Structure of Some von Neumann Algebras with Isolated Discrete Modular Spectrum

From a finite von Neumann algebra f$ with a faithful normal trace and its normal injective * endomorphism $ satisfying 0(S) = 0(1)$0(1), we construct another von Neumann algebra M(3f, 0) by a method which reduces to groupmeasure construction when % is commutative and is an automorphism. If $ satisfies 0(z) = (l)z for all central elements z of % and 0(l)"i = e~ for a positive number 3 (%) = 0j(l) % 0 j ( l ) , j(z) = z(?>j(l) for all central element z of % and $j(l)* = e~j. If p is a KMS state under time translation of a C* algebra, which is asymptotically abelian with respect to (either discrete or continuous) space translation and if the spectrum of generator of time translation has exclusively an isolated point spectrum in the representation associated with p, then the associated von Neumann algebra has the above structure where the asymptotic ratio set of f£jas well as that of a possible finite summand (if non-zero) is {1} and roo(Af(Sfy, ) for a commutative semigroup of endomorphisms of a finite von Neumann algebra, instead of one 0, is given. § 1. Notation and Main Results Let 3Ji be a von Neumann algebra, Q be a cyclic and separating unit * Received August 4, 1972. 2 HUZIHIRO ARAKI vector, A be the modular operator for Q, r(t)Q = A^QA"*, /be the modular conjugation operator for J2 and j(Q) = JQJ. $Qa denotes the eigenspace of log A belonging to an eigenvalue a and 3Wa denotes the set of @e2ft such that r(t)Q = e Q. It is known that 2J£0 is a finite von Neumann algebra containing the center Q of Wl and J2 is a cyclic and separating trace vector for 2J10 restricted to £>0. 3o denotes the center of 2TC0. 3o ^3We are interested in the structure of 5K and we can analyze it when log A has exclusively an isolated point spectrum and 3o—3For any given finite von Neumann algebra ff with a faithful normal trace vector W such as 5F10 with Q and its normal injective * endomorphism 0 satisfying ($) = 0(1)^0(1), we present in section 2 a method of constructing another von Neumann algebra, denoted as M(£?, 0), with a cyclic and separating vector fi(f$, 0). If (F, (z)¥) = e~(¥, z¥) for all regc (the center of g)» then the spectrum of modular operator J(§, 0) for fl(g, 0) is Sea = {Q}\J{e ; 71 = 0, ±1,...}. If 0(*) = *0(1) for ^egc, then 30 = 3. The following main result proved in section 3 gives a converse. Theorem I. Let 971 be a von Neumann algebra with a cyclic and separating vector Q such that log A has exclusively an isolated point spectrum {0, ±al9 ±a2, ...}, 0n of PnW,Q satisfying STRUCTURE OF SOME VON NEUMANN ALGEBRAS if Pw^0, where \ is the canonical tl mapping in 2ft0. In a special case where 2ft is a factor, the spectrum of logJ is necessarily an additive group. (This statement is always true if all subspaces tQ(I^ = (E/3-Q — Ea+0)fQ for 4 = \AdEx.> I=(a, /?), is cyclic for 2ft (and hence separating for 2ft due to /§(/) = §( — /)) even when J has a continuous spectrum.) Examples where the center of 2ft0 does not coincide with the center of Tl are tensor product of ITPFI of the class 501 (type III or II) with any finite von Neumann algebra where the vector & is product of a defining product vector of the ITPFI with a cyclic and separating trace vector for the finite von Neumann algebra. Another example for 3o^3 is RX®R with @ = QX§§® where Qx is the defining product vector of Rx, R is type 12 and spectrum of modular operator for (R, 0) is {y~ l , 1, j} with y£ Sx. The condition 3o—3 * satisfied for a KMS state of an asymptotically abelian C* algebra. More precisely, a net of operator Qa in a von Neumann algebra 2ft is called strongly central if there exists a weakly total self-adjoint subset SB of 2ft such that lim \jQa, uT\ = Q strongly for every a we26. A subset §1 of 2ft is called strongly ra central relative to a net ra of * automorphisms of 2ft if raQ is strongly central in 2ft for each @e§J. We have Theorem 2. Let 2ft be a von Neumann algebra, ra be a net of * automorphisms of 2ft, p be a faithful normal ra invariant state of 2ft and SI be a weakly dense C* subalgebra of 2ft, which is invariant under modular automorphism r p(t) for p and is strongly ra central. Assume that modular operator A p for p is such that log A p has exclusively an isolated point spectrum. Then 3o—3 d there exists central projections Pn satisfying (i)~(iii) of Theorem 1. If the representation space is separable, in addition, then {1} if Po^O, = {1} if Pw^0, HUZIHIRO ARAKI SXn if Pn*Q, where xn = e~ «, n>Q. This theorem is applicable to a situation where p is a KMS state for time translation of a C* algebra A, which is asymptotically abelian for (discrete or continuous) space translation, the generator of time translation has exclusively an isolated point spectrum in the representation associated with p and the representation space is separable. §2. Construction of M(g, 0) Let f$ be a finite von Neumann algebra acting on a Hilbert space ^ with a cyclic and separating unit trace vector W and %c be the center of %. Let \ be the canonical \ mapping on $c, Jw be the modular conjugation operator for W and jw(Q}=JyQJ^. Let $ be a normal injective * endomorphism of ££> &w be the vector state by 3F, and 0*<% be the normal positive linear functional defined by Both <% and $*<% are faithful and tracial. By the Radon-Nikodym theorem, there exists a strictly positive selfadjoint operator A^=\kdE^ such that E£^$c and lim *u In other words, W is in the domain of 0(^) = limz0(^4z) and (2.1) WGW 0(*)?K^)20 = (F, *F), *EE&. (These equations hold for ze£5 as will be seen in the following proof.) Lemma 1. There exists a unique isometric operator V satisfying (2.2) It satisfies (2.3) V*V=\, (2.4) STRUCTURE OF SOME VON NEUMANN ALGEBRAS (2.5) C/r, F>0. Proof. Because $*a)¥ and o)¥ are tracial, we have -«¥(((A^)¥ ', which is invariant under 0©). Hence VV* commutes with 0(@),

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