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Extension of the structure theorem of Borchers and its application to half-sided modular inclusions

A result of H.-W. Wiesbrock is extended from the case of a common cyclic and separating vector for the half-sided modular inclusion of von Neumann algebras to the case of a common faithful normal semifinite weight and at the same time a gap in Wiesbrock's proof is filled in.


Introduction
J. Bisognano and E. Wichmann ( [4]) made a discovery about the connection of the modular operator and the modular conjugation for the von Neumann algebra generated by quantum fields in a wedge region of the Minkowski space-time with kinematical transformations, namely pure Lorentz transformation and the TCP 1 operator.
H. J. Borchers ([6]) formulates an important feature of this connection in the abstract setting of a pair of von Neumann algebras N ⊂ M with a common cyclic and separating vector Ω , and a one-parameter group of unitaries U (λ) having a positive generator, which induces a semigroup of endomorphisms of (M , Ω) , obtaining a commutation relation of U (λ) with the modular operator and the modular conjugation for (M , Ω) , which reproduces the kinematical commutation relations in the Bisognano-Wichmann situation.
A further development has been achieved by H.-W. Wiesbrock ([37], [38], [39], [40], [41], [42]), who introduces the notion of the half-sided modular inclusion and obtains an underlying group structure (cf. [31]), as well as an imbedding of the canonical endomorphisms of the subfactor theory into a one-parameter semigroup of endomorphisms in this specific situation. Thus he gets a correspondence between 2-dimensional chiral conformal field theories and a class of type III 1 subfactors.
Unfortunately, there is a gap in Wiesbrock's proof of his basic theorem ( [37], Theorem 3, Corollary 6 and Corollary 7, [42]). We will fill in this gap in Wiesbrock's proof and further generalize the result to the case of a common normal semifinite faithful weight.
As a basic tool to prove general half-sided modular inclusion results, we generalize a structure theorem of H. J. Borchers ([7], Theorem B) considerably, making Borchers' proof at the same time more transparent.
The extension from the state case to the weights turns out to be not straightforward. For this purpose we introduce as a basic tool the notion of a Hermitian map by using modular structure.
It seems that before the summer of 1995, when we independently noticed the gap in the proof of Wiesbrock's half-sided modular inclusion theorem, this gap was generally overlooked. Thanks to Prof. Detlev Buchholz, who has been visiting the first named author in the fall of 1995, we learned about each others insights and started to collaborate on this paper. The first version of the paper, containing a complete proof of the General Half-sided Modular Inclusion Theorem, Theorem 2.1, was available already at the end of 1995. It had a restricted circulation, but it was presented at several conferences. Other topics, like Theorem 2.2 on the structure and type of the involved von Neumann algebras and Proposition 2.4 on pathologies of the analytic extension of orbits of one-parameter automorphism groups, are of more recent date.
We notice that since 1995 a number of papers appeared, containing proposals for a complete proof of the half-sided modular inclusion theorem (see, for example, [17], Section 3 and [10], pp. 608 and 609). None the less, till now we have no knowledge of a completely elaborated proof, even in the state case.

(a) Notations and facts from the Modular Theory of von Neumann
Algebras (see, for example, [35], Chapter 10).
For two Hilbert spaces H and K we denote by B(K, H) the Banach space of all bounded linear maps from K to H . B(H, H) will be denoted simply by B(H) . If T is a not necessarily everywhere defined linear operator from K to H , then Dom (T ) will stand for the domain of T .
We denote the weak and the strong operator topology on B(K, H) respectively by wo and so . The weak topology defined on B(K, H) by all linear functionals belonging to the norm-closure of the wo-continuous linear functionals in the dual of B(K, H) will be denoted by w . Further, the locally convex vector space topology defined on B(K, H) by the seminorms where ϕ ranges over all w-continuous positive linear functionals on B(K) , will be denoted by s . We notice that on any bounded subset of B(K, H) wo = w and so = s .
