- 1 - Spectra of states, and asymptotically abelian C*- algebras Erling St0rmer University of Oslo, Oslo, Norway 1. Introduction. If ()t is an asymptotically abelian C*-algebra and is an extremal invariant state with cyclic representation p rrp , the structure of p and rr (Cit ) " is quite well understood p if rr (0{) 11 is a semi-finite von Neumann algebra [8,13,15,16]. p It is the purpose of the present paper to study the general case when rr (O't) 11 may also be of type III • This is best done if p we define the spectrum Spec(p) of a state p of a C*-algebra to be - roughly - the set of real numbers u such that there is 01 A E with p(A*A) = 1 such that up(BA) is approximately equal to o (AB) for all B E <n. (Definition 2. 1). For exampel; p is a trace if and only if Spec(p) = [ 1 } and if p is a pure ' state and not a homomorphism then Spec(p) = {0' 1 } • If xp is the cyclic vector such that P (A) = (rr (A)x ,x) for A E Ot, p p p we may cut down rr ((}()" by the support E of the state wx , p p p and define the modular operatDr of Tomita of x relative to p this smaller von Neumann algebra. I£ we extend the modular oper- ator to be 0 on the complement of E it turns out that its p spectrum equals Spec(o) (Theorem 2.3). Together with the resent results of Connes [2, 3] this result gives us a useful tool for - 2 - studying the spectrum of p • Now assume ~ is asymptotically abelian and that p is a strongly clustering invariant state, e.g. if p is an invariant factor state. Then our main result (Theorem 3.1) states that the nonzero elements in Spec(p) form a closed subgroup of the multiplicative group lli+ of positive real numbers. Furthermore, if w is a state of 01 quasi-equiva- lent to p then Spec(p) c Spec(w) • This last statement shows in particular that Spec ( p) is a ,'(·-isomorphic invariant for rr P ( ot)" • Since every proper closed subgroup of IR+ is cyclic we have obtained an isomorphism class for each u E [0,1] , where + 1 correspond to the group [ 1 } and 0 to JR • It seems that Spec(p) most often equals :m+ • This is in particular the case 6t when is asymptotically abelian with respect to a one parame ter group and p is an extremal Kl\18-state (Corollary 4. 5). We shall follow the theory of asymptotically abelian 0*-al Of gebras as developed in [15]. Thus we shall say a 0*-algebra is asymptotically abelian with respect to a group G of *-auto- morphisms if there is a sequence [g } 2 in G such that n n= 1 , , ••• , liml![gn(A),B]!I = 0 for all A,B E (}(. This definition is suffi- n ciently general to take care of most cases of physical interest and extends in particular the original one of Doplicher, Kastler, and Robinson [5] and Ruelle [12], in which case G is the trans lation group IRn • We refer the reader to [6] for a general sur- vay of the theory of asymptotically abelian 0*-algebras. It is unclear at the present whether our results can be generalized to other definitions of asymptotically abelian systems. As indicated above the main part of our analysis will be con cerned with the modular operator of Tomita. We refer the reader to the notes of Takesaki [17] for the theory of Tomita and Takesaki. - 3 - For the general theory of von Neumann algebras the reader is re ferred to the book of Dixmier [4]. We only remark that the strong -* topology on a von Neumann algebra is generated by the semi I! norms A - !lAx + I!A*xll and that the usual density theorems hold 9 for this topology. The author is indebted to A. Connes for very helpful corres- pondence. 