A Note on Eigenvalues of Liouvilleans

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1 A Note on Eigenvalues of Liouvilleans V. Jak»sić and C.-A. Pillet 2;3 Department of Mathematics and Statistics McGill University 805 Sherbrooke Street West Montreal, QC, H3A 2K6, Canada 2 PHYMAT Université detoulon, B.P. 32 F La Garde Cedex, France 3 CPT-CNRS Luminy, Case 907 F-3288 Marseille Cedex 9, France June 29, 200 Abstract. Let L be the Liouvillean of an ergodic quantum dynamical system (M;fi;!). We give anew proof of the theorem of Jadczyk that eigenvalues of L are simple and form a subgroup of R. If! is a (fi;fi)-kms state for some fi 6= 0we show that this subgroup is trivial, namely that zero is the only eigenvalue of L. Hence, for KMS states ergodicity is equivalent to weak mixing. Key words. mechanics Liouvillean, ergodic theory, spectral theory, weak mixing, quantum statistical

2 2 Introduction Let M be a von Neumann algebra on the Hilbert space H and M Λ its predual. The positive elements of M Λ satisfying!() =are called states. Let fi t be a ff(m Λ ;M)-continuous group of automorphisms of M. The pair (M;fi) is called a W Λ -dynamical system. In the algebraic formalism of quantum statistical mechanics, the elements of M describe observables of the physical system under consideration and the group fi specifies their time development. A functional 2 M Λ is fi-invariant if ffi fi t = for all t. Atriple (M;fi;!), where! is a fiinvariant state, is called quantum dynamical system. For our purposes we may assume without loss of generality that! is a vector state, namely that!(a) = (Ω;AΩ) for some Ω 2 H, and that Ω is a cyclic and separating vector for M. Inwhat follows, (M;fi;!) is a given quantum dynamical system satisfying these properties. The system (M;fi;!) is called ergodic if for all A; B 2 M, Z T lim T! 2T T!(fi t (A)B)dt =!(A)!(B); and weak-mixing if lim T! 2T Z T T j!(fi t (A)B)!(A)!(B)j 2 dt =0: Clearly, weak-mixing implies ergodicity. It is known that ergodic properties of quantum dynamical systems can be characterized in spectral terms, in analogy with Koopman s lemma of classical ergodic theory [BR, JP]. There exists a unique self-adjoint operator L on H such that for A 2 M, fi t (A) =e itl Ae itl ; LΩ =0: The operator L is a non-commutative analog of the classical Koopman operator. One can show (see Theorem 4.2 in [JP]) that the quantum dynamical system (M;fi;!) is ergodic iff zero is a simple eigenvalue of L and weak-mixing iff L has no other eigenvalues except for a simple eigenvalue zero. We denote by ff p (A) the set of eigenvalues of a self-adjoint operator A. Inthis paper we give a new proof of the following result of Jadczyk [J] (see also [BR], Theorem ). Theorem. Assume that zero is a simple eigenvalue of L. Then all eigenvalues of L are simple and ff p (L) is a subgroup of R.

3 3 Remark. For dynamical systems which arise in classical ergodic theory this result goes back to Halmos and von Neumann, see [HN, W]. The first results in the non-commutative case go back to [RR]. Our proof of Theorem. is somewhat simpler and perhaps more transparent then the argument in [J, BR]. Moreover, the method of the proof yields some additional information. Let be the modular operator associated to Ω, L =log and ff t (A) =e itl Ae itl the group of modular automorphisms of M. Since LΩ =0the triple (M;ff;!) is also a quantum dynamical system. Theorem.2 Assume that zero is a simple eigenvalue of L and L. Then ff p (L) =ff p (L) =f0g: Let fi 6= 0 be a real parameter.! is called (fi;fi)-kms state if for all A; B 2 M there is a function F A;B, analytic inside the strip fz j 0 < signfi Imz < jfijg, bounded and continuous on its closure, and satisfying the KMS-boundary condition F A;B (t) =!(Afi t (B)); F A;B (t +ifi) =!(fi t (B)A): A (fi;fi)-kms state describes a physical system in thermal equilibrium at inverse temperature fi. By a theorem of Takesaki [BR],! is a (fi;fi)-kms state iff L = fil. Therefore, Theorem.2 implies the following somewhat surprising result. Theorem.3 Assume that the system (M;fi;!) is ergodic and that! is a (fi;fi)-kms state for some fi 6= 0. Then the system is weak-mixing. Theorems.2 and.3 show how modular structure associated to! confines spectral structure of Liouvillean. Besides general interest, we expect that these theorems will be technically useful in the study of concrete models in quantum statistical mechanics. For an application of these theorems to the study of ergodic properties of Pauli-Fierz systems we refer the reader to [DJP]. Acknowledgments. We are grateful to Jan Dereziński for insisting on the question which has led to this note. Part of this work has been done during the visit of the first author to Aarhus University. V.J. is grateful to E. Skibsted and J. Dereziński for their hospitality. The research of the first author was partly supported by NSERC. 2 Proofs We assume that the reader is familiar with Tomita-Takesaki theory. For notational purposes we recall some basic results of this theory.