For a weight ϕ on a von Neumann algebra M we use the standard notations: We notice that for a ∈ M + we have a ∈ M ϕ ⇐⇒ ϕ(a) < +∞ . A von Neumann algebra M on a Hilbert space H is in standard form with respect to a normal semifinite faithful weight ϕ on M if there is a linear map with dense range N ϕ ∋ x −→ x ϕ ∈ H such that ϕ(x * x) = x ϕ 2 , (a x) ϕ = a x ϕ , x ∈ N ϕ , a ∈ M . In particular, by the faithfulness of ϕ , the map x −→ x ϕ is injective. We notice also that the above map x −→ x ϕ is unique up to natural unitary equivalence. If ϕ is bounded, then ξ ϕ = (1 H ) ϕ is a cyclic and separating vector for M and x ϕ = x ξ ϕ for all x ∈ M = N ϕ . In this case ϕ is the vector form M ∋ x −→ ω ξϕ (x) = (x ξ ϕ | ξ ϕ ) .
Let M be a von Neumann algebra on a Hilbert space H , in standard form with respect to a normal semifinite faithful weight ϕ on M . Then the antilinear operator H ⊃ {x ϕ ; x ∈ A ϕ } ∋ x ϕ −→ (x * ) ϕ has closure S ϕ and the invertible positive selfadjoint operator ∆ ϕ = S * ϕ S ϕ is called the modular operator of ϕ . If S ϕ = J ϕ ∆ 1/2 ϕ is the polar decomposition of S ϕ , then J ϕ is an involutive antiunitary operator (antilinear surjective isometry with J 2 ϕ = 1 H ), called the modular conjugation of ϕ . The operators ∆ ϕ and J ϕ satisfy the commutation relation in particular, If ϕ is bounded and ξ ϕ = (1 H ) ϕ is the associated cyclic and separating vector, then S ϕ ξ ϕ = ξ ϕ , ∆ ϕ ξ ϕ = ξ ϕ , J ϕ ξ ϕ = ξ ϕ .
The fundamental result of the modular theory claims that so that M ∋ x −→ σ ϕ t (x) = ∆ it ϕ x ∆ −it ϕ ∈ M , t ∈ R (2.4) defines an so-continuous one-parameter group of automorphisms (σ ϕ t ) t∈R of M , called the modular automorphism group of ϕ , and so that M ∋ x −→ J ϕ x * J ϕ ∈ M ′ is a * -antiisomorphism. Moreover, the weight ϕ is invariant under the action of the modular automorphism group : ϕ σ ϕ t (a) = ϕ(a) , a ∈ M + , t ∈ R . (2.6) The center Z(M ) of M is contained in the fixed point von Neumann subalgebra {x ∈ M ; σ ϕ t (x) = x , t ∈ R} ⊂ M , which is usually denoted by M ϕ . On the other hand, J ϕ z J ϕ = z * for all z ∈ Z(M ) . We recall also (see the proof of [28], Lemma 5.2 or [45], Corollary 1.2) : x ∈ N ϕ , y ∈ M ϕ =⇒ x y ∈ N ϕ , (x y) ϕ = J ϕ y * J ϕ x ϕ . (2.7) Let e ∈ M ϕ be a projection and let ϕ e denote the restriction of ϕ to eM e . By [28], Proposition 4.1 and Theorem 4.6 (see also [33], Propositions 4.5 and 4.7), ϕ e is a normal semifinite faithful weight and its modular group is the restriction of the modular group of ϕ to eM e . Thus, if π e : eM e −→ B(eH) is the faithful normal * -representation which associates to every x ∈ eM e the restriction x | eH considered as a linear operator eH → eH , then the modular group of the weight ϕ e • π −1 e on π e (eM e) , that is (π e • σ ϕe t • π −1 e ) t∈R , is implemented by the unitary group (∆ it ϕ | eH) t∈R on eH . Nevertheless, π e (eM e) is not always in standard form with respect to ϕ e • π −1 e (indeed, if M ⊂ B(C 4 ) is a type I 2 factor, in standard form with respect to its trace, and e ∈ M is a minimal projection, then π e (M ) is one-dimensional, while its commutant π e (M ) ′ is four-dimensional, so π e (M ) and π e (M ) ′ are not antiisomorphic).