2, The spectrum of a state. In this section we shall give two equivalent definitions of the spectrum of a state and then obtain some simple properties of the spactrQrn. s..:!.· Definition Let 0(_ be a c~(--algebra and p a state of 01_. Then the sEectrum of p denoted by Spec(p) is the set of real 9 9 numbers u such that given s > 0 there is A E ~ for which p(A*A) = 1 such that ~ 1 u p ( BA) - p cAB) 1 < € p cB -;<-:s ;-2- Dt . for all B E VIe shall soon show that u must be non negative. A modifi- cation of the same shows that in the definition we might argw~ent as well have assumed u to be a complex nun1ber. It is clear that the definition can be generalized to other linear functionals. Let p and 0( be as ar)ove. Let n be a representation p of p on a Hilbert space d{ and X a unit vector in d{p p p m. cyclic for TI (()1) such that P (A) = (n (A)x ,x ) for A E p p p p con" Let d{p denote the von Neumann algebra n p • Let Ep = r ~'x] Then x is a se~arating and cyclic vector for the '- 11\.p p • p l:' von Neumann algebra E fR. E acting on E(r( Let be the p p p p p• - 4 - modular operator of X relative to E 0{ E and consider it p p p (J 9 as an operator on d{p by defining it to be 0 on (I-E )d( • p p Definition 2.2. With the above notation we call the modular operator of the state p • ot Theorem 2. 3. Let be a C-J<--algebra and p a state of a{ with modular operator 6 • Then Spec(p) = Spec(6 ) . 0 p I Proof: Suppose u 0 and u E Spec(p) • In the notation intro- duced above drop the subscripts p so 6<.= ~P,E = E X = X 9 p ' p' 6 = 6p ' n = TT p • We first shovv u belongs to the spectrum of wx considered as a state on Ed<E Since n(Ol ) is dense in CR in the strong-* topology it is clear that u belongs to the cRv. spectrum Spec(wx) of w as a state of X Let 6 > 0 be given. Choose e: , 0 < e: < 1 , so small that 2!u!-1 max{e,e:(u+e)1 < o. We assert that if A E 6< is such that \IAxl! = 1 and 1 ) < eiiBxl! ;, .l for all B E tR_, then I!E A Ex I! 2 > 1 - o • ' ,. For this let ~ = max[e,e:(u+e)} • Let B =A* • Then 1) gives 2) 11 u - !!, A* xI!, 2 11 < e !!. A * x !,'I ' hence !JA-ll-xl! 2 < u + e: I!A*xll If !IIJ .LAi. 7v ~x 1~ 1: > 1 we have since € < 1 I,.I A* X I'! < 'I. A*uX 1' I + € < u + 8 • ·' • 1,, 11 = Thus !!A*x!! ,::: max{1,u+e:} • Now apply 1) to B EA* • Then we have - 5 - 3 ) I u - II E A* x 'I 2 I < e: !I E A* x I\ _:: 8 !I A * x \1 < '1'"1 In particular, since is arbitrarily small we have that u > 0. ~ Now apply 1) to B = E A*E • Then we have I Since u 0 we then have by 3) and 4) 0 < 1 - liE A E X !1 2 = u - 1 !u ljE A I~ X !!2 - u I _:: u-1 !u!!EAEx 1!2 - \!EA*Exll2! + u-1l '!EA-)(-Exl12 - u I -1 -1 <u n+u o. 'Yl~ The assertion follows. Note that if B E E6?E then I u(EAEx~B*x) - (Bx,EA-l(-Ex)! = ! I u ( Ax , B* x) - ( Bx , A* x ) < e: lj Bx!! • 2 Since 1 > IIEAEx ! > 1-6 it follows that u E Spec(wxiElRE) , as 1 we wanted to show. Restricting attention to EtRE we may thus assume x is separating and cyclic for (f( (so E =I). Let J be the con juga- tion so that J62~ Bx = 6-l2 JBx = B~fx for BE ct:r)v [17,Thm.7.1]. Since the Tomita algebra (called modular algebra in [17]) is 6{ strong-* dense in we may assume A belongs to the Tomita al- _l gebra, and thus Ax belongs to the domain of 6 2 (see e.g. proof of [17,Thm.10.1]). Then 1) becomes ~ ~ I !u(Ax~ 6-2JBx) - (Bx,Jt-.