Szűcs, József M.

English

Szűcs József M.

University of Szeged

Acta Sci. Math., 52 (1988), 141—144 Normalcy is a superfluous condition in the definition of (7-finiteness* JOSEPH M. SZŰCS Dedicated to Professor Károly Tandori on his 60th birthday Let M be a W*-algebra and let G be a group of *-automrophisms of M. In [2] 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° and the mapping /—/ff is normal. (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 [2]. If M is <r-finite, for example, if M is an operator algebra in a separable Hilbert space, then this assumption ob-viously implies the existence of a faithful G-invariant normal state on M.) In the pre-sent paper we shall prove that the assumption of normalcy of cp is superfluous in this theorem (cf. Theorem). Under additional hypotheses, we shall also prove that (p itself is a normal state (cf. Corollary 1). Furthermore, we shall prove some converse results (cf. Corollaries 2 and 3). For the general theory of H/*-algebras, we refer the reader to [1] and [3]. At the end of the paper we shall make two comments on our paper [4]. Theorem. Let Mbea W*-algebra and let Gbea group of * -automorphisms ofM. If there exists a faithful G-invariant state cp on M, then there exists a faithful G-in-variant normal state i¡/ on M, i.e., M is G-finite. Proof**. Let (p=<p„+(ps be the canonical decomposition of <p into normal part <p„ and singular part cps [3]. Consider an element g£G. Then <p„(g-) is normal due to * This work was supported in part by organized research money granted by Texas A&M Uni-versity at Galceston. ** The author's first proof of this theorem was much more complicated. This proof originated from a comment by R. R. Smith at a seminar at Texas A&M University, College Station. Received January 8, 1985. 142 Joseph M. Szűcs the continuity properties of g. On the other hand, (ps(g-) is singular, since a positive linear form p on M is singular if and only if every nonzero projection p£M majorizes a nonzero projection qdM such that p(q)=0 [3]. Since (p is ^-invariant and the decomposition into normal and singular parts is unique, we obtain that <p„ is ¿^in-variant (for all g£G). Furthermore, <pn is faithful. For let p be a nonzero projection in M. Since <ps is singular, there exists a nonzero subprojection q of p in M, such that (ps{q)=0. Then (pn(j>)=<p„(q)=<p(q)-(ps(q)=<p(q)>0 because q> was assumed to be faithful. Summing up, we can choose ip = (pn. Coro l l a ry 1. Let M be a W*-algebra and G a group of * -automorphisms of M. Suppose that for every t£M, the norm-closed convex hull of the orbit Gt of t under G contains at least one G-invariant element. If q> is a G-invariant faithful state on M and the restriction of cp to the fixed-point algebra MG is normal, then (p is normal. Proof . According to Theorem, M is G-finite [2]. Consequently, the G-invariant element, say tG, in the norm-closed convex hull of Gt is unique [2]. Moreover, the mapping t—tG: M—MG is normal [2]. Since cp is G-invariant and norm-continuous, q>{t)—(p{tc) (t£M). Therefore, the mapping t-^cp(t): C is the composite mapping of t-*t°: M—MG and t-~cp(t): MG-~ C. Since both of these mappings are normal, cp is normal. Coro l l a ry 2. Let M be a W*algebra and G a group of * -automorphisms of M. Suppose that for every t£M, the w*-closed (norm-closed) convex hull of the orbit Gt of t under G contains exactly one G-invariant element, say t°. If for isO, t7±0, then M is G-finite. Proof . The mapping t->-ta: M->-MG is linear. In the case of the norm-closed convex hull, this can be proved as follows. The homogeneity of the mapping t-*tG is obvious. To prove its linearity, let t, s£M and let £>0 be a given number. There exists a v0 in the convex hull conv G of G, such that ||u0(/)— ZG|| <e/2. Similarly, there exists Did conv G, such that ||u1i;0(5)—iG||<£/2. Since every element of G has norm 1, we have ||i>i»0(f)— tG\\<e/2. Consequently, | |^v0(t-hs)—(tG+Jg)||^ ^l |f i I ,o(0~ i G l l + ll i ;i t 'o(5 ')—jGll<e- Since £>0 was arbitrary, this proves that In the case of the w*-closed convex hull, the linearity of i—iG can be proved as follows. The homogeneity of i—iG is obvious. Let us verify its additvity. Let s, t£M. Then there exists a net vn in conv G, such that lim vn(s)=sG. Since the unit ball of n M is iv*-compact, there exists a subnet vk of vn, such that i f t=lim vk(t) exists. Then lirri i)=limyfc(5)+lim t > f e ( i ) = l i m b e l o n g s to the w*-closed convex hull of G ( J + t). By the definition of (tk)G, there is a net w„ in conv G, such that lim«'n(/jt)=(iJt)c. Then 1 i m w n ( + =1 im [ iv„()+w„(/ t ) ]=l im [ i G + H n n n Normalcy is a superfluous