ENDOMORPHISMS OF B(H), II:FINITELY CORRELATED STATES ON OnO. BRATTELI AND P.E.T. JORGENSENAbstract. We identify sets of conjugacy classes of ergodic endomorphisms of B(H) whereH is a �xed separable Hilbert space. They correspond to certain equivalence classes ofpure states on the Cuntz algebras On where n is the Powers index. These states, called�nitely correlated states, and strongly asymptotically shift invariant states, are de�nedand characterized. The subsets of these states de�ning shifts will in general be identi�edin [BJW], but here an interesting cross section for the conjugacy classes of shifts calleddiagonalizable shifts is introduced and studied.1. IntroductionLet H be a given separable in�nite-dimensional Hilbert space. If � is a unital endomor-phism of B(H), the (Powers) index of � is de�ned as the n 2 f1; 2; : : : ;1g such that thecommutant of �(B(H)) is isomorphic to the factor of type In, [Pow2]. Throughout thispaper, we will always let \endomorphism" mean unital �-endomorphism. It is well known(see [Arv], [Lac1, Theorem 2.1, Proposition 2.2] and [BJP, Theorem 3.1]) that there is aone{one correspondence between endomorphisms of B(H) of index n, and non-degenerate�-representations (henceforth called representations) of On on H, up to the canonical actionof U(n) on On, where On is the Cuntz algebra of order n. We say that two endomorphisms �,� in End(B(H)) are conjugate if there is an automorphism of B(H) such that �� = ��;and this means that they have the same index n, and that the corresponding representationsof On are unitarily equivalent up to the action of U(n), see [Lac1, Proposition 2.4] and [BJP,Theorem 3.3]. We are interested in two subclasses of the class of endomorphisms of B(H),namely the class of ergodic endomorphisms (i.e., those such that C 1 are the only invari-ant elements) and the even smaller class of shifts (i.e., those endomorphisms � such thatT1n=1 �n(B(H)) = C 1). The �rst of these families corresponds to irreducible representationsof On, and the classi�cation of their conjugacy classes thus amounts to the classi�cation of1991 Mathematics Subject Classi�cation. 46L10, 46L50, 47A58, 81S99.Key words and phrases. Endomorphisms, shift, index, in�nite tensor product, spin chains, Hilbert space.Research supported by the Norwegian Research Council.Research supported by the U.S. National Science Foundation.1

2 O. BRATTELI AND P.E.T. JORGENSENpure states of On, up to the action of U(n) and unitary equivalence. Since On is an antilim-inal C�-algebra, this classi�cation is therefore non-smooth, [BJP, Theorem 1.1], [Dix], [Gli].We show here in Sections 3{6 that the smaller set of �nitely correlated states (de�nitionbelow) on On gives both a \rich" set of conjugacy classes of ergodic endomorphisms, and atthe same time these states lend themselves to explicit calculations. They form a union of�nite-dimensional manifolds. The conjugacy classes can be calculated. Using recent conceptsand results of Fannes et al. [FNW2] we will in a forthcoming paper, [BJW], identify those�nitely correlated states on On which correspond to shifts on B(H).Although our main concern is with pure states of On which give rise to shifts, i.e., purestates such that the canonical UHF-subalgebra UHFn is weakly dense in the operators onthe representation Hilbert space, a generic pure state of On will of course not have thisproperty. In fact, UHFn is the �xed point algebra of the gauge action of T of On, and thisis a quasi-product action by condition 11 of the main theorem in [BEEK]. By condition 9 ofthat theorem, or, more explicitly by [Eva], On has gauge invariant pure states !, and then!jUHFn is pure, but UHFn is not dense, so these de�ne ergodic endomorphisms which arenot shifts. For the case n =1, see [Lac2, Theorem 4.3].Let S be an isometry on a Hilbert space H , and let n := dimN(S�). Then for every k,we have a canonical decompositionH = C n � � � � � C n| {z }k times � SkH:If S is a shift, i.e., TSmH = f0g, we say that n is the multiplicity of the shift. It is knownthat n is a complete unitary invariant for the shifts. For an endomorphism � of B(H) of�nite index n we similarly have a canonical decompositionB(H) = Mn � � � Mn| {z }k times �k(B(H))where n denotes the Powers index. But now as noted, even when � is a shift on B(H), n isnot a complete conjugacy invariant. In fact, in [BJP], we display a nonsmooth continuumof nonconjugate B(H) shifts for each value of the Powers index n � 2.In Section 6, we characterize the pure states ! on On with the property ! � �k+1 = ! � �kfor some k 2 N , where � is the canonical shift on On, see (3.1). The set Sk of these stateshas a natural structure as a �nite-dimensional di�erentiable manifold, and as a manifold itis di�eomorphic to the manifold Ln;k consisting of all pairs (L;R), whereL 2 L(C n ;B(C nk ));R 2 B(C nk );and, with Li = L(jii);

ENDOMORPHISMS OF B(H). II 3we have the following properties: R � 0 and Tr(R) = 1;nXi=1 LiL�i = Pwhere P is a projection in B(C nk ),PLi = LiP = Li; PR = RP = R;RP � �P for some � > 0; nXi=1 L�iRLi = R;and, up to a scalar, R is the unique solution of this equation. See Theorem 6.1 for otherversions of the latter conditions.In Section 7, we show that the action ! ! ! � �g�1 of U(n) on the state space of On givesrise to an action Rn of U(n) on the manifold Ln;k by(Rn(g)L)(x) = Adk(g)L(g�1x)(Rn(g)R) = Adk(g)Rfor x 2 C n , g 2 U(n), where Adk(g) = Ad(g) � � � Ad(g)| {z }k timesand � is the canonical action of U(n) on On, see end of Section 2. The associated orbitscorrespond 1{1 to conjugacy classes of shifts with Powers index n. (In Section 7, the actionRn will actually be replaced by the coaction g ! Rn(g�1).)Of course, by linearization, we may embed Ln;k as a closed submanifold of a Hilbert spacewith inner producth(L;R) j (L0; R0)i = TraceMnk 0@ nXj=1L�jL0j1A+ TraceMn(R�R0)and the action of U(n) then extends to a unitary representation.We are concerned in Section 7 with elements in a closed subset of S1k=1 Pk, where Pk isde�ned in the introduction to Section 3. Section 8 is about the complement of the closureof Sk Pk. Suppose ! 2 Pk, then ! � �k+1 = ! � �k, and so ! � �k is �-invariant. This statetherefore extends canonoically to a shift invariant state on the UHF-algebraOZ Mn = 1O�1Mn

4 O. BRATTELI AND P.E.T. JORGENSENwhich will be denoted !1. The space Ln will be de�ned in Section 7 such that the mapping(Ln 3 (L;R))! !1(L;R)| {z }1 is 1{1. If �1g denotes the U(n)-action�1g := 1O�1Ad(g)on N1�1Mn, then the representation Rn(g) is given by!1(L;R) � �1g�1 = !1(Rn(g)(L;R)):Also the assignment ! ! !1 is such that the two shifts ��! and ��!0 (for given !, !0 2 P )are conjugate i� there is a g 2 U(n) such that!01 = !1 � �1g ;or equivalently, for the corresponding elements L, L0 2 Ln, we have L0 = Rn(g)L.To identify these in�nite families of nonconjugate shifts we introduce in Section 7 a classof elements ! 2 P which we call diagonalizable. If �0 denotes the Haar representation (see[BJP]) of On acting on H0 = L2(X; �0), where X = ZNn, and �0 denotes the correspondingHaar measure on X, then we say that � is diagonalizable if there is a measurable functionu : X ! T1 such that �(si) = Mu�0(si) where Mu is the multiplication operator de�nedfrom u. The diagonalizable elements will be denoted by PD. The result in Section 7 is theassertion that PD is a \section" for the U(n)-orbits under the representation Rn describedabove: Speci�cally, PD intersects a generic set of U(n)-orbits in a �nite dimensional manifolddi�eomorphic to a disjoint union of n! copies of Tn. This means that by just varying thefunctions u : X ! T we get a set of distinct conjugacy classes in P .2. Preliminaries and NotationLet H = Hn ' C n be a �nite-dimensional complex Hilbert space. The dimension n willbe �xed throughout, and the inner product on H will be the usual onehx j yi = nXi=1 �xiyi(2.1)for elements x, y 2 H with coordinate representation x = (x1; : : : ; xn); and the norm k � k isgiven by kxk2 = hx j xi = nX1 jxij2:Consider the free unital �-algebra generated by H, i.e., the �-algebra of all polynomials ofh 2 H and h� 2 �H, where �H is the conjugate Hilbert space of H. If one adds the relationh�k = hh; ki1(2.2)

ENDOMORPHISMS OF B(H). II 5then the C�-envelope of the resulting �-algebra is the familiar Cuntz-Toeplitz C�-algebra,[Eva], [JSW]. If feigni=1 is a basis for H, e.g.,ei = (0; : : : ; 0;| {z }i�1 places 1; 0; : : : ; 0);(2.3)and one adds the relation nXi=1 eie�i = 1(2.4)then the resulting C�-algebra On is the Cuntz-algebra. It is well known [Cun] to be sim-ple, and it plays a crucial role (see [Lac1], [Lac2], [Arv], and [BJP]) in the study of theendomorphisms of B(H).To stress the distinction between elements in H, and elements in one of the involutivealgebras generated by H and �H, we adopt the notation sh and s�h for the correspondingelements in the algebra. With the speci�c choice of basis, we write si for sei. The relation(2.2) may then be written in the familiar forms�i sj = �ij1;(2.5)or in a basis free form s�hsk = hh; ki1:(2.6)The second relation (2.4) becomes nXi=1 sis�i = 1:Let K be the C�-algebra of the compact operators (on a separable Hilbert space). Then wehave the familiar short exact sequence0! K ! Tn ! On ! 0where Tn denotes the Cuntz-Toeplitz algebra. See [Eva] and [BEGJ] for details. In fact K isisomorphic to the two-sided ideal in Tn generated by 1�Pni=1 sis�i .Let � be a representation of On on a Hilbert space H, and set Si = �(si). Then theformula �(A) = nXi=1 SiAS�i(2.7)for 8A 2 B(H) de�nes an endomorphism of B(H), of Powers index n (see [Pow2] and [BJP]).As mentioned in the introduction, every endomorphism of B(H) arises this way. (The result(see [Lac2]) may be modi�ed to apply also to the case when the Powers index is in�nite.)Recall that � 2 End(B(H)) is ergodic if the subalgebrafA 2 B(H)) : �(A) = Ag

6 O. BRATTELI AND P.E.T. JORGENSENis one-dimensional; and that � is a shift if1\k=1�k(B(H))is one-dimensional, i.e., if the intersection is on the form C 1 where I is the identity in B(H).There is an action � by automorphisms of the group T on On, given by �z(sh) = zsh, forz 2 T and h 2 H. The corresponding subalgebraO�n = fa 2 On : �z(a) = a; 8z 2 Tg(2.8)is denoted by UHFn, and has the formMn Mn � � �| {z }1 to 1 :(2.9)Recall from [Cun] that UHFn is generated linearly by the following elementssi1si2 � � � sims�jm � � � s�j2s�j1:(2.10)In fact, the isomorphism between (2.8) and (2.9) is given by letting the element (2.10)correspond to e(1)i1j1 e(2)i2j2 � � � e(m)imjm(2.11)where eij denote the usual matrix units in Mn. We will sometimes use the Dirac notationeij = jei >< ejj :(2.12)As mentioned in the introduction, the action �z of T naturally extends to an action � ofthe unitary group U(n) of C n . For g 2 U(n), the automorphism �g on On is determined by�g(sx) := sgx for 8x 2 C n :The restriction of �g to the subalgebra UHFn is just the product actionAd(g) Ad(g) � � �| {z }1 to 1 on 1O1 Mn:As we pointed out in the introduction, a given � 2 End(B(H)) is ergodic i� the corresponding� 2 Rep(On;H) is irreducible. We also showed in [BJP] that � is a shift i� the restriction�jUHFn is already irreducible. As a consequence, we found, in [BJP], that a classi�cationof the shifts up to conjugacy is given by equivalence classes in the set P of all pure states! on UHFn such that ! is quasi-equivalent to the shifted state, given by x 7! !(1 x),8x 2 UHFn. This equivalence relation is quasi-equivalence up to the action of U(n). But theclassi�cation problem is di�cult in the sense that the classi�ers P= � form a non-smoothspace.