However, for any projection e ∈ M ϕ , π = π eJϕeJϕ : eM e ∋ x −→ x | eJ ϕ eJ ϕ H ∈ B(eJ ϕ eJ ϕ H) is a faithful normal * -representation, such that the von Neumann algebra π(eM e) is in standard form with respect to ϕ e • π −1 (cf. [19], Lemma 2.6). Moreover, ∆ ϕ and J ϕ commute with eJ ϕ eJ ϕ and we have the identifications ∆ ϕe•π −1 = ∆ ϕ | eJ ϕ eJ ϕ H , J ϕe•π −1 = J ϕ | eJ ϕ eJ ϕ H . (2.9) For the convenience of the reader, let us outline the proof of (2.9). For the faithfulness of π , let x ∈ eM e be such that x eJ ϕ eJ ϕ = 0 . Then To see that π(eM e) is in standard form with respect to ψ = ϕ e • π −1 , first we notice that, according to (2.7), N ψ = π(N ϕe ) = e N ϕ e . Next, the linear map = eJ ϕ eJ ϕ x ϕ ∈ eJ ϕ eJ ϕ H has dense range. Indeed, every vector in eJ ϕ eJ ϕ H belongs to the closure of Finally, for every π(x) ∈ π(N ϕe ) = N ψ and π(a) ∈ π(e M e) hold true: The commutation of J ϕ with eJ ϕ eJ ϕ follows immediately from the commutation of e with J ϕ eJ ϕ . Let J e denote the involutive antiunitary operator Further, using (2.2) and e ∈ M ϕ , we obtain for every t ∈ R : Thus also ∆ ϕ commutes with eJ ϕ eJ ϕ , so is an invertible positive selfadjoint operator ∆ e , whose positive selfadjoint square root is we deduce that S ψ ⊂ J e ∆ 1/2 e . For the equality S ψ = J e ∆ 1/2 e , which will imply (2.9), let ξ ∈ Dom (∆ 1/2 the sequence (ex n e) n≥1 belongs to A ϕe and we have (ex n e) ϕ = eJ ϕ eJ ϕ (x n ) ϕ −→ eJ ϕ eJ ϕ ξ = ξ , Now the closedness of the graph of S ψ yields ξ ∈ Dom (S ψ ) . For a projection p ∈ Z(M ) ⊂ M ϕ we have J ϕ pJ ϕ = p , so (2.9) yields Let M = {0} be a von Neumann algebra, in standard form with respect to a normal semifinite faithful weight ϕ on M . Then the Connes spectrum Γ(σ ϕ ) of the modular automorphism group σ ϕ of ϕ is the intersection of the Arveson spectra of all modular automorphism groups σ ϕe , where e ranges over all non-zero projections e ∈ M ϕ . By [13], Lemme 1.2.2 and Théorème 2.2.4 (see also [33], Theorem 3.1 and Proposition 16.3), Γ(σ ϕ ) is a closed additive subgroup of R and it does not depend on the choice of ϕ , so it can be denoted (like in [27], 8.15) by Γ(M ) . Furthermore, by [13], Lemme 3.2.2 (see also [33], Proposition 28.1), λ ∈ Γ(M ) if and only if e λ belongs to the spectrum σ(∆ ϕe ) of ∆ ϕe for all non-zero projections e ∈ M ϕ .
Let M be a von Neumann algebra on a Hilbert space H , in standard form with respect to a normal semifinite faithful weight ϕ on M . Let further N ⊂ M be a von Neumann subalgebra such that the restriction ψ of ϕ to N is semifinite. If {y ϕ ; y ∈ N ψ } is dense in H , then N is in standard form with respect to ψ such that y ψ = y ϕ for all y ∈ N ψ . This happens, for example, if N ⊂ M ⊂ B(H) are von Neumann algebras having a common cyclic and separating vector ξ o , and ϕ is In the above situation, owing to (2.5), we have considered by R. Longo ([24], [25]) and called the canonical endomorphism of the inclusion N ⊂ M . The canonical endomorphism γ , in particular the tunnel plays an important role in the Subfactor Theory (see [26] and [22]).