condition in the definition of G-finiteness 143 + wn (01=-s0+lim vv„ ( 4 )=s 5 + ( t k ) G . Consequently, (5 c +/ i t ) c =5 G +( / J c . Since tl sG + tk belongs to the w*-closed convex hull of G ( j + t), we have ( j + i ) G = ( j G + 0 G -Therefore, (s+t)G=iF+(tJG. Similarly, since tk belongs to the vi>*-closed convex hull of G(t), we have tG=(tk)G. Summing up, we have obtained that ( j + i ) G = = j G + ( i f c ) G = i c + / G , which was to be proved. So far we have proved that i—tG: M-*MG is linear. On the other hand, it is evident that [g(t)]G=tG for every g£G, t£M and tG=t for t£MG, the G-fixed-point algebra in M. Now let <p„ be a normal state on MG. Let <p(t)=(p0(tG) for t£M. Then cp is a G-invariant state on M. Le tp be the support of <p„. Then p£MG and (ptp)G=ptGp. Consequently, <p is faithful on pMp, by the hypotheses of the corollary and by the faithfulness of <p0 on pMGp. Since <p is invariant under the restriction of G to pMp, Theorem can be applied. We obtain that pMp is finite with respect to the restriction of G to pMp. This implies [2] that cp is a G-invariant normal state on M with support p. Since sup p= 1 if (p0 runs over all normal states of MG, we obtain that M is G-finite [2]. C o r o l l a r y 3. Let M be a W*-algebra and G a group of *-automorphisms of M. If T : M^MG is a G-invariant faithful positive linear mapping which leaves M° elementwise fixed, then M is G-finite. Proof . It is similar to the end of the proof of Corollary 2. Remarks . 1. The proof of one half of Corollary 2 does not require Theorem: Let M be a W*-algebra and G a group of * -automorphisms of M. Suppose that for every t£M, the norm-closed convex hull of the orbit Gt of t under G contains exactly one G-invariant element, say tG. If tGji0 for t^0, then M is G-finite (and t-»tG is a normal positive linear mapping of M onto MG). Proof . As in the proof of Corollary 2, we first prove that is a linear mapping. This done, let (p0 be a normal positive linear form on MG and let p denote the support of <p0. Then (jptp)G=ptGp and ;—<p0(/G) is a faithful positive linear form (p on pMp, invariant under the restriction of G to pMp. Let e be a nonzero projection in pMp, such that q>(e -e) is normal [1]. Then <p(- e)dMG. Let i;„£convG be such that ||i>„(e)—eG\\ — 0 as «-"<*>. We have q>(-v„(e))£Ma by the G-invariance of cp and by the fact that (p(- e)Mjf. Then the norm limit of (p( • v„(e)) in MM is q>(- eG), since cp£M*. Therefore, Consequently, t-+(p(eGteG)=(p0((eGteG)G)=(p0(eGtGeG) is a normal positive linear form on M. Since eG^p, eG£MG, we obtain that t-»eGtGeG is normal on M. If cp0 runs over all nor-mal forms on MG, we obtain that every nonzero projection p£M° majorizes a 144 Joseph M. Szűcs: Normalcy is a superfluous condition in the definition of G-finiteness nonzero projection e£M (it is e c ( M c ) such that tece is normal. This implies that i—ta is normal on M and thus M is G-finite [2]. 2. The assumption of Theorem that q> is faithful is essential. Indeed, let G be an abstract infinite Abelian group. Then G acts naturally on M=/~(G) as a group of *-automorphisms. A G-invariant state on M is noting else but an invariant mean on G. We know that there are ¡finitely many invariant means on G, none of wich are normal (actually, they are singular). 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 c-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 £/„cG is a summing sequence [5], then it is easy to prove that under the hypotheses of Lemma 1, the sequence —— 2geu S( 0 w*-converges to tG for every I u„\ t£B*. The remaining results of the paper can be extended to amenable groups G without altering the proofs. References [1] J. DIXIMIER, Les algèbres d'opérateurs dans l'espaces Hilbertiens (Algèbres de von Neumann), 2nd ed., Gauthier-Villars (Paris, 1969). [2] I. KOVÁCS and J. Szűcs, Ergodic theorems in von Neumann algebras, Acta Sei. Math., 27 (1966) 233—246. [3] SH. SARAI, C*-algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 60, Springer-Verlag (Berlin—Heidelberg—New York, 1971). [4] J. Szűcs, Some weak-star ergodic theorems, Acta Sei. Math., 45 (1983), 389—394. [5] F. P. GREENLEAF, Ergodic theorems and the construction of summing sequences in amenable locally compact groups, Comm. Pure Appl. Math., 26 (1973), 29—46. TEXAS A&M UNIVERSITY AT GALVESTON DEPARTMENT OF GENERAL ACADEMICS MITCHELL CAMPUS GALVESTON, TEXAS 77553, U.S.A. 

Normalcy is a superfluous condition in the definition of G-finiteness

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Normalcy is a superfluous condition in the definition of G-finiteness

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