ENDOMORPHISMS OF B(H). II 73. Strongly Asymptotically Shift Invariant States and FinitelyCorrelated StatesThe present paper deals with a smaller problem. Let � denote the canonical shift on On,de�ned by �(x) = nXi=1 sixs�i ; 8x 2 On:(3.1)We will be considering pure states ! on On such that, for some k,! � �k+1 = ! � �k:(3.2)These states are said to be strongly asymptotically shift invariant (of order k). If k is given,the corresponding set of pure states will be denoted Sk. If ! is a pure state on the subalgebraUHFn with the invariance property (3.2), we say that ! 2 Pk. In the latter case, it followsfrom [BJP, Lemma 5.2] that ! �q ! � � on UHFn, and ! corresponds to a shift on B(H).Note that if ! 2 Sk restricts to a pure state on UHFn, then the restriction is contained in Pk.If then � = !jUHFn, we proved in [BJP, Lemma 5.2] that � extends to a pure state ' on Onsuch that �'(UHFn) is weakly dense in B(H'), and it is easily checked that the extensionhas the invariance property (3.2). It is also clear from the construction in [BJP, Lemma 5.2]that the extension ' is unique up to the gauge action � of T (see [Lac1, Theorem 4.3] for thecorresponding result when n = 1), and it follows from [BEEK] that the extensions ' � �z,z 2 T, are mutually disjoint in the strong sense that�Z �T � � �z dz� (On)00 = B(H') L1(T):In fact, this is equivalent to �'(UHFn) being dense in B(H') (see [BEEK] for details). Wewill show in [BJW] that ! is one of these extensions, when ! 2 Sk and !jUHFn is pure.We will now introduce a class of states on On which will be called �nitely correlated states,and in Section 4 we will show that Sk Sk is contained in these states.For a given state ! on On, the GNS-representation will be denoted by (�!;H!;!) orsimply (�;H;), i.e., � is the cyclic representation of On on H, with cyclic vector , suchthat !(x) = h j �(x)i for 8x 2 On:(3.3)Extending a de�nition in [FNW1, FNW2], we say that the state ! is �nitely correlated ifthe subspace V � H generated linearly by and the vectors�(s�h1s�h2 � � � s�hm)(3.4)for hi 2 H, and m = 1, 2, : : : ; is �nite-dimensional.The space generated linearly by the vectors (3.4) with a �xed m will be denoted by Vm,and V0 = C. If ! is �nitely correlated, there is a smallest k such that V = Pki=0 Vi. Ifthen Vk is left invariant by all S�i , we say that ! 2 FCk. (We say this whenever Vk is leftinvariant, even if k is not the minimal such k.) Note that FCk is not necessarily increasing

8 O. BRATTELI AND P.E.T. JORGENSENin k, and the union of the FCk's is not necessarily the set of all �nitely correlated states.The set of pure states in FCk will be denoted by PFCk.The de�nition above is new, as [FNW2] is concerned with a di�erent C�-algebra, viz.,the two-sided in�nite tensor product N1�1Mn (see details in section 8 below). Our presentde�nition for On is on the face of it unrelated, but a main point in our paper is to show thatour states may in fact be described with a set of labels which is directly related to thoseused in [FNW2] for N1�1Mn.The case when V from above is one-dimensional, yields the identity�(s�h) = hh; 'i for 8h 2 H(3.5)where ' is some �xed vector in H such that, k'k = 1. The corresponding states are calledCuntz states. When ! = !' is a Cuntz-state, its restriction to UHFn is the pure productstate ' ' � � �| {z }1 to 1(3.6)corresponding to the representation (3.3) of UHFn, so it follows that the Cuntz states arein P0 ' S0. We showed conversely in [BJP, Theorem 4.1] that every element in P0 is aCuntz-state.Hence the set P0 is parametrized by the unit-ball in the Hilbert space H = C n , and weshall show that a corresponding result is also true for Sk. Since clearly Pk � SkjUHFn , theresults in [BJW] then give a parametrization of Pk.For states ! on On, the condition (3.2) is important because of a result which we nowproceed to describe. We showed in [BJP, Lemma 5.2] that the pure states ! on UHFn de�neshifts on B(H) i� ! � � �q ! where �q denotes quasi-equivalence, [Dix]. If ! is given, and(�!;H!) is the GNS-representation (extended to On on the same Hilbert space as in [BJP])then the corresponding shift �! on B(H!) is given by�!(A) = nXi=1 �!(si)A�!(si)�(3.7)for 8A 2 B(H!). If ! and !0 are two such pure states, we showed ([BJP, Lemma 5.4]) thatthe corresponding shifts �! and �!0 are conjugate, i.e., that �!0 = � � �! � ��1 for some� 2 AutB(H), i� 9g 2 U(n) such thatlimm!1 k!0 � �m � ! � �g � �mk = 0:(3.8)The following result is immediate from this:Proposition 3.1. If !, !0 2 S1k=1 Pk, then the corresponding shifts �! and �!0 are conjugatei� 9m 2 N and g 2 U(n) such that!0 � �m = ! � �g � �m:(3.9)

ENDOMORPHISMS OF B(H). II 9Remark. Our main use of the more restricted family of states is the fact that the condition(3.9) in Proposition 3.1 above is easier to verify than the corresponding asymptotic property(3.8) for the general case. We also show in Section 6 below that (3.9) lends itself to explicitcomputations for the examples of conjugacy classes of shifts which we studied in the precursor[BJP].Proof. The proof is the assertion that if the limit of an eventually constant sequence iszero, then the terms in the sequence must be identically zero from a step on.4. Strongly Asymptotically Shift Invariant States Are FinitelyCorrelatedOne main object of the present paper is the set of shifts on B(H), and the correspondingconjugacy classes. More generally, we shall consider endomorphisms which are not necessarilyshifts; but we will also be more speci�c in that we look at those states ! on On which areinvariant from a certain step on, i.e., satisfying (3.2) above. For each k, we show that thesestates form a �nite-dimensional manifold, thus simplifying considerably the classi�cationproblem for the corresponding subclass of ergodic endomorphisms of B(H).Theorem 4.1. Let k and n be positive integers, and let ! be a pure state on On suchthat ! 2 Sk. It follows that ! is �nitely correlated and, moreover, the space Vk spannedby the vectors �!(s�h1 � � � s�hk), h1; : : : ; hk 2 C n , is invariant under each of the operatorsS�i = �!(s�i ).Proof. Since ! is a pure state on On, the corresponding GNS-representation � is irre-ducible. If Si := �(si), then !(�(x)) = nXi=1 hS�i j �(x)S�iifor all x 2 On. More generally, seti1i2���im := S�im � � �S�i2S�i1:(4.1)Then !(�m(x)) =Xi1 � � �Xim hi1���im j �(x)i1���imi for 8x 2 On:(4.2)It follows that the GNS-representation of ! � � identi�es with the subrepresentation of then-fold direct sum �� �� � � �� � de�ned by the cyclic subspace generated by the free directsum of the i vectors, i.e., 1 � � � � � n, and that of ! � �m is unitary equivalent to thesubrepresentation of the nm-fold sum with cyclic vectorXi1 � � �Xim �i1���im

10 O. BRATTELI AND P.E.T. JORGENSENwhere each index ij runs over f1; : : : ; ng. Since � is irreducible, it follows that the commutant�!��(On)0 is naturally embedded inMn. This is because the commutant of the representationA 7! A� A� � � � � A| {z }n times(4.3)consists of the operator matrices on Ln1 H of the form Pij zijEij, with scalar indices zij 2 C .The same result holds for ! � �m with the obvious modi�cation coming from considerationof multi-indices.Using (4.2) and (3.2), we now conclude that each of the states!i1���ikik+1 = Di1���ikik+1 j �i1���ikik+1E(4.4)is dominated by ! � �k; and so, by using Segal's Radon-Nikodym theorem [Seg2], [Br-Rob,Theorem 2.5.19], or [KR], we conclude that, for each (i1; : : : ; ik; ik+1) there are positiveoperators Z = Zi1���ik+1 in the commutant �!��k(On)0 such that!i1���ik+1(A) = ! � �k(AZ)(4.5)where, on the right hand side, we have extended ! � �k to B(H!��k) in the obvious manner.By the above argument, the representation �!��k is a subrepresentation of the nk fold directsum of �!, and the commutant of the latter representation is isomorphic to Mnk . Thesubrepresentation corresponds to a projection E in Mnk , and the operators Z live inside thisprojection. We may extend Z to operators in Mnk by setting (1 � E)Z = Z(1 � E) = 0.The formula (4.5) may now be written in multi-index summation form, p = (p1; : : : ; pk),q = (q1; : : : ; qk), with pj and qj in f1; : : : ; ng. The matrix Z and its entries zp;q still dependon (i1; : : : ; ik+1), but the latter multi-index is �xed for the moment. We get!i1���ik+1(A) =Xp Xq zpq hp j Api for 8A 2 B(H).(4.6)But the matrix Z is positive, so on the form Z = Y �Y where Y = [yrp] 2 Mnk , e.g., takeY := pZ. Now set, �r :=Xp yr;pp 2 H(4.7)where r = (r1; : : : ; rk) is also a multi-index. Formula (4.6) then takes the form!i1���ik+1(A) =Xr h�r j A�ri(4.8)for A 2 �!(On), and thus, by closure, for all A 2 B(H). But !i1���ik+1 is a vector functionalon B(H) and thus proportional to a pure state, and it follows from (4.8) that each of thevector functionals h�r j ��ri are proportional to !i1���ik+1 , and thus each of the �r are a scalarmultiple of i1���ik+1 . Thus i1���ik+1 is a scalar multiple of some �r. But the vectors �r arelinear combinations of the vectors p = S�pk � � �S�p2S�p1, and thus i1���ik+1 are so. This provesTheorem 4.1.