Let P ↑ + (1) denote the two-dimensional Lie group generated by the hyperbolic rotations and the lightlike translations , Ch. 17, § 2, A), which is the Poincaré group on the light-ray. Furthermore, the commutation relation . Therefore, endowing R 2 with the Lie group structure defined by the composition law the mapping R 2 ∋ (s, t) −→ T s L t ∈ P ↑ + (1) becomes a Lie group isomorphism. In particular, P ↑ + (1) is connected and simply connected. On the other hand, the map . If we identify P ↑ + (1) with G along the above isomorphism, the Lie algebra p ↑ + (1) of P ↑ + (1) will be identified with the Lie algebra g of G , and the exponential map p ↑ + (1) −→ P ↑ + (1) with the exponential map g −→ G , that is with the usual exponentiation of the matrices belonging to g . We notice that g is the set of all 2 × 2 real matrices X such that exp(t X) ∈ G , t ∈ R , and [X, Y ] = XY − Y X for all X , Y ∈ g . The elements of g ≡ p ↑ + (1) are of particular interest: we have any two of X 1 , X 2 , X 3 is a basis for g ≡ p ↑ + (1) and for j = 1 , 2 , 3 , respectively. According to the general theory of unitary representations of Lie groups (see e.g. [2], Ch. 11, § 1, B or [30], Section 10.1), if π is an so-continuous unitary representation of G ≡ P ↑ + (1) on a Hilbert space H and D G (π) denotes the Gårding subspace of H for π , then the formula defines a representation of the Lie algebra p ↑ + (1) into the Lie algebra of all skewsymmetric linear mappings D G (π) → D G (π) . Moreover, for every X ∈ p ↑ + (1) , the linear mapping i d π(X) : H ⊃ D G (π) −→ D G (π) ⊂ H is essentially selfadjoint (see e.g. [2], Ch. 11, § 2, Corollary 4 or [30], Corollary 10.2.11). Therefore, if X ∈ p ↑ + (1) and A is the selfadjoint linear operator in H , then (the first exp is the exponential map of the Lie group P ↑ + (1) , while the second one indicates functional calculus). Indeed, by the definition of d π(X) we have d π(X) ⊂ i A , so the selfadjoint operator −i d π(X) is contained in the selfadjoint operator A , which implies their equality.
We notice for completeness that, according to [16], Theorem 3.3, the Gårding subspace D G (π) is actually equal to the set of all C ∞ -vectors for π .
We notice also the following simple fact concerning the essential selfadjointness of sums of symmetric operators: If H is a Hilbert space, D ⊂ H is a dense linear subspace and A , B : D −→ D are linear operators, then Consequently also A + B is essentially selfadjoint and A + B = A + B .
(2.17) implies that, for any so-continuous unitary representation π of P ↑ + (1) , and assume the following half-sided modular inclusion :

20)
defined on the intersection of the domains of log ∆ N and log ∆ M , is an essentially selfadjoint operator with positive selfadjoint closure P and, letting we have the following : s ∈ R} generates a group of unitary operators on H , which is the image of an so-continuous unitary representation π of P ↑ + (1) on H , uniquely determined by any two of the relations where X 1 , X 2 , X 3 ∈ p ↑ + (1) are the Lie algebra elements defined in (2.13) .
Remark. This theorem is a generalization of Wiesbrock's statement, where ϕ is assumed to be bounded.
The strategy of our proof. The proof will be given in Section 7 in several steps.
(i) First we study ∆ −it N ∆ it M by using the Modular Extension Theorem which will be proved in Section 5.
(ii) Then we show that ∆ −it N ∆ it M has a strong operator limit T for t −→ −∞ . For the existence of the wave operator T we use our generalization of the Borchers Structure Theorem which will be proved in Section 6.
(iii) Using the above ingredients, we define an so-continuous one-parameter family U (s) , s ∈ R , of unitaries.
(iv) We prove that the defined family U (s) , s ∈ R , is a one-parameter group having positive generator P and we verify that for it the statements (1) -(10) in Theorem 2.1 hold.
(v) Using that the generator P of R ∋ s → U (s) satisfies (10), (2.18) will imply that P is the closure of the operator (2.20).
We also prove the following result about the structure of half-sided modular inclusions: Under the assumptions and with the notation as in Theorem 2.1, the following hold : (1) γ s (z) = z for s ≥ 0 and z ∈ Z(M ) , so Z γ s (M ) = Z(M ) for all s ≥ 0 . (3)

Remarks.
(1) is proved in the case of bounded ϕ in [9], Theorem 2.4, and for the general case we shall essentially repeat the same proof. If ϕ is bounded, then the equality (3) follows from [24], Corollary 2.2, but for the proof of (3) in our setting we need a different method. Finally, for bounded ϕ , the inclusion M ϕ ⊂ {x ∈ M ; γ s (x) = x , s ≥ 0} and the type of M (1 H − p) and N (1 H − p) were established in [37] if M is a factor, and in [8], [9] in the case of a general M . However, there is a gap in the proof in [8], [9]: it is shown only that, for every e ∈ M ϕ majorized by 1 H − p , the spectrum of ∆ ϕ | eH is [0, +∞) , while the right proof requires that the spectrum of the modular operator of ϕ e , which is ∆ ϕ | eJ ϕ eJ ϕ H (see (2.9)), be equal to [0, +∞) .