4 4 Let and J be the modular operator and the modular conjugation associated to the vector Ω. For all A 2 M, J =2 AΩ=A Λ Ω. Set L =log. Then Je itl =e itl J; Je itl =e itl J; (2.) By Tomita-Takesaki theorem, JMJ = M 0. e itl e isl =e isl e itl : The natural cone P is the closure of the set fajajω j A 2 Mg ρh. The cone P is self-dual, namely P = fψ 2 Hj(Ψ; Φ) 0forall Φ 2 Hg. For every state 2 M Λ, there is a unique vector Ω 2 P such that (A) =(Ω ;AΩ ). Moreover, the state is fi-invariant iff LΩ = 0. In particular, if zero is a simple eigenvalue of L, then! is the unique fi-invariant state in M Λ. Proof of Theorem.. Let E be an eigenvalue of L and Ω E a (normalized) eigenvector associated to E. Weshow first that Ω E is a cyclic and separating vector for M. Note that since JL = LJ, Ω E := JΩ E is an eigenvector of L associated to the eigenvalue E. The states! ±E (A) =(Ω ±E ;AΩ ±E ) are fi-invariant, and hence for all A 2 M, (Ω;AΩ) = (Ω ±E ;AΩ ±E ): (2.2) Thus, if AΩ E = 0, then AΩ = 0 and A = 0 (since Ω is separating). Hence Ω E is separating. To prove that Ω E is cyclic, let P 0 be the orthogonal projection on MΩ E and Q 0 = P 0. Then Q 0 2 M 0 and (Ω E ;Q 0 Ω E ) =0. Let Q := JQ 0 J. Then Q is an orthogonal projection, Q 2 M, and 0=(Ω E ;QΩ E ) =(Ω;QΩ) = kqωk 2 : Since Ω is separating, Q =0and P 0 =. Hence Ω E is cyclic. Let U 0 be the linear map defined on MΩ E by U 0 AΩ E = AΩ. Since Ω E is separating, the map U 0 is well-defined. Cyclicity of Ω E and (2.2) yield that U 0 extends to a unitary map on H. Since U 0 ABΩ E = ABΩ =AU 0 BΩ E ; U 0 2 M 0. Let L E be the Liouvillean associated to Ω E. Clearly, L E = L E and U 0 LU 0Λ = L E = L E: (2.3)

5 5 Since zero is a simple eigenvalue of L, E is also a simple eigenvalue. Hence eigenvalues of L are simple. Note that if U := JU 0 J, then (2.3) implies ULU Λ = L + E: (2.4) Let now E i, i =; 2 be two eigenvalues of L and let U 0 i, U i be as in (2.3), (2.4). Set W := U 0 2 U. Then W is unitary and WLW Λ = L + E E 2 : It follows that E 2 E is an eigenvalue of L and ff p (L) is a subgroup of R. 2 Proof of Theorem.2. Theorem. and the third relation in (2.) yield that ff p (L) = ff p (L), and so it suffices to prove that ff p (L) =f0g. Let E 2 ff p (L), U 0, U be as in (2.3), (2.4), and W = UU 0. W is unitary, W Ω 2 P and W L = LW. Since zero is a simple eigenvalue of L we must have W Ω = e i Ω for some real phase. Since P is a self-dual cone, =0and Using that JΩ =Ωand U 2 M we derive UJUJΩ=Ω: JUΩ=U Λ Ω=J =2 UΩ = Je L=2 UΩ: It follows that LUΩ =0, and since LUΩ = EUΩ, E =0. Hence ff p (L) =f0g. 2 References [BR] Brattelli O., Robinson D. W.: Operator Algebras and Quantum Statistical Mechanics, Springer-Verlag, Berlin, second edition (987). [DJP] Dereziński J., Jak»sićV., Pillet C.-A.: In preparation. [HN] [J] [JP] Halmos, P.R., von Neumann, J.: Operator methods in classical mechanics II. Ann. Math. 43, 332 (940). Jadczyk A. Z.: On some groups of automorphisms of von Neumann algebras with cyclic and separating vector. Commun. Math. Phys. 3, 42 (969). Jak»sić, V., Pillet C.-A.: On a model for quantum friction III. Ergodic properties of the spin-boson system. Commun. Math. Phys. 78, 627 (996).

6 6 [RR] Robinson D.W., Ruelle D.: Extremal invariant states. Ann. Inst. Henri Poincaré 6, 299 (967). [W] Walters, P.: An Introduction to Ergodic Theory. Springer-Verlag, New-York (982).