-2Ax) < 81!Bx\l , or I ~ ~ I< (u6-2Ax,JBJx)- (62-Ax~JBJx) e:!JJB,Jxll • Since Ji<J =cR.' by [17,Thm.12.1] and x is cyclic for R.1 we 9 have ! ~ ~ I (u6 -2 Ax- 62Ax,y) < 8 !\y!\ 6<.. for all y E Thus we have - 6 - l!6-2~ (ui- L\)Ax!! = l!u6-2~ Ax- 62~ Axl! < e ~ .l ~ Now 6-2 (u2 I+ 62 ) ~I • Hence we have I!Cu2~ I- 62~ )Ax!! < ll6-2J_ (u21. I + 621 )(u21 I- 6-12 )Axll = !!6--21. (ui- L\)Ax!! < e ~ ~ Since Ax is a unit vector and e is arbitrary u2 E Spec(L\2 ) hence u E Spec(L\) . Now suppose u = 0 E Spec(p) • If 0 I Spec(L\) E = I ~ so is separating and cyclic for R._. Furthermore since 0 I Spec(L\) X 1 there exists k > 0 such that L\2 -> ki By 1 ) we can for each • integer n find A E tR_ such that !lA x!l = 1 and n "n ·' I I (Bx~A~x) < 1/n!\BxJ! 6Z. for all B E Since x is cyclic we have I!A~xl! < 1 /n for each n . Thus 1/n > i!,A n* x'!, This is a contradiction for n sufficiently large. Therefore 0 E Spec(L\) , and we have shown Spec(p) c Spec(L\) . . Conversely assume u E Spec(L\) We assert that 0 E Spec ( L\ -21 ( u I - L\ ) ) • Indeed~ if u = 0 then 0 E Spec(L\2~ ) = . - Spec(6-2~ (0I- L\)) so the assertion holds for u = 0 If u/0 ' choose a spectral projection F for L\ such that FL\ and F L\ -~ are bounded and u E Spec(F6) . Let e > 0 and choose a unit vector y E Fa-{ such that !!(ui-L\)y!J < E:/I!F6-21 II. Then we have 1 ~ ll6-2 (ui- 6)y'! = I!6-2 F(ui- L\)ylj ~ < \!6-2FII!I(ui- 6)Y!I < e • ~ Thus 0 E Spec(6-2 (ui-6)) as asserted. Now the Tomita algebra l is dense in the domain of 6-2 (ui- 6.) , (see proof of [17 ,Thm.10.1]. Therefore if E: > 0 is given there exists A in the Tomita alge-:. - 7 - bra such that II Ax!! = 1 and 1 1 . !!uli-2 Ax- L:I2 Axll < € Therefore if B Eo;( we have I !u(Ax,B*x) - (Bx,A*x) = I 1 1. I< (ull-2 Ax,JBx)- (ll2 Ax,JBx) e\\JBx!l = ~:;!]BI! • Thus u E Spec(wx) Since n((n) is strong-* dense in ~' u E Spec(p) • The proof is complete. Corollary 2.4. Let 0( be a c~<-algebra and p a state of Of, 01.. p(A) = (np(A)xp,xp) for A E Then i) Spec(p) is a closed subset of the non negative real num bers such that 1 E Spec(p) • -1 ii) If ul 0 u E Spec(p) then u E Spec(p) • iii) Spec(p) = [ 1 } if and only if p is a trace. iv) Spec(p) = [ 0' 1} if and only if wx is a trace on TT ((}l)' p p but p is not a trace on 01.~ Proof: i) Since 1 E Spec(ll ) and Spec(6 ) is a closed subset p p of the non negative reals, the same is true for p by Theorem 2.3. ii) Since u I 0 , u E Spec(6 ) implies u-1 E Spec(6 ) by p p [17,Thm.7.1], ii) follows from Theorem 2.3. = iii) If p is a trace then p(AB) p(BA) for all A,B E (}{. Let u E Spec(p) • Then I I I II l up (BA) - p ( AB) = u - 1 p ( AB) for all A,B E C)(. If u I 1 let e: = -fr!u- 1! • Choose A E <n = such that p(A*A) 1 and such that - 8 - !u- 11 I I tiu- 1! 1 p(AB) < p(B7(·B)2 I l f:t for all B E 0( • Thus p (AB) < tP (B·X-B 2- for all B • In particular if B = A* we get 1 = p (A«-A) = p(AA*)< ip(AA*)~ = t , = a contradiction. Thus u 1 = Conversely~ if Spec(o) 1 then by Theorem 2.3 Spec(6 ) p = (1] ~ so wx is a trace on n P (rV"L¥ 1)", see e.g. proof of r~ 17,Thm. 13.1], hence p is a trace on (}(_. iv) Assume Spec(p) = [0,1} Then the spectrum of act- 6PEP ing on Epd{p is [ 1 } where E = [TT (01) 'x ] Thus, as above, p p p W is a trace on E TT (0t) "".R0 ~ hence a trace on TTP(O{)'. By X p p p p iii) p is not a trace. Conversely, if w is a trace on n (0l) 1 xp 0 , but p is not a trace, then as above the spectrum of is £1L 6PEP hence Spec(Ll 0 ) = [ 0' 1 } ' so by Theorem 2.3 Spec(p) = [ 0 9 1] • The proof is complete. 