We don't know if the above inclusion still holds without assuming the strong operator density of M ϕ ∩ M ϕ in M ϕ . We notice that, by (3) and (4) in the above theorem, if M ϕ ⊂ s≥0 γ s (M ) would hold in general, then we would always have: (c) The Analytic Extension Theorem.
Let β ∈ R , β = 0 . Set H ∞ (S β ) will denote the Banach algebra of all bounded analytic complex functions on S β .
Most parts of the following Analytic Extension Theorem are known, but we shall give a proof for the convenience of the reader: (1) R ∋ s −→ A is T B −is has a uniformly bounded so-continuous extension There exists a Borel set Ξ o ⊂ R of non-zero Lebesgue measure such that, for every ξ ∈ K and η ∈ H , there exists an f ξ,η ∈ H ∞ (S β ) satisfying for almost all s ∈ Ξ o . (4) A −β T B β is defined and bounded on a core of B β . Moreover, if the above conditions are satisfied, then, for every z ∈ S β , We notice that, for A and B as in Theorem 2.3, R ∋ s −→ α s = A is · B −is is an so-continuous one-parameter group of linear isometries on B(K, H) . If, for some T ∈ B(K, H) and z ∈ C , the orbit which is analytic in S ℑz , then we say that T belongs to the domain of α z and define α z (T ) = T (z) . According to Theorem 2.3, for T ∈ B(K, H) the conditions are equivalent and, if they are satisfied, then α z (T ) = A iz T B −iz . For a more detailed study of the analytic extension operator α z , especially in the most relevant case z = −i , when it is called the analytic generator of the group α , we refer to [12], [43] and [44]. Let us point out that, in general, the five equivalent statements in Theorem 2.3 are not equivalent with (4 ′ ) A −β T B β is densely defined and bounded. Indeed: Remark. Actually the proof of Proposition 2.4, which will be given in Section 3, works to prove the next more general statement: If A and B are invertible positive selfadjoint linear operators on a non-zero Hilbert space, such that , t ∈ R , and further, then, for every s > 0 , Nevertheless, there are situations in which the above statement (4 ′ ) is equivalent with the statements (1) − (5) in Theorem 2.3. One of such situations occurs in [36], Lemma 15.15 and Theorem 15.3, namely: Let M ⊂ B(H) be a von Neumann algebra, in standard form with respect to a normal semifinite faithful weight ϕ on M . Then, for any a ∈ N ϕ , the following are equivalent : Using again Theorem 2.3, we obtain that (c) holds. Finally, the implication (c) ⇒ (a) is trivial.
Let M ⊂ B(H) be a von Neumann algebra, in standard form with respect to a normal semifinite faithful weight ϕ on M .
The next lemma shows that 1 H can be approximated by particularly regular elements of M ϕ with respect to the so-topology: Nets {a ι } ι as in Lemma 2.5 (called in [45], § 1 , regularizing nets for ϕ) will be used to prove the following description of N ϕ : (2) For x ∈ M and ξ ∈ H , Using the above lemma, we get immediately wheref is the inverse Fourier transform of f : ( In the next variant of [35], 10.21, Corollary 1, certain standard properties of the Tomita algebra T ϕ are formulated.
and, for every y ∈ A ϕ , there exists a sequence {y n } n≥1 in T ϕ such that We notice that the set S ϕ of all x ∈ T ϕ , for which with c(x) ≥ 0 a constant depending only on x , is a * -subalgebra of T ϕ and for every y ∈ A ϕ there exists a sequence {y n } n≥1 in S ϕ such that y n so −→ y and y * n so −→ y * , -(y n ) ϕ −→ y ϕ and (y * n ) ϕ −→ (y * ) ϕ in the norm-topology of H (see [35], 10.22).
Let H , K be Hilbert spaces and M ⊂ B(H) , N ⊂ B(K) von Neumann algebras, in standard form with respect to the normal semifinite faithful weights ϕ on M and ψ on N . An essential role will be played by the fixed point real linear subspaces of K and H under S ψ and S ϕ , respectively: They have been used by various authors earlier: see, for example, [29] and [15].