ENDOMORPHISMS OF B(H). II 115. A Reconstruction TheoremIn this section, we �rst, in Theorem 5.1, describe a map from the set of all �nitely correlatedstates on On into a system consisting of a state on a matrix algebra and a partition of unity.The hypotheses of this theorem are in particular ful�lled for ! 2 Sk, by Theorem 4.1.Subsequently, we show in Theorem 5.2 that such a system de�nes a state on On. Finally, inTheorem 5.3, we give necessary and su�cient conditions on the system for the state to bepure. In Section 6 we will specialize to the case ! � �k+1 = ! � �k.The �rst result is a corollary to our previous theorem. Let Ak be the subalgebra of UHFnspanned linearly by the elements si1si2 � � � siks�jk � � � s�j1, where im, jm = 1; : : : ; n. As explainedaround (2.10), (2.11), Ak is isomorphic toMn � � � Mn| {z }k times 'Mnk . If x = (x1; : : : ; xk), wherexi 2 H, we will use the notation sx = sx1sx2 � � � sxk , and if x, y 2 Hk, exy = sxs�y = jx >< yj.Theorem 5.1. Let k and n be positive integers, and let ! 2 FCk; i.e., ! is a �nitelycorrelated state such that each S�i leaves the subspace Vk � H! invariant. Then there areelements Li 2 Ak (i = 1; : : : ; n) such that the state ! is given by!(sxsi1 � � � sim1s�jm2 � � � s�j1s�y) = !(Lim1 � � �Li1exyL�j1 � � �L�jm2 )(5.1)for x, y 2 Hk, il, jl 2 f1; : : : ; ng. In particular, restriction of ! to UHFn is given by! �A e(k+1)i1j1 � � � e(k+m)imjm � = ! �Lim � � �Li2Li1AL�j1 � � �L�jm� for 8A 2 Ak:(5.2)Hence ! is determined by its restriction to Ak and the elements fLigni=1 in Ak, and we havenXi=1 !(LiAL�i ) = !(A) for all A 2 Ak:(5.3)Furthermore, if P 2 Ak is the support projection of the restriction of the state ! to Ak, theelements Li 2 Ak may be chosen such that PLiP = Li, and with this choice the Li's areunique.Remark. Since Pi sis�i = 1, the algebra On is the closed linear span of operators of theform sxsi1 � � � sim1 s�jm2 � � � s�j1s�y, and so (5.1) de�nes ! uniquely from � := !jAk and fLigni=1.The following useful formula follows immediately from (5.1):�!(�k(s�j))��(X) = ��(XL�j)(5.4)for X 2 Ak, j = 1; : : : ; n, as follows: By (5.1)!(sxs�js�y) = !(exys�j)= !(sxs�ys�j)But since S�jVk � Vk we obtain from hereS�jS�y = S�y�(L�j):

12 O. BRATTELI AND P.E.T. JORGENSENMultiplying to the left by Sy and summing over y in an orthonormal basis for �Hk, we obtain�(�k(s�j)) = �(L�j)and since �k(s�j) 2 Ack, the formula (5.4) follows.This can also be used to give an alternative de�nition of L�j 2 PAkP , where P is thesupport projection of !jAk . One has�!(�k(s�j)) = Xi1���ik Si1 � � �SikS�jS�ik � � �S�i1:But by assumption, S�jS�ik � � �S�i1 is in Vk, and hence the sum above is a linear combinationof elements of the form Si1 � � �SikS�jk � � �S�j1, i.e., of elements in �(Ak). Hence there existsan L�j 2 Ak such that �(�k(s�j)) = �(L�j)and then (5.4) is valid for all X 2 Ak.Now, as P is the smallest projection in Ak such that �(P ) = , it follows that we mayreplace L�j by L�jP in the last formula. Furthermore, as �k(s�j) 2 A0k, we have�(P )�(�k(s�j)) = �(�k(s�j))�(P )= �(�k(s�j))so the formula is unchanged if L�j is replaced by PL�j . Thus, we may assume Lj = PLjP .But as is separating for �(PAkP ), the Lj is then uniquely determined by the formula.Next iterating the formula as�i = s�i�(a);valid for all a 2 On, one obtains as�x = s�x�k(a)for all x 2 Hk. Combining this with an iteration of (5.4) givesS�im1 � � �S�i1S�x = S�x�(Li1) � � ��(Lim1 )and (5.1) follows. We now proceed to another proof.Proof of Theorem 5.1. Since ! 2 FCk, the space Vk spanned by nS�i1 � � �S�iko isinvariant under each of the operators S�i := �(s�i ) where � is the GNS-representation of !.We also have an antilinear map from the k-fold tensor product H � � � H into Vk whereH ' C n . This map is given by(x1 � � � xk) := S�xk � � �S�x2S�x1(5.5)for xi 2 H, i = 1; : : : ; k. The antilinearization of formula (5.5) may be abbreviated (x) =S�x, x 2 C nk ; so we get Vk as a quotient space; C nk divided out with a linear subspace Nconsisting of vectors x such that k(x)k2 = 0, i.e., C nk =N ' Vk.

ENDOMORPHISMS OF B(H). II 13Let Li be some lifting to C nk of the induced operator on the quotient,S�i(x) = (Lix):(5.6)for all x 2 C nk . We conclude that N must be invariant for each Li. Each Li may be identi�edwith an element in Ak ' B(C nk ) in the following way: Once the basis ei for H has beenchosen as in (2.3), then the element e(1)i1j1 � � � e(k)ikjk in Ak acts on Hk in a canonicalfashion, giving a �-isomorphism between Ak and B(Hk). Transporting Li back with this�-isomorphism, Li identi�es with an element in Ak. Doing this, one veri�es the formulaLiexyL�j = eLix;Ljy(5.7)for x, y 2 C nk , as follows: If u, v 2 C nk , thenDu j LiexyL�jvE = hu j Lixihy j L�jvi= hu j Lixi hLjy j vi= Du j eLix;Ljy j vE :Let us now verify formula (5.2). We note that the element A in Ak may be taken to be inthe form A = exy where x, y 2 C nk . Then!(exy e(k+1)i1j1 � � � e(k+m)imjm ) = hS�im � � �S�i1S�x j S�jm � � �S�j1S�yi= hS�im � � �S�i1(�x) j S�jn � � �S�j1(�y)i= h(Lim � � �Li1 �x) j (Ljm � � �Lj1 �y)i= !(eLim ���Li1x;Ljm ���Lj1 �y)= !(Lim � � �Li1exyL�j1 � � �L�jm):which is the desired formula.Formula (5.3) follows by putting m = 1 and j1 = i1 = i in (5.2), and then summing overi = 1 to n, using nXi=1 eii = 1:Let us now prove the last statement of Theorem 5.1. So far, the Li's are only unique upto their action on N . But note thatN = nx 2 C nk j (x) = 0o= nx 2 C nk j h(x);(x)i = 0o= nx 2 C nk j !(exx) = 0o :

14 O. BRATTELI AND P.E.T. JORGENSENMoreover exx ranges over all multiples of one-dimensional projections in B(C nk ) when xranges over C nk , and it follows from the above formula thatN = (1� P )C nkwhere P is the support projection of !. But as LiN � N , we have Li(1�P ) = (1�P )Li(1�P )and hence PLi(1� P ) = 0.Now S�i(x) = (Lix)= (PLix + (1� P )Lix)= (PLix)since (1� P )Lix 2 N = ker (�). ThusS�i(x) = (PLix)= (PLiPx) + (PLi(1� P )x)= (PLiPx)since PLi(1� P ) = 0. Thus, if Li is replaced by PLiP , one still has the formula S�i(x) =(Lix), and hence one derives (5.2) as before. Thus Li may be chosen such that Li = PLiP .But since the map induced by (�) from PB(C nk )P to B(Vk) is an isomorphism, this choiceof Li is unique.We will now show conversely that if k and n are given, then every system fLigni=1 ofmatrices in B(C nk ), together with a positive matrix R in B(C nk ) of trace 1, determine a stateon On by the formula (5.1), if the pair fR; fLigg satisfy a certain normalization condition(5.8).The question becomes one of extending the �xed state � = Tr(R�) on Ak to On such thatthe extended state ! is given by (5.1). For 8x, y 2 C nk , we then have�(exy) = hxjRjyi :We shall say that the operators fLigni=1 are normalized ifXi �(eLixLiy) = �(exy) 8x; y 2 C nk ;or, equivalently, Xi L�iRLi = R:(5.8)This is again equivalent to (5.3).The normalization is a condition on the combined system consisting of the Li's and R,or equivalently the Li's and �. We will see during the proof of the next theorem thatnormalization is a translation of the Cuntz property Pni=1AiA�i = IVk to the Li's.

ENDOMORPHISMS OF B(H). II 15Theorem 5.2. Let k and n be positive integers, and � be a state on the subalgebra Ak � On.Let fLigni=1 be a system of elements in Ak which are normalized relative to �. Then theformula !(sxsi1 � � � sim1s�jm2 � � � s�j1s�y) = �(Lim1 � � �Li1exyL�j1 � � �L�jm2 )de�nes a state ! on On which extends �. Furthermore, ! 2 FCk.Proof. If exy 2Mnk ' Ak � UHFn � Onwe have, with the cyclic vector in the GNS representation � of Ak,�(exy) = h j �(exy)i = hx j Ryi= trace ����R1=2y >< xR1=2���� = trace(R1=2exyR1=2)(5.9)Since the Li operators are normalized relative to R, we haveXi (R1=2Li)�(R1=2Li) = (R1=2)2and hence R1=2x = 0) R1=2Lix = 0. Thus each operator Li passes to the quotient spaceVk := C nk .nx 2 C nk : R1=2k x = 0o :(5.10)For each i, we denote the corresponding induced operator on Vk by A�i . Speci�callyA�i (x + ker(R1=2k )) = (Lix) + ker(R1=2k ):(5.11)Relative to the norm, x 7! R1=2k x onC nk .ker �R1=2k � ;the normalization property (5.8) then translates intonXi=1AiA�i = IVk :(5.12)Using [Pop1, Theorem 2.1] we conclude the existence of a representation (�;H�) of On suchthat Vk is isometrically embedded in H�, and�(s�i )jVk = A�i :(5.13)(See the remarks before (6.3) for more details on this.) Let Pk denote the orthogonal pro-jection of H� onto Vk, and consider the completely positive mapping' : On ! B(Vk)given by '(a) := Pk�(a)jVk for 8a 2 On:(5.14)