3. Asymptotically a bell§!.! C- x-_ algebras. This section is devoted to the main result on asymptotically abelian C*-algebras and its proof. Following [ 15] if ()'( is a c-)~-algebra and G a group of -;~-automorphisms of ()(, we say crt is asymptotically abelian with respect to G if there is a sequence [g } in G such n n>1 01. that whenever A,B E then lim II [ g ( A) , B ] = 0 , I\ n-+:::o n where [ , ] is the Lie commutator. A G-invariant state p of ~ is said to be ~ronglX clustering (or strongly mixing) if for A B E 01 we have 9 lim p ( g (A) B ) = p ( A) p ( B ) • n-+x n - 9 - We shall need a concept which is slightly more general than that of quasi-equivalence. If p and w are states of a( we say w is quasi-contained in p if the cyclic representation TT of w w is quasi-contained in that of p ; in other words is TT TT p w quasi-equivalent to a subrepresentation of It is easy to iT p see that w is quasi-contained in p if and only if W=W 0 TT, p - where w is a normal state of rrp((r() 11 • 3.:..J.. ot Theorem Let be a c~\-algebra which is asymptotically abelian with respect to a group G of *-automorphisms. Suppose p is a strongly clustering G-invariant state. Then the nonzero elements in Spec(p) form a closed subgroup of the multiplicative group of positive real numbers. Furthermore, if w is a state of ~ quasi-contained in p then Spec(p) c Spec(w) . We shall first prove a few lemmas. Let as in the proof of Theorem 2.3 rr be a *-representation of Oi on a Hilbert space a-e X a unit vector in d{ cyclic for rrCOO such that P (A)= ' (rr(A)x,x) for A E at. Let a<= TT ( 01.) 11 9 let g .... ug be a uni- a-e tary representation of G on such that U X = X and rr(g(A)) g = Ugrr(A)U~1 for g E G A E 01.. Let E be the orthogonal ' 0 projection on [y E J{_ ~ Ug y = y for all g E G} • Then E = [x] 0 is the one dimensional projection on the subspace spanned by x , since p is extremal G-invariant by [15,Thm.4.4] and therefore E = [x] Let [gn} be a sequence in G such 0 that 1 im \1 [ gn (A) , B] II = 0 and 1 im p ( g ( A) B ) = p(A)p(B) Then n n n by [15,Thm.4.4] Ugn .... [x] weakly, and if A E c}( then Ugnrr(A)u~: ~ p(A)I weakly. Let E = [6< I x] be the support of wx R. on Let 6 be the modular operator of the state p (De fin- ition 2.2) and J the conjugation of the Hilbert space E~ de- - 10 - fined by x, so JE!1<EJ = E 6<.1 by [17,Thm.12.1]. Extend J to all of a'( by defining it to be 0 on (I -E) &e.. Thus J = JE = EJ Since wx is invariant under the automorphisms T - u T u-1 its support E is invariant. Therefore E U = U E for g g g g all g E G • Lemma 3.2. Let A E n(6t) • Let y E ~. Then lim 'lu-1 Au y!l = !I Axil !!y!l n .... oo gn gn · ' ·' · ·1 Proof. For B,C E 01 we have and 1 im p ( g~ 1 ( C ) B ) = 1 im p ( B g~ 1 ( C ) ) = 1 im p ( gn ( B ) C ) = p ( B ) p ( C ) , so that the sequence (gn-1 } have the same properties as the se- quence (gn} . Thus for B E D1_ we have weak l~m U~:n(B)Ugn = p(B)I • Thus we have for A E n(~) = lim,r ug -1 A ug y , u-g 1 Aug y ) n n n n 1 ua- = 1 im ( ug- .A* A y, y) n °ll The proof is complete. Lemma 3.3. Let e: > 0 be given. __ Let A E n(Ot) be chosen so that 1 = !1Axl! < !lEAxll + e • Let y E E of. . ,: ,[ I Then we have