Let us formulate the basic properties, for example, of K S ψ : Statement (3) in the next lemma explains why we call the fulfilment of the implication in Definition 2.10.(2) "implementation of ψ in ϕ by T ".
is an injective * -homomorphism and (4) For bounded ψ and ϕ and the corresponding cyclic and separating vectors The following result provides important criteria for Hermiticity: (1) T is Hermitian with respect to (ψ, ϕ) ; analytic in S −1/2 and satisfying Moreover, if the above equivalent conditions are satisfied, then, with the notation from the Modular Extension Condition (8), we have and T (s) is Hermitian with respect to (ψ, ϕ) for all s ∈ R .
(f) Generalization of the structure theorem of Borchers.
Let M , N , ϕ , ψ be as in the preceding subsection, and T ∈ B(K, H) Hermitian with respect to (ψ, ϕ) . Then, by Theorem 2.12, the orbit R ∋ s −→ ∆ is ϕ T ∆ −is ψ of T has a bounded so-continuous extension T ( · ) to S −1/2 , analytic in S −1/2 , which satisfies the boundary conditions T (s) is Hermitian with respect to (ψ, ϕ) for all s ∈ R , The next extension of a structure theorem of H. J. Borchers ([7], Theorem B, see also [6], Theorem 11.9 and [37], Theorem 2) shows, in particular, that also the converse statement holds, that is any bounded so-continuous map S −1/2 −→ B(K, H) , which is analytic in S −1/2 and satisfies the above boundary conditions, arises from a Hermitian T ∈ B(K, H) as above.
be a bounded map which is analytic in S β and satisfies the boundary conditions Then, for some T ∈ B(K, H) which is Hermitian with respect to (ψ, ϕ) , Hence the given map z → T (z) extends to an so-continuous map on the whole S β and, with the same notation T ( · ) for the extension, it satisfies Remark. Our theorem owes much to Borchers' work, its proof being based on the main idea of the proof of Theorem B in [7]. Nevertheless, our approach has several features of generality: (a) z −→ T (z) is not assumed to be so-continuous on the whole S β , but only the existence of radial limits are assumed almost everywhere on the boundary. In our application to the proof of Theorem 2.1 we shall use Theorem 2.13 with Ξ o = {0} and Ξ 1 = ∅ .
(b) We are considering the case of arbitrary normal semifinite faithful weights ϕ and ψ , without assuming their boundedness.
(c) On the boundary we assume only the Hermiticity of T (s) and J ϕ T (s+ iβ) J ψ rather than the implementation of ψ in ϕ by these operators. The advantage of our assumption consists in its linearity, which allows "mollification", while Borchers' assumption is of quadratic nature, more difficult to handle.
(d) Our proof is made more elementary, avoiding most arguments of the twodimensional complex analysis and using instead of the Malgrange-Zerner Theorem only the elementary Osgood Lemma (the Hartogs Theorem for continuous functions) along with the Morera Theorem (one-dimensional edge-of-the-wedge theorem).
(g) Complements to the implementation theorem of Borchers.
Based on the ideas from [1], an invariant subspace theory was developed in [44] for the "bounded analytic" elements associated to an so-continuous one-parameter group (α t ) t∈R of * -automorphisms of a von Neumann algebra M ⊂ B(H) . This theory allows, starting with an already existent one-parameter group of unitaries on H which implements α , to construct canonically a new implementing group of unitaries on H , which has a minimality property and inherits certain properties of the * -automorphism group α (see [1], Proposition in Section 3, where the idea is formulated in the realm of a particular situation, and [44], Theorem 5.3, Corollary 5.4, Lemma 5.11 for the general theory).
The above method yields a proof for the one-parameter version of the celebrated implementation theorem of Borchers [5], claiming the innerness of α whenever it is implemented by a one-parameter group of unitaries U (s) s∈R having positive generator (see [1], Theorem 3.1 and [44], Corollary 5.7). Moreover, as we shall see in the next theorem, the obtained canonical inner implementing group of unitaries inherits certain commutation properties of the * -automorphism group α .
We recall that, if M is a von Neumann algebra and (α s ) s∈R is an so-continuous one-parameter group of * -automorphisms of M , then the spectral subspace of α corresponding to a closed set F ⊂ R is defined by wheref denotes the inverse Fourier transform (2.28) of f (see [1], Definition 2.1).