16 O. BRATTELI AND P.E.T. JORGENSENViewing the Ai's as operators on H� by setting them equal to zero on the orthogonalcomplement of Vk, we have from (5.13):S�i Pk = PkS�i Pk = A�iand we conclude that'(si1 � � � sils�jm � � � s�j1) = PkSi1 � � �SilS�jm � � �S�j1Pk= PkSi1Pk � � �PkSilPkS�jmPk � � �PkS�j1Pk= Ai1 � � �AilA�jm � � �A�j1(5.15)for all l, m 2 N and all corresponding multi-indices (see [BEGJ, Proposition 2.1] for a similarargument). We may de�ne a state ! on On by the formula!(a) := h j �(a)i = h j '(a)i for 8a 2 On:Speci�cally !(si1 � � � sils�jm � � � s�j1) = DA�il � � �A�i1 j A�jm � � �A�j1E ;(5.16)and it follows that ! on On does restrict to the given state � on Ak. Let us introduce theoperator V = Pni=1 L�i ei from C nk into C nk C n = C nk+1 . A calculation yields!(a eij) = �(V �(a eij)V ) = �(LiaL�j)for 8a 2 Ak, 8i; j 2 f1; : : : ; ng, where as usual eij denotes the matrix entries in Mn. Thenotation a eij is short for a e(k+1)ij , with the eij-term sitting in the tensor slot k + 1relative to the in�nite tensor product representation (2.9). The asserted formula (5.1) nowfollows precisely as in the proof of Corollary 5.1 above. This formula immediately impliesthat ! 2 FCk.Theorems 5.1 and 5.2 say that there is a one-one correspondence between states ! 2 FCkand pairs �(�) = Tr(R�), fLigni=1 consisting of a state � on Ak (alias density matrix R) withsupport projection P (alias range projection of R), and n operators Li 2 PAkP satisfying thenormalization condition Pi L�iRLi = R. We now address the question on when ! 2 PFCk.The answer is:Theorem 5.3. Let ! 2 FCk, and let Li 2 PAkP , �(�) = Tr(R�) be the objects associated to! by Theorems 5.1 and 5.2. The following conditions are equivalent:(i) ! is pure.(ii) The operator equation Xi L�ixLi = xhas a unique positive solution x 2 Ak with Tr(x) = 1 (namely, x = R).Remark. We defer a more detailed discussion of the condition (ii) until the Theorem 6.1,but note that the condition is at least as strong as irreducibility of the system fLi; L�i g ofoperators on P C nk , given that the equation has a solution.

ENDOMORPHISMS OF B(H). II 17Proof. The state ! is pure if and only if any state ' for which there exists a � > 0 with�' � ! is a multiple of !, so we must characterize those '. The starting point is the relation(5.4) �!(�k(s�i ))! = �!(L�i )!which can be written !((�k(si)� Li)(�k(si)� Li)�) = 0:Since �' � !, we obtain '((�k(si)� Li)(�k(si)� Li)�) = 0and thus �'(�k(s�i ))' = �'(L�i )':If A 2 Ak, this implies �'(�k(s�i ))�'(A)' = �'(A)�'(�k(s�i ))'= �'(AL�i )'and iterating this, we obtain�'(�k(s�j1) � � ��k(s�jm)�'(A)' = �'(AL�jm � � �L�j1)':Thus '(�k(si1 � � � sim1 s�jm2 � � � s�j1)A) = '(Lim1 � � �Li1AL�j1 � � �L�jm2 )for all A 2 Ak, and hence ' 2 FCk, and the Li's associated to ' are the same as thoseassociated to !, and ' is determined by its restriction to Ak. This restriction is determinedby the density matrix x 2 Ak of ' : '(A) = Tr(xA)for A 2 Ak. But the Cuntz relationPi sis�i = 1 implies as before the normalization conditionXi L�ixLi = xand as ' is determined by x and fLigni=1, the equivalence of (i) and (ii) is clear.Corollary 5.4. If ! 2 FCk with associated objects R, fLig, then the face generated by !in the state space of On is �nite dimensional, and a�nely isomorphic to the convex set ofmatrices x 2 Ak with the propertiesx � 0; Tr(x) = 1; and Xi L�ixLi = x:

18 O. BRATTELI AND P.E.T. JORGENSENProof. We showed during the proof of Theorem 5.3 that if ' is a state dominated by amultiple of !, then ' 2 FCk and has the same fLig as !, and the density matrix has theproperties stated in the corollary. Conversely, if x has the properties in the corollary, thenthe support of x is contained in P , and if ' 2 FCk is the corresponding state, it follows from�nite dimensionality that there exists a � > 0 such that �'jAk � !jAk . But as the Li's arethe same for ' and !, this inequality extends to On.6. Asymptotically Shift Invariant StatesIn this section we specialize the theorems in Section 5 to the case ! 2 Sk. We alreadynoted in Theorem 4.1 that ! is �nitely correlated and that Sk � PFCk; and we will nowstudy which additional requirements the fact that ! 2 Sk places on fLig and �.Theorem 6.1. Let n, k 2 N, let � be a state on Ak � On, and let fLigni=1 be elements inAk satisfying the normalization condition (5.8). Then the corresponding state ! on On fromTheorem 5.2 satis�es ! � �k = ! � �k+1(3.2)if nXi=1 LiL�i = 1 on the support of �:(6.1)Conversely, if ! 2 Sk, then the associated operators Li (which exist by Theorem 4.1 andTheorem 5.1) satisfy (6.1).Moreover, let ! be a state on On de�ned by � and fLig as in Theorem 5.2, such that boththe normalization conditions (5.8) and (6.1) are satis�ed, and PLiP = Li where P is thesupport projection of �, so thatnXi=1 LiL�i = P and nXi=1 L�iRLi = Rwhere R is the density matrix of �. Let Pk be the projection from H! onto Vk. The followingconditions are equivalent:(i) ! is pure on On.(ii) fLi; L�i g acts irreducibly on P C nk (i.e., S�i jVk acts irreducibly on Vk) and Pk 2 �!(On)00.(iii) The only positive solutions of the operator equationXi L�ixLi = xare the positive scalar multiples of R.(iv) The operator Ak 7! Ak : x 7! Pi L�ixLi has 1 as eigenvalue of multiplicity one.(v) The only positive solutions of the operator equationXi LixL�i = x

ENDOMORPHISMS OF B(H). II 19are the positive scalar multiples of P .(vi) The operator Ak 7! Ak : x 7! Pi LixL�i has 1 as eigenvalue of multiplicity one.Proof. From (5.1), we get! � �k(ei1j1 � � � eimjm) = �(Lim � � �Li1L�j1 � � �L�jm)and ! � �k+1(ei1j1 � � � ) =Xi �(Lim � � �Li1LiL�iL�j1 � � �L�jm):It is clear from this that (3.2) holds if Pi LiL�i = 1 on the support of �. But when the Lioperators act irreducibly on P C nk , then this condition is also necessary, as follows from therespective formulas for ! � �k and ! � �k+1.We next show that the purity of !, or equivalently the irreducibility of the representation� from (3.3), is equivalent to irreducibility of the fLig system, together with the conditionPk 2 �!(On)00. But this follows from the commutant lifting theorem (see [NaFo]) which ispart of the conclusion of [Pop1, Theorem 2.1]; see also [BEGJ] for more details. Speci�cally,we need to use the formula (5.11) which relates the Li's to the Ai's. When the Ai's are given,and � is a representation of On which serves as a minimal dilation, i.e.,[�(On)Vk] = H�(6.2)and (5.13), then we �rst observe by GNS representation techniques that the representation� is determined up to unitary equivalence by the system Ai in the sense that if A0i is anothersystem of operators on a �nite dimensional Hilbert space V 0k, and there is a unitary U :Vk ! V 0k such that A0iU = UAi, then the associated minimal dilations � and �0 are unitaryequivalent representations of On. This is proved in the same way as one proves that thecyclic representation associated to a state is determined up to unitary equivalence.More nontrivially, the commutant lifting theorem states that there is a canonical iso-morphism between the commutant of the operator system fAig and the commutant of therepresentation �. In view of the uniqueness of the minimal dilation, in order to prove this itsu�ces to prove it for a particular explicit construction of the minimal dilation which we arenow going to describe. We emphasize that by the commutant of the operator system fAigwe mean those operators that commute both with Ai and A�i for i = 1; : : : ; n, i.e., the vonNeumann algebra generated by those unitaries U 2 B(Vk) such that UAiU� = Ai.Speci�cally, let the operator system fAigni=1 on Vk be given. Let A be the operator-rowmatrix [A1; : : : ; An], and set DA := (In � A�A)1=2, and D := DA(Lni=1 Vk). (Note thatsince AA� = 1, we have that kA�Ak = kAA�k = 1, and hence DA is well de�ned.) LetF(C n) = C � C n � (C n C n) � � � � be the unrestricted Fock space over C n , and de�neoperators �i on F(C n) by �i(�1 � � � �k) = ei �1 � � � �kfor �j 2 C n , where ei is the standard basis. The �i then generate a representation of theToeplitz-Cuntz algebra, [Eva], [BEGJ]. Let 0 = (1� 0� (0 0)� � � � ) denote the vacuum

20 O. BRATTELI AND P.E.T. JORGENSENvector in F(C n), and for i 2 f1; : : : ; ng de�ne �i : Vk ! D by�iv = DA(0; : : : ; 0| {z }i�1 times ; v; 0; : : : ; 0)for v 2 Vk. De�ne also Ti : DF(C n)! DF(C n) by Ti = 1�i, and Di : Vk ! DF(C n)by Div = �iv 0:De�ne Si on Vk � (D F(C n)) bySi(v + f) = Aiv � (�iv 0 + (1 �i)f)= Aiv � (Div + Tif)= Ai 0Di Ti ! vf!(6.3)for 8v 2 Vk and 8f 2 D F(C n). Then it can be checked (and follows from [Pop1] and[BEGJ]) that the Si's satisfy the Cuntz relations,S�i Sj = �ijI and nXi=1 SiS�i = I(6.4)where I denotes the identity operator on Vk�DF(C n). Hence they de�ne a representation� of On which is easily checked to be a minimal dilation.To return to the proof of Theorem 6.1, note that the following version of the commutantlifting theorem is true. (For a general background on \commutant lifting" see e.g., [Pop2]and [DMP].)Lemma 6.2. Adopt the general assumptions of Theorem 6.1 . If U is a unitary on Vkcommuting with the Ai's, then U has a unitary extension to H! commuting with the Si's.Moreover this extension is unique.Proof. As AiU = UAi, U commutes with all A�iAj, and hence U In commute with(In � A�A)1=2 on Vk C n . In particular U In leave the subspace D invariant, and if therestriction is called UD, thenUD(In � A�A)1=2 = (In � A�A)1=2UDand hence (UD IF(Cn ))Di = DiU:Thus, de�ning U 0 on Vk � (D F(C n)) byU 0 = U � (UD IF(Cn ))one has U 0Si = SiU 0so U 0 is the sought-after extension.

