automorphisms. As suggested by R. R. Smith, this can be done even in a simpler manner by appealing to the decomposition of a state into singular and normal parts. On G-finite W*-algebras 481 References [1] J. DIXMIER, Les algèbres d'opérateurs dans l'espace Hilbertien (Algèbres de von Neumann), 2nd éd., Gauthier-Villars (Paris, 1969). [2] F. P. GREENLEAF, Ergodic theorems and the construction of summing sequences in amenable locally compact groups, Comm. Pure Appl. Math., 26 (1973), 29—46. [3] I. KOVÁCS and J. Szűcs, Ergodic theorems in von Neumann algebras, Acta Sci. Math., 27 (1966), 233—246. [4] J. Szűcs, Some weak-star ergodic theorems, Acta Sci. Math., 45 (1983), 389—394. TEXAS A&M UNIVERSITY DEPARTMENT OF GENERAL ACADEMICS MITCHELL CAMPUS GALVESTON, TEXAS 77553, U.S.A. 31 

On G-finite W*-algebras

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On G-finite W*-algebras

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