ENDOMORPHISMS OF B(H). II 21To prove uniqueness of the extension, note that any unitary extension of U must have theform U 0 = U 00 W !on H! = Vk � (D F(C n)), where W is unitary in D F(C ). That U 0 commute withSi = Ai 0Di Ti !means UAi = AiUWDi = DiUWTi = TiW:The �rst relation is ful�lled since U 2 fAig0. Since the representation i! �i of the Toeplitzalgebra is irreducible, the last relation implies that W has the formW = w 1F(Cn )where w is unitary on D. Now, the second relation meansw�i = �iU:But this means that w is uniquely de�ned on the sum of the ranges of the �i's by U , andsince the sum of these ranges in D, it follows that w is uniquely determined (in fact wecomputed earlier that w = UD). Thus the extension U 0 is unique, and Lemma 6.2 is proved.Let us now continue the proof of Theorem 6.1 by establishing the equivalence of the twostatements(ii) Pk 2 �!(On)00 and fAig is irreducibleand(i) �! is irreducible.Clearly (i) ) (ii), since Ai = PkS�i Pk = S�i Pk. Conversely, assume (ii) and let U bea unitary in �!(On)0. Then UPk = PkU , and UPk 2 fAig0 thus UPk = PkU = Pk byirreducibility of fAig. But by the uniqueness part of Lemma 6.2 it follows that U = 1. Thisends the proof of (i) , (ii).It remains to show that each of the conditions (iii){(vi) are equivalent to (i):(i) , (iii): This follows from Theorem 5.3.(iii), (iv): Clearly (iv) ) (iii). To prove the converse implication, assume thatXi L�ixLi = xfor some x 2 Ak. Then Xi L�ix�Li = x�and hence if x1 = 12(x + x�), x2 = 12i(x � x�) then x1, x2 are eigenelements of eigenvalue 1,x = x1 + ix2 and x1 = x�1, x2 = x�2. To show that x is a scalar multiple of R, it therefore

22 O. BRATTELI AND P.E.T. JORGENSENsu�ces to assume that x is self-adjoint. But as PLiP = Li, it follows from x = Pi L�ixLithat Px = xP = x, and hence �x � �0P for some �0 > 0. But since P is the supportprojection of R it follows from �nite dimensionality of Ak that P � �00R, where �00 is theinverse of the smallest nonzero eigenvalue of R. Hence�x � �0P � �0�00R = �Rwhere � > 0. Thus �R + x � 0, and since �R + x is an eigenelement of y 7! Pi L�i yLi ofeigenvalue 1, it follows from (iii) that �R + x is a scalar multiple of R. Thus x is a scalarmultiple of R, and (iv) is valid.(v) , (vi): This is proved as (iii), (iv), with P playing the role of R.To �nish the proof of Theorem 6.1 it remains to establish (iv) , (vi), and this followsfrom the following lemma.Lemma 6.3. Let A be a unital C�-algebra with a faithful trace state tr, let L1; : : : ; Ln beelements in A and let R, S be positive invertible elements in A withnXi=1 L�iRLi = R and nXi=1 LiSL�i = S:For any x 2 A and any � 2 C with j�j = 1, the following statements are equivalent:(i) Pni=1 LiSxL�i = �Sx.(ii) Pni=1 L�ixRLi = ��xR.Proof. Let us �rst consider the case S = 1, and de�ne�(x) = nXi=1LixL�i :Then � is a completely positive map with �(1) = 1, and hence the generalized Cauchy-Schwarz inequality is valid �(x)��(x) � �(x�x);[Br-Rob, pp. 229{230]. We may assume that R is normalized such that tr(R) = 1 and thenwe may de�ne a state � on A by �(x) = tr(Rx):Then �(�(x)) = Xi tr(RLixL�i )= Xi tr(L�iRLix)= tr(Rx) = �(x):So � is �-invariant, and then�(�(x)��(x)) � �(�(x�x)) = �(x�x)

ENDOMORPHISMS OF B(H). II 23by Cauchy-Schwarz. If (�;H;) is the GNS-representation associated to �, it follows thatwe may de�ne a contraction W on H byW�(x) = �(�(x)):Let us suppress the notation � from now on, and show thatW �x = nXi=1 L�ixRLiR�1for all x 2 A: hW �x j yi = hx j Wyi= hx j �(y)i= �(x��(x))= Xi tr(Rx�LiyL�i )= Xi tr(L�iRx�Liy)= Xi tr(R(R�1L�iR)x�Liy)= Xi DL�ixRLiR�1 j yE ;which shows the desired formula.Now, choose a speci�c x 2 A such that�(x) = �xwhere j�j = 1, and put � = x. Then W� = �(x) = �x = ��. Now one computes W �� � ��� 2 = kW ��k2 � j�j2k�k2and as kW �k = kWk � 1 and j�j = 1 one deducesW �� = ���:

24 O. BRATTELI AND P.E.T. JORGENSENUsing the explicit formula for W �, one thus has the equivalences�(x) = �xmWx = �xmW �x = ��xmnPi=1L�ixRLiR�1 = ��xmnPi=1L�ixRLiR�1 = ��xmnPi=1L�ixRLi = ��xRwhere the next last equivalence follows from faithfulness of tr, and thus of �. This provesLemma 6.3 in the case S = 1.For a general S, introduce li = S�1=2LiS1=2and R0 = S1=2RS1=2.Then Xi lil�i = 1and Xi l�iR0li = R0:Using the lemma with S = 1, we thus have the equivalence, for j�j = 1;Pi liyl�i = �ymPi l�i yS1=2RS1=2li = ��yS1=2RS1=2or Pi LiS1=2yS1=2L�i = �S1=2yS1=2mPi L�iS�1=2yS1=2RLi = ��S�1=2yS1=2R:

ENDOMORPHISMS OF B(H). II 25Introducing x = S�1=2yS1=2, this saysPi LiSxL�i = �SxmPi L�ixRLi = ��xRand Lemma 6.3 is proved.To prove the �nal equivalence (iv) ,(vi) of Theorem 6.1 we just apply Lemma 6.3 onA = PAkP and with S = P and � = 1, to deduce that the dimensions of the eigensubspacesof x 7! Pi L�ixLi and x 7! Pi LixL�i corresponding to eigenvalue 1 must be the same. Thisends the proof of Theorem 6.1.Remark. Let �(x) = Pni=1 LixL�i , Wx = �(x), x 2 PAkP , be the operators intro-duced in the proof of Lemma 6.3. From [Al-HK], we know that�(W ) \ T = �(�) \ Tis a subgroup of T, in the present case a �nite group, called the Frobenius Group G�.For the decomposition W = U � V on L2(�), with U unitary, and V completely non-unitary (see [NaFo]), we have �(U) = G� and the spectrum of V is contained in the interiorof f� 2 C : j�j � 1g. This means that we have the following clustering i� G� = f1g:8m 2 N , 8A 2Mnm , 8B 2 On:limr!1!(A�m+r(B)) = !(A)!(B)and the convergence is exponential.In [BJW] we will establish that a state ! 2 Sk will actually de�ne a state in Pk if and onlyif (in addition to the properties (i){(vi) of Theorem 6.1) the peripheral spectrum of � consistsof a 1 alone, i.e., G� = f1g. In general, if G� � Zm, the state !jUHFn has a decompositioninto pure states \over Zm". We will illustrate this with an example in Example 6.2, where!jUHFn = !1jUHFn = mXi=1 1m' � �i�����UHFnand ' is a pure state on Mn1 which is periodic with period m under the two-sided shift.The fact that !jUHFn = !1jUHFn is of course very special for this example. We defer thegeneral discussion to [BJW].The following example is a preamble to the class of examples analyzed in Section 7.Example 6.1. We consider the setting in Theorem 5.2 and Corollary 6.1 above. Wehave n 2 N , but set k = 1. In [BJP, Theorem 8.1] we gave a concrete example of a state! in P1, i.e., a state ! on On such that ! � � = ! � �2, and the restriction !jUHFn is pure.The corresponding shift on B(H) we showed was not conjugate to any shift de�ned froma product state on UHFn. Note that the algebra A1 is now just a copy Mn of the n byn complex matrices and the space V1 from (3.4) has dimension n. Using Theorems 4.1and 5.2 we note that the state !, and therefore, the corresponding shift on B(H), may be

26 O. BRATTELI AND P.E.T. JORGENSENcalculated directly from the elements fLigni=1 in A1 ' Mn, and a simple calculation, using[BJP, Chapter 8] yields the formulaAi = n�1=2 nXj=1hi; jieji(6.5)where hi; ji := exp �2�p�1 ij=n� for 8i; j 2 f1; : : : ; ng;(6.6)and e(1)ij denote the usual matrix units for Mn (see (2.12) above). As a result, we note thatthere are vectors hi 2 C n , khik = 1 8i, such thatLi = jei >< hij ;hi(j) : = n�1=2hi; ji:(6.7)It is easy to check from (6.5) that nXi=1L�iLi = nXi=1 LiL�i = Inwhere In is the unit-matrix in Mn. Note also, in this case, that the set f;1; : : : ;ng isorthogonal, where i = S�i .For this example, it is also easy to check the minimality condition from [FNW2, De�nition1.2]. It amounts to the assertion that there is no proper subalgebra of A1 ' Mn whichcontains the unit, and is invariant under all the operatorsA 7! LiAL�j on A1:(6.8)Let us discuss this condition a bit further in the present context, where we have normal-ization Xi L�iRLi = Rand strong asymptotic invariance Xi LiL�i = P:By [FNW2, Theorem 1.5] minimality then means that the only eigenvalue of the operatorx 7! Pi LixL�i of absolute value 1 is 1, and the corresponding eigenspace is one-dimensional,i.e., the only eigenvector in PAkP of this operator with eigenvalue of modulus 1 is P . Butthen a simple argument (see the proof of Lemma 7.8) shows that the only solutions ofXi L�ixLi = xare the scalar multiples of R, and hence Theorem 5.3 implies that minimality of the fLigsystem implies purity of !.

ENDOMORPHISMS OF B(H). II 27It can be shown that minimality of the fLig-system on PAkP is equivalent with irre-ducibility of the corresponding systemfLim � � �Li1L�j1 � � �L�jmgm = 1; 2; : : : , i1; : : : ; jl = 1; : : : ; n, on P C nk , [FNW2].Example 6.2. Let us end by exhibiting a state in S1 on O3 where fLig is irreducible,but not minimal. Here P = 1, A1 ' M3 andL1 = 0B@ 0 0 10 0 00 0 0 1CA ; L2 = 0B@ 0 0 01 0 00 0 0 1CA ; L3 = 0B@ 0 0 00 0 00 1 0 1CA :Then Xi LiL�i =Xi L�iLi = 1and Lim � � �Li1L�j1 � � �L�jm = �i1j1�i2j2 � � � �imjmeim�1;jm�1so the linear span of these consists of all diagonal 3�3 matrices. Hence fLig is not minimal,albeit irreducible.Now, if x = 0B@ x11 x12 x13x21 x22 x23x31 x32 x33 1CAone computes that Xi L�ixLi = 0B@ x22 0 00 x33 00 0 x11 1CA :Thus the operator x! Pi L�ixLi has 0 as an eigenvalue of multiplicity 6, and the three cuberoots �m = 1; �; �2 of 1 as simple eigenvalues with corresponding eigenvectors0B@ 1 0 00 �2m 00 0 �m 1CA :In particular, the only possible choice of R is R = 131, and it follows from Theorem 5.3 thatthe corresponding state ! is pure, i.e., ! 2 S1.Let us compute the restriction of ! to UHF3. IfI = (i1; i2; : : : ; im)where il 2 Z3, put�(I) = ( 1 if ip+1 = ip + 1mod3; p = 1; : : : ; m� 10 otherwise.

28 O. BRATTELI AND P.E.T. JORGENSENThen a calculation using Li = e i;i�1 shows that! �ekl e(2)i1j1 � � � e(m+1)imjm � = 3�1�(I)�(J)�k;i1�1�l;j1�1�imjm= 3�1�(I; J)�(I)where �(I; J) = ( 1 if I = J0 otherwise.Thus ! restricted to UHF3 is a convex combination of three pure states! = 13(!1 + !2 + !3)where !i is the pure product state on UHF3 = N1m=1M3 de�ned by the in�nite productvector ei ei+1 ei+2 � � � (cyclic notation from Z3)where feigi2Z3 is the canonical basis of C 3 . In particular, this shows that if ! 2 S1, then!jUHF3 is not necessarily pure. Note that in the example !jUHF3 is actually �-invariant, itis a convex combination of 3 pure states of period 3 under �, which form an orbit of length3 under the action of �� on UHF�3. That the �-invariant state ! is not pure then also followsfrom the fact that the peripheral spectrum of x 7! Pi LixL�i consists of more than the point1, namely the three cube roots of 1.7. U(n)-orbits and a Cross SectionLet n 2 N be �xed. From Proposition 3.1, we know that given states ! and !0 on UHFn,both in S1k=0 Pk, determine conjugate shifts on B(H) i� there is a g 2 U(n) such that!01 = !1 � �g, where !1 and !01 are the associated translationally invariant pure states onN1�1Mn. (For more details on the state !1, see Section 1.) Each ! (and !1) is associatedwith elements L 2 L(C n ;Mnk) for some k. We will now show that these elements L span aHilbert space which in turn carries a unitary corepresentation of U(n), g 7! Lg, such that Lgis associated with the state ! � �g for g 2 U(n). We thus get the conjugacy classes of shiftson B(H) labeled by orbits for this unitary corepresentation. The examples we give beloware a set of shifts (for �xed Powers index n) which are labeled by functionsu : 1Y1 Zn ! T(7.1)depending only on a �nite number of variables. When k > 0, and u is a nonconstant function,then the corresponding shift �u is not conjugate to any of the shifts which correspond to aproduct state on UHFn, and which were considered in [Lac1], [BJP].We will show in Theorem 7.5 that generically our u-function examples form a cross sectionfor the U(n)-orbits in the L space in the sense that each U(n)-orbit intersects the set of u-function examples in at most a manifold homeomorphic to a disjoint union of n! copies of

ENDOMORPHISMS OF B(H). II 29Tn: that is, when the conjugacy class is given then there is only at most this manifold offunctions u which represent the shifts from the conjugacy class.Now to details: Let k, n 2 N , and letu : Zn� � � � � Zn| {z }(k+1)-times ! Tbe a given function. Let X = Q11 Zn with Haar measure, and let H = L2(X) be thecorresponding Hilbert space. Then in [BJP] we have considered the � 2 Rep(On;H) givenby (�(si)�)(x1; x2; : : : ) = n1=2u(x1; : : : ; xk+1)�x1i�(x2; x3; : : : )(7.2) (�(s�i )�)(x1; x2; : : : ) = n�1=2�u(i; x1; : : : ; xk)�(i; x1; x2; : : : )(7.3)Now, let ! be the state corresponding to the vector = 1 2 L2(X). A calculation, usingthe formula for �(s�i ), now shows that! �e(1)i1j1 e(2)i2j2 � � � e(m)imjm� = ! �si1 � � � sims�jm � � � s�jm�= DS�im � � �S�i1 j S�jm � � �S�j1E= n�k�mFk;m (i1; : : : ; im; j1; : : : ; jm)where Fk;m(i1; : : : ; im; j1; : : : ; jm) = Xx1;::: ;xk �k;m(i; x)�k;m(j; x)(7.4)and �k;m(i; x) = �k;m(i1; : : : ; im; x1; : : : ; xk)= u(im; x1; : : : ; xk)u(im�1; im; x1; : : : ; xk�1)� � �u(im�k; : : : ; im; x1)u(im�k�1; : : : ; im) � � �u(i1; : : : ; ik+1):This means that �k;m is given by the expression above if m � k + 2, but if m < k + 1 theproduct de�ning �k;m just truncates after the factoru(i1; : : : ; im; x1; : : : ; xk+1�m):Using � �e(1)i1j1 � � � e(m)imjm� = nXi=1 e(1)ii e(2)i1j1 � � � e(m+1)imjm ;and the formuli above, one now calculates (for m > k + 1)! � � �e(1)i1j1 � � � e(m)imjm� = 1n nXi=1 u(i; i1; : : : ; ik)u(i; j1; : : : ; jk)! �e(1)i1j1 � � � e(m)imjm� :Thus if i1 = j1; : : : ; ik = jk, then ! � � is equal to !, i.e.,! � �jAck = �jAck :

30 O. BRATTELI AND P.E.T. JORGENSENwhich amounts to the invariance ! � �k+1 = ! � �k. To show ! 2 Pk, we must check that !is pure on UHFn.We denote the state de�ned by on On by �! when it becomes important to distinguishit from the corresponding state ! on UHFn.Proposition 7.1. The restricted state �!jUHFn is pure on UHFn.The proof will be based on a lemma (below) and some calculations which we proceed todescribe.Remark. It follows from [BJP, Lemma 5.2] that�!(A) := nXi=1 ��!(si)A��!(si)�is a shift on B(H).Proof. Set Si := ��!(si) and i1���ik := S�ik � � �S�i1. We then havei1i2���ik(x1; x2; : : : ) = n�k=2�(i1; i2; : : : ; ik; x1; x2; : : : )and therefore S�ji1;::: ;ik = n�1=2u(i1; i2; : : : ; ik; j)i2;::: ;ik;j:Let fLjgnj=1 be the associated elements in Ak ' Mnk . Let ei1���ik := e(1)i1 � � � e(k)ik denotethe canonical basis vectors in C nk = C n � � � C n| {z } .k The operators Lj may be expanded inthe vectors ei1���ik as follows: Let h(i1 ���ik�1)j 2 C nk be given byh(i1���ik�1)j (�1; : : : ; �k) = De�1 ����k j h(i1���ik�1)j ECnk= �i1�2�i2�3 � � � �ik�1�kn�1=2u(�1; i1; : : : ; ik�1; j):Then a small calculation, using the de�ning relation(Lix) = S�j(x)for Li, where x = x1 � � � xk 2 C nk , and the expansion(x) = Xi1���ik �xi11 � � � �xikk i1���ikand the formula for S�ji1;::: ;ik , above, shows thatLj = Xi1���ik�1 ���ei1���ik�1j >< h(i1���ik�1)j ���(7.5)where j >< j is the Dirac notation.In this example, the general formulas from Theorem 6.1 can be veri�ed directly:

ENDOMORPHISMS OF B(H). II 31Lemma 7.2. Let (Lj) and (hi1���ik�1j ) be as above and de�neR = Xi1���ik ���hi1���ik�1ik >< hi1���ik�1ik ���= n�1 Xi1���ikX�1 X�1 u (�1; i1; : : : ; ik) �u (�1; i1; : : : ; ik) e(1)�1�1 e(2)i1i1 � � � e(k)ik�1ik�1:We have nXj=1LjL�j = 1and nXj=1L�jRLj = Rwhen 1 is the identity element in Ak 'Mnk .Proof. The adjoints of Lj 2 Ak areL�j = Xi1���ik�1 ���h(i1���ik�1)j >< ei1���ik�1j���and it follows thatnXj=1LjL�j = Xj ����ei1���ik�1j >< h(i��� )j ���� ����h(i0��� )j >< ei01���j����= Xi Xi0 Xj Dhi1���j j hi01���j E �i1i01 � � � �ik�1i0k�1e(1)i1i01 � � � e(k�1)ik�1i0k�1 e(k)jj= n�1 X�1i1���ik�1j ju(�1; i1; : : : ; ik�1; j)j2 e(1)i1i1 � � � e(k�1)ik�1ik�1 e(k)jj= Xi1���ik�1j e(1)i1i1 � � � e(k)jj = I:The two systems Lj and L�j represent shift operators as follows:Lj ji1 � � � iki = n�1=2u(i1; : : : ; ik; j) ji2 � � � ikji(7.6)and L�j ji1 � � � iki = n�1=2Xp �u (p; i1; : : : ; ik�1; ik) �ik ;j jpi1 � � � ik�1i(7.7)The density matrix R 2 Ak,R = Xi1���ik ���h(i1���ik�1)ik >< h(i1���ik�1)ik ���

32 O. BRATTELI AND P.E.T. JORGENSENthen satis�esXj L�jRLj= Xj Xi1���ikX� X�����h(�1����k�1)j >< e�1����k�1j���� ����h(i1���ik�1)ik >< h(i1���ik�1)ik ���� ����e�1����k�1j >< h(�1����k�1)j ����= X � � �X ��1�1 � � � ��k�1�k�1�j�k�j�kh(i1���ih�1)j (�)h(i1���ik�1)(�) ���h�1����k�1j >< h�1����k�1j ���= X�1����k�1Xj ���h�1����k�1j >< h�1 ����k�1j ��� = R:It follows from Theorem 5.2 that the system (Lj; R) determines a state ! on On whoserestriction to UHFn satis�es (5.2).To �nish the proof of Proposition 7.1, we need only check that the representation (7.2){(7.3) is irreducible on H = L2(X) when restricted to UHFn.Let T 2 B(H) and assume T�(a) = �(a)T , 8a 2 UHFn. Recall UHFn contains thecanonical m.a.s.a. generated bye(1)i1i1 � � � e(m)imim � si1 � � � sims�im � � � s�i1and the representation yields�(si1 � � � sims�im � � � s�i1)�(x1; x2; : : : ) = �i1x1�i2x2 � � � �imxm�(x1; x2; : : : ); 8� 2 L2(X):It follows that there exists f 2 L1(X) such that T = mf , i.e., that T is a multiplicationoperator on L2(X), � 7! f�. But T also commutes with the other operators in UHFn, andthese act as:� �si1 � � � sims�jm � � � s�j1� �(x1; x2; : : : )= � �e(1)i1j1 � � � e(m)imjm� �(x1; x2; : : : )= u(x1; : : : )u(x2; : : : ) � � �u(xm; : : : )�u(jm; xm+1; : : : ) � � � �u(j1; : : : ; jm; xm+1; : : : )�i1x1 � � � �imxm�(j1; : : : ; jm; xm+1; : : : )= Fk;m(i; j; x)�i1x1 � � � �imxm�(j1; : : : ; jm; xm+1; : : : )(see (7.4) above).Since T is a multiplication operator, it also commutes with� 7! �i1x1 � � � �imxm�(j1; : : : ; jm; xm+1; : : : ):This is because �(si1 � � � sims�jm � � � s�j1) is the product of a unitary multiplication operatorand the latter operator, and T commutes with the former, and thus with the latter. A littlecomputation then shows that the function f in T = mf must satisfyf(i1; : : : ; im; xm+1; : : : ) = f(j1; : : : ; jm; xm+1; : : : )

ENDOMORPHISMS OF B(H). II 33for all i, j multi-indices x 2 X, and therefore be constant on X. It follows that the com-mutant of �(UHFn) on L2(X) is one dimensional, which is the asserted irreducibility. Thisends the proof of Proposition 7.1.We showed that when u is given as in (7.8) and ! is the corresponding state, then �!jUHFnis irreducible. Thus ! de�nes a shift on B(H), and by [BJP, Lemma 5.4] two shifts de�nedfrom ! and !0 coincide i� there exists g 2 U(n) such that !0 = ! � �g.In conclusion, there is associated with every k, n 2 N and functionu : Zn � � � � � Zn| {z }(k+1) times ! T(7.8)the following complementary data:(i) !u 2 Pk.(ii) �!u 2 RepS(On;H) = f� 2 Rep(On;H) j �jUHFn is irreducibleg.(iii) �u(A) = P�!u(si)A�!u(si)� for all A 2 B(H), a shift on B(H).(iv) Lu 2 L(C n ;Mnk), Ru 2Mnk .For (iv), note that a system fLjg 2 Mnk determines an L 2 L(C n ;Mnk) by setting fory 2 C n L(y) := nXj=1 yjLj:De�nition 7.1. We say that an element ! in Pk is diagonal if it can be represented by afunction u as in (7.8).Speci�cally, there is a function u : Zn � � � � � Zn| {z }k+1 times ! Tand a basis for C n such that, in the basis, L is represented as follows L (jji) = Lj (j 2 Zn)and Lj ji1 � � � iki = n�1=2u(i1; : : : ; ik; j)� ji2i3 � � � ikjiand R ji1 � � � iki = n�1X� u(�; i2; : : : ; ik)�u(i1; i2; : : : ; ik) j�i1 � � � ik�1i :We showed in Proposition 3.1 that two diagonal (or arbitrary) states !, !0 2 S1k=0 Pkdetermine conjugate shifts i� there is a g 2 U(n) such that !01 = !1 � �g. This means thatconjugacy classes of shifts correspond to U(n)-orbits with the group U(n) acting on the datain any one of the forms (i) or (iv).We now describe the diagonal elements in S10 Pk as a \cross section" for the associatedorbit space.

34 O. BRATTELI AND P.E.T. JORGENSENTheorem 7.3. Consider two diagonal elements in S10 Pk (relative to the same basis in C n)corresponding to functions u and u0. Then the corresponding shifts are conjugate i� thereexists a k such that u and u0 are both functions of k + 1 variables:Zn � � � � � Zn| {z }k+1 times ! Tand there exists a g 2 U(n) such thatu0(i0; i1; i2; : : : ; ik)�i1j1�i2j2 � � � �ikjk(7.9) =Xj0 Xp1 � � �Xpk g(j0; i0) g(p1; i1)g(p1; j1) g(p2; i2)g(p2; j2)� � � g(pk; ik)g(pk; jk)� u(j0; p1; p2; : : : ; pk)for all (i0; : : : ; ik) 2 Zn� � � � � Zn| {z }k+1 .For the proof we need the following result which relates the state ! and the correspondingtensor L, and the transformation rule for the U(n)-coaction.Lemma 7.4. If a state ! 2 Sk is given by the tensor L 2 L(C n ;Mnk) and g 2 U(n), thenthe elements Lg(x) := (g�1 � � � g�1)| {z }k L(gx)(g � � � g) 8x 2 C n ;(7.10) Rg := (g�1 � � � g�1)R(g � � � g)determines the state ! � �g.Proof. Let � := !jMnk where Mnk is viewed as the subalgebraAk 'Mnk � UHFn � Onand let Adk(g) = g � � � g| {z }k � g�1 � � � g�1| {z }k . Then it follows that(! � �g)jMnk = � � Adk(g):We have for 8x, y 2 C n ,�(L(x)L(y)�) = XXxi�yj�(LiL�j)= XXxi�yj(! � �k)(eij)= (! � �k)(sxs�y) = ! � �k(exy) = !1(exy):

ENDOMORPHISMS OF B(H). II 35If g 2 U(n), and Lg is as in (7.10), then(� �Adk(g))(Lg(x)Lg(y)�) = �(Adk(g)(Adk(g�1)L(gx) Adk(g�1)L(gy)�))= �(L(gx)L(gy)�)= (! � �k)(sgxs�gy)= (! � �k)(�g(sxs�y))= (! � �g) � �k(exy);and this formula shows that ! � �g is determined by the tensor Lg as speci�ed.The formula for Rg is computed in a similar fashion.Remark. We say that some L as in the lemma is in reduced form if L(x) 2 PMnkP ,8x 2 C n , where P is the support projection for � := !jMnk . If L and L0 are in reduced formand ! and !0 are the respective states, then (for g 2 U(n)) we have Lg = L0 i� !1��g = !01.When u � 1 the elements fL(y)gy2Cn � Mnk are represented on C nk = C n � � � C n| {z }k asfollows, see (2.10){(2.11) above: Let w := n�1=2(1; : : : ; 1)| {z }n 2 C n . ThenL(y)(x1 � � � xk) = hw j x1ix2 � � � xk yand L(y)�(x1 � � � xk) = hy j xkiw x1 � � � xk�1:When u : Zn � � � � � Zn| {z }k+1 ! T is introduced, the formulas hold with the following modi�-cation: The vector w = (wi)ni=1 becomeswi := n�1=2u(i; � � �| {z }k ) =: u0(i; � � � )and u(i; i1; i2; : : : ; ik)is viewed as a diagonal matrix for each (i1; : : : ; ik)Lu(y)(x1 � � � xk) = D�u0(� � � ) j x1Ex2 � � � xk ywith the variables � � �| {z }1 to k acting on the tensor x2 � � � xk y. Similarly u(i0; i1; : : : ; ik)can be interpreted as the corresponding dual operator for Lu(y)�.Proof of Theorem 7.3. Elements in C n will be denoted y, x1, : : : ; xk. A basis fjiigni=1for C n will be �xed such that x� =Xi x�i jii ; � = 1; : : : ; k

36 O. BRATTELI AND P.E.T. JORGENSENwith summation indices i ranging over Zn. Ifu : Zn � � � � � Zn| {z }k+1 ! T;the contracted function: Zn� � � � � Zn| {z }k times ! C is de�ned byD�x1 j u(�; i2; : : : ; ik+1)E :=Xi1 x1i1u(i1; i2; : : : ; ik+1):Functions f on Zn will be identi�ed with diagonal matrices 0BB@ f(1) 0. . .0 f(n) 1CCA, and simi-larly functions on Zn � � � � � Zn| {z }k will be identi�ed with diagonal elements inMnk = Mn � � � Mn| {z }k .We then haveL(y)(x1 � � � xk) = Xj Xi1 � � �Xik yjx1i1 � � �xkik � Lj ji1 � � � iki= n�1=2Xj Xi1 � � �Xik yjx1i1 � � �xkik � u(i1; : : : ; ik; j) ji2; : : : ; ik; ji= n�1=2 *�x1 j u(�; � � �| {z }k + ���x2 � � � xk yE :Hence, for g 2 U(n), we haveLg(y) ���x1 � � � xkE= (g�1 � � � g�1| {z })k L(gy) ���gx1 � � � gxkE= n�1=2(g�1 � � � g�1| {z })k �gx1 j u(�; � � �| {z }� ����gx2 � � � gxk gyE�= n�1=2 *gx1 j Adk(g�1)u0@�; � � �| {z }k 1A+����x2 � � � xk yE�= n�1=2 D�x1���Adk(g�1)u�g 0@�; � � �| {z }k 1A����x2 � � � xk yE�where u�g 0@�; � � �| {z }k 1A =Xj g(j; i)u0@j; � � �| {z }k 1A :

ENDOMORPHISMS OF B(H). II 37Recalling the formulaAd(g�1)0BB@ f1 0. . .0 fn 1CCA = 0@ nXp=1 g(p; i)g(p; j)fp1A ;(7.11)the desired formula (7.9) in the theorem follows.Remark. When u is given as in (7.8), then it is only for a very special subset in U(n)that ! � �g is diagonal in the same basis.Let k, n 2 N be �xed. The transformation rule from the expression on the right hand sidein (7.9) holds for general diagonal elements in Pk. The U(n) coaction refers to the manifoldL of all tensors subject to the conditions in Theorem 6.1 above. We may de�ne an innerproduct for elements L and L0 in L ashL j L0i = traceMnk 0@ nXj=1L�jL0j1Aand Lemma 7.4 then implies that the U(n)-coaction L 7! Lg extends to a unitary coactionon the linearization, i.e., we haveDLg j L0gE = hL j L0i for 8L; L0 2 L:By a slight abuse of notation, we will use Tn � Sn to denote the subgroup of U(n) withthe property that g 2 Tn � Sn i� each row and each column of g has only one nonzeroelement, and this element is then necessarily a phase factor. Thus Tn � Sn identi�es withthe semidirect product of the n-torus Tn by the permutation group Sn of n elements, actingon Tn by permuting coordinates.Theorem 7.5. For any u 2 C(Zk+1n ;T), the U(n)-orbit fLgu j g 2 U(n)g in Ln;k intersectsthe diagonal elements for g 2 Tn � Sn, and if g = (�1; : : : ; �n)� � 2 Tn � Sn, we haveug(i0; i1; : : : ; ik) = ��(i0)u(�(i0); �(i1); : : : ; �(ik)):Conversely, for a dense open subset of C(Zk+1n ;T), the U(n)-orbit in Ln;k intersects thediagonal elements only for g 2 Tn � Sn.Remark 1. For a general u 2 C(Zk+1n ;T) the intersection could be larger. For example,if u(x0; x1; : : : ; xk) = 1 for all x0, x1, : : : , xk, then the set of g such that Lgu is diagonal isthe set of all g 2 U(n) transforming0BBBB@ 11...1 1CCCCA into a vector of the form 0BB@ �1...�n 1CCAwhere j�ij = 1 for i = 1, : : : , n.

38 O. BRATTELI AND P.E.T. JORGENSENRemark 2. For the dense open subset of C(Zk+1n ;T) we shall take the set of u with theproperty that for any (i1; : : : ; ik) 2 Zkn there exists a pair i, j 2 Zn such thatu(i; i1; : : : ; ik) 6= u(j; i1; : : : ; ik);but if k � 2 this is not the optimal choice.Proof. Fix the function u = u(x0; : : : ; xk) and assume that g 2 U(n) is an element suchthat !u � �g is diagonal. This means that the u0 de�ned by formula (7.9) is a function ofk + 1 variables such that ju0(x0; : : : ; xk)j = 1for all x0, : : : , xk 2 Zn. Now, identify u with the �nite sequence�U(i0) = [u(i0; i1; : : : ; ik)�i1j1 � � � �ikjk]of nk � nk unitary diagonal matrices, i.e., i0 labels the n matrices, and (i1; : : : ; ik; ; j1 � � � jk)labels the matrix entries. Formula (7.9) in conjunction with formula (7.11) then says thatg 2 U(n) is such that !u � �g is diagonal if and only if�U 0(i0) =Xj0 g(j0; i0)(g�1 � � � g�1| {z }k factors ) �U(j0)(g � � � g| {z }k factors )(7.12)is a new family of nk � nk unitary diagonal matrices.From this formula we �rst see that if g 2 Tn�Sn, then �U 0(i0) is diagonal since �U(j0) is so,and the �rst part of the theorem follows. Next note that g = (�0; : : : ; �n) � � correspondsto the matrix g(i; j) = �i�i;�(j)in U(n), and, inserting this into the formula (7.9), the formula for ug follows.Now, multiply (7.12) by g(k0; i0) and sum over i0 to obtainXi0 g(k0; i0) �U 0(i0) = (g�1 � � � g�1| {z }k factors ) �U(k0)(g � � � g| {z }k factors )for all k0 2 Zn. But by Stone-Weierstrass's theorem, if u has the property in Remark 2, the�-algebra generated by �U(1), : : : , �U(n) is the �-algebra D of all diagonal operators in Mnk .Since �U 0(1), : : : , �U 0(n) are assumed to be diagonal, it thus follows from the relation abovethat (g�1)kDgk � D:From a standard result of Weyl, [Hel], it follows that gk 2 Tnk�Snk , and hence g 2 Tn�Sn.This ends the proof of Theorem 7.5.Theorem 7.6. Let k, n 2 N be given, and letu : Zn � � � � � Zn| {z }k+1 ! T

ENDOMORPHISMS OF B(H). II 39be a function and let the system Lj = Luj depending on u be given as in (7.6). Then theelements 1; LiL�j ; : : : ; Lip � � �Li1L�j1 � � �L�jp ; : : :span all of Mnk , i.e., the system is minimal in the sense of [FNW2].Proof. The result follows from a brute force calculation, or from the clustering for !1,which in turn follows from (7.13), below, and [FNW2, Theorem 1.5]. When applied to thepresent example, [FNW2] yields the asserted minimality property for fLigni=1 if we check that,for 8p 2 N , 8A 2 Ap 'Mnp and 8B 2 UHFn � On, limj!1 !1(A�p+j(B)) = !1(A)!1(B).Recall, since ! = !u satis�es !1 = ! � �k, the desired clustering property is implied by thefollowing:Lemma 7.7. Let u : X ! T be given and suppose it is a function of k+1 variables, and let! = !u be the corresponding state. Then for all p 2 N, all A 2 Ap, and all B 2 UHFn � On,we have ! �A�p+2k(B)� = !(A)(! � �k)(B):(7.13)Proof. Let m > 2k. Then! �e(1)i1j1 � � � e(m)imjm� = n�m ZX 1Yh=0 u(�h(i; x))u(�h(j; x)) d�(x)(7.14)where d�(x) is the Haar measure on X which here involves on a �nite number of summations,and where u is viewed as a function on X = Q11 Zn but depending only on the �rst k + 1variables; (i; x) := (i1; i2; : : : ; im; x1; x2; : : : ) 2 X;and �(y1; y2; : : : ) := (y2; y3; : : : ) for 8y 2 X:The \in�nite" product is really �nite, i.e., the last factors u(�h(i; x)) 6= 1 areu(im�k+1; : : : ; im; x1) � � �u(im; x1; x2; : : : ; xk):For the evaluation of the left hand side in (7.13) we may restrict to terms e(1)i1j1 � � � e(m)imjmwith m > 2k, and the subindices on the form(i1 � � � iprp+1 � � � r2p+2ki2p+2k+1 � � � im)and (j1 � � � jprp+1 � � � r2p+2kj2p+2k+1 � � � jm):We take A = e(1)i1j1 � � � e(p)ipjp and similarly for B. Then the result follows where thefactors are written out in !(A�p+2k(B)) and terms on the form u(rq � � � rq+k)�u(rq � � � rq+k)are cancelled. (Recall u maps into T so u(x)�u(x) = ju(x)j2 = 1 for 8x 2 X.)

40 O. BRATTELI AND P.E.T. JORGENSEN8. Density of Strongly Asymptotically Shift Invariant States in theAsymptotically Shift Invariant StatesLet us use the terminology that a pure state ! of UHFn is asymptotically shift invariantif it is in P , i.e., if limm!1 (! � � � !)jAcm = 0or limm!1 ! � �m+1 � ! � �m = 0:We say that ! is strongly asymptotically shift invariant if there is a k 2 N such that ! 2 Pk,i.e., ! � �k+1 = ! � �k:We will now address the question how large Sk Pk is in P . The answer is that it is less thannorm dense:Proposition 8.1. There is a state ! 2 P such that if ' 2 Sk Pk, then (! � ')jAcm = 1for all m 2 N.Proof. Let !m be a sequence of pure states on Mn such that1Xm=1 k!m � !m+1k2 < +1but f!mjm � Mg is dense in the pure state space of Mn for all M 2 N (so in particularP1m=1 k!m � !m+1k = +1). (Such a sequence may be constructed as follows: Let 'm beany dense sequence in the pure state space of Mn. The 'm are vector states given by unitvectors in C n , and we may assume h�m; �m+1i � 0 where �m is a unit vector correspondingto 'm. By rotating �m into �m+1 through a sequence of m2 equal angles, we obtain m2 + 1pure states 'm;0 = 'm, 'm;2, : : : , 'm;m2 = 'm+1 such that k'm;k � 'm;k+1k � �=m2 fork = 0; : : : ; m2 � 1, and thus Pm2�1k=0 k'm;k � 'm;k+1k2 � m2 (�=m2)2 = �2=m2. Now let !mbe the sequence '1, '2;0, : : : , '2;4 = '3;0, : : : , '3;9 = '4;0, : : : . Then f!mg is dense, andPm k!m � !m+1k2 � Pm �2=m2 < +1.)Let ! be the corresponding in�nite product state on UHFn =N1m=1Mn,! = 1Om=1!m:By [BJP, Example 5.5], ! 2 P . Let � > 0 and choose ' 2 Pk such that ('� !)jAcl � � forsome l 2 N . But this would imply k!m1�!m2k � 2� for allm1, m2 � l, and as f!m j m �Mgis dense, it follows that 2� � 2. The proposition follows.Remark. By a simple argument, one may replace 1 by 2 in Proposition 8.1.

ENDOMORPHISMS OF B(H). II 41Acknowledgments. The present paper was started while the two authors visited theFields Institute for Research in the Mathematical Sciences, and the main body of work wasdone while the authors visited the Centre for Mathematics and Its Applications, School ofMathematical Sciences, Australian National University. The paper was �nished when the�rst author visited the Department of Mathematics, University of Iowa. We are very gratefulfor hospitality from the respective hosts, Professors G.A. Elliott (the Fields Institute), D.W.Robinson (ANU) and P.S. Muhly (University of Iowa). The research also bene�tted frommany helpful conversations with B.V.R. Bhat, G.A. Elliott, A. Kishimoto, M. Laca, P. Muhly,G. Price, D.W. Robinson and from e-mail exchanges with R.F. Werner.The �rst named author was supported by the Norwegian Research Council, and bothauthors by the National Science Foundation (U.S.A.), and the second author was also sup-ported by a University of Iowa Faculty Scholar Fellowship and travel grant, and by a grantfrom the Australian National University. This support is gratefully acknowledged.References[Al-HK] S. Albeverio, and R. H�egh-Krohn, Frobenius theory for positive maps on von Neumann algebras,Commun. Math. Phys. 64 (1978), 83{94.[Ara1] H. Araki, On quasi-free states of CAR and Bogoliubov automorphism, Publ. Res. Inst. Math. Sci.6 (1970), 385{442.[Ara2] H. Araki, On quasi-free states of the canonical commutation relations II, Publ. Res. Inst. Math.Sci. 7 (1971), 121{152.[ACE] H. Araki, A.L. Carey, and D.E. Evans, On On+1, J. Operatory Theory 12 (1984), 247{264.[Ar-Woo] H. Araki and E.J. Woods, Complete Boolean algebras of type I factors, Publ. Res. Inst. Math. Sci.2 (1966), 157{242.[Arv] W.B. Arveson, Continuous analogues of Fock space I, Mem. Amer. Math. Soc. 80 (1989), no. 409.[AK] W. Arveson and A. Kishimoto, A note on extensions of semigroups of �-endomorphisms, Proc.Amer. Math. Soc. 116 (1992), 769{774.[BEEK] O. Bratteli, G.A. Elliott, D.E. Evans, and A. Kishimoto, Quasi-product actions of a compactabelian group on a C�-algebra, Tohoku Math. J. 41 (1989), 133-161.[BEGJ] O. Bratteli, D.E. Evans, F.M. Goodman, and P.E.T. Jorgensen, A dichotomy for derivations onOn, Publ. Res. Inst. Math. Sci. 22 (1986), 103{107.[BJP] O. Bratteli, P.E.T. Jorgensen, and G.L. Price, Endomorphisms of B(H), Proceedings of Symposiain Pure Mathematics, Amer. Math. Soc. 1995 (ed., I.E. Segal), to appear.[BJW] O. Bratteli, P.E.T. Jorgensen, and R. Werner, Pure states on On, in preparation.[Br-Rob] O. Bratteli and D.W. Robinson, Operator algebras and quantum statistical mechanics, vol. II,Springer-Verlag, Berlin{New York, 1981.[Bra] O. Bratteli, Inductive limits of �nite dimensional C�-algebras, Trans. Amer. Math. Soc. 171(1972), 195{234.[Cob] L.A. Coburn, The C�-algebra generated by an isometry, Bull. Amer. Math. Soc. 73 (1967), 722{736.[Cun] J. Cuntz, Simple C�-algebras generated by isometries, Commun. Math. Phys. 57 (1977), 173{185.[Dae] A. van Daele, Quasi-equivalence of quasi-free states on the Weyl algebra, Commun. Math. Phys.21 (1971), 171{191.[Din] H.T. Dinh, On discrete semigroups of �-endomorphisms of type I factors, Internat. J. Math. 3(1992), 609{628.

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ENDOMORPHISMS OF B(H). II 43[PR] R.T. Powers and D.W. Robinson, An index for continuous semigroups of �-endomorphisms ofB(H), J. Funct. Anal. 84 (1989), 85{96.[Seg1] I.E. Segal, The structure of a class of representatives of the unitary group on a Hilbert space, Proc.Amer. Math. Soc. 8 (1957), 197{203.[Seg2] I.E. Segal, Irreducible representations of operator algebras, Bull. Amer. Math. Soc. 61 (1947),69{105.[Sta] P.J. Stacey, Product shifts on B(H), Proc. Amer. Math. Soc. 113 (1991), 955{963.[Voi] D. Voiculescu, Symmetries of some reduced free product C�-algebras, LNM 1132, Operator Alge-bras and Their Connection to Topology and Ergodic Theory, Springer-Verlag, Berlin, 1985, pp.556{588.[Wor] S.L. Woronowicz, Compact matrix pseudogroups, Commun. Math. Phys. 111 (1987), 613{665.Mathematics Institute, University of Oslo, PB 1053-Blindern, N-0316 Oslo, NorwayE-mail address : bratteli@math.uio.noDepartment of Mathematics, University of Iowa, Iowa City, IA 52242, U.S.A.E-mail address : jorgen@math.uiowa.edu