Theorem 2.14. Let M ⊂ B(H) be a von Neumann algebra and P a selfadjoint operator in H , such that P is bounded below and Ad exp(isP ) leaves M invariant for all s ∈ R , defining thus an so-continuous one-parameter group (α s ) s∈R of * - where χ (0,e λ ] stands for the characteristic function of (0, e λ ] . Moreover, The above theorem will be used in the proof of Theorem 2.2. (h) Summary of the remaining part of the paper.
The remainder of this paper presents proofs for the above results: -Theorem 2.3 and Proposition 2.4 in Section 3, -Lemma 2.5, Lemma 2.6, Proposition 2.7 and Proposition 2.8 in Section 4, -Lemma 2.9, Lemma 2.11 and Theorem 2.12 in Section 5, -Theorem 2.13 in Section 6, -Theorem 2.1 in Section 7 and, finally, -Theorem 2.14 and Theorem 2.2 in Section 8.

The Analytic Extension Theorem
The aim of this section is to prove Theorem 2.3 and Proposition 2.4.
Proof of Theorem 2.3.
For the proof of the remaining part, we introduce the following notation: let K c (B) and H c (A) be the set of all vectors ξ ∈ K and η ∈ H , respectively, with compact spectral support for log B and log A , respectively. For such ξ and η , C ∋ z −→ B iz ξ and C ∋ z −→ A iz η are analytic functions of exponential type with respect to Im z and they are uniformly bounded in S β . Furthermore, K c (B) ⊂ K and H c (A) ⊂ H are dense linear subspaces and they are cores of B iz and A iz for every z ∈ C , respectively.
Proof of (2) ⇒ (1). The uniform boundedness of S β ∋ z −→ T (z) and its socontinuity are to be proved. The latter is automatic on S β , where T ( · ) is analytic.
Let ξ ∈ K c (B) and η ∈ H c (A) . Then , which is analytic in the interior, coincide on R . By (2.24) and by the density of H c (A) in H , it follows that (5), we have by the Three Line Theorem obtaining thus the uniform boundedness of S β ∋ z −→ T (z) . Due to this uniform boundedness, it suffices to prove the convergences for ξ ∈ K c (B) . We give a proof explicitly only for β > 0 , the treatment of the case β < 0 being completely similar. Let E denote the spectral projection of log A corresponding to (−∞, 0] . Owing to (3.1) we can split T (z) ξ as follows: We note that Proof of (3) ⇒ (4). We proceed in three steps.
Step 1. First we quote some results from the theory of the Hardy spaces on the disc. Let H ∞ (D) be the Banach algebra of all bounded analytic complex functions on the unit disc D = {z ∈ C ; |z| < 1} . Any g ∈ H ∞ (D) has a non-tangential limit for almost all ζ in the boundary ∂ D of D (the unit circle) due to Fatou's Theorem (see, for example, the second Corollary on page 38 of [20] or the theorems on pages 5 and 14 of [21]). Furthermore, the map H ∞ (D) ∋ g −→ g ∈ L ∞ (∂ D) obtained this way is an isometric algebra homomorphism. On the other hand, the range { g ; g ∈ H ∞ (D)} of the above homomorphism is equal to ψ(e is ) e iks ds = 0 for all k = 1 , 2 , . . . } and hence it is weak * closed (see, for example, [14], § 20.1). We notice also that, according to a uniqueness theorem of the Riesz brothers (see, for example, the second Corollary on page 52 of [20] or the Theorem on page 76 of [21]), if for some g ∈ H ∞ (D) the boundary function g vanishes almost everywhere on a Borel subset of ∂ D with non-zero arc length measure, then g = 0 .
We consider the one point compactification of the right half and the left half of S β and denote each added point by +∞ and −∞ , respectively. We extend the function to be +∞ at ζ = +1 and −∞ at ζ = −1 . Then the extended function is a homeomorphism, mapping D onto S β conformally, and the boundary ∂ D onto ∂ S β ∪ {−∞ , +∞} absolutely bicontinuously with respect to the arc length measures: if Ξ is a Borel set in ∂ D , then Ξ has arc length measure 0 if and only if Φ β (Ξ) has arc length measure 0 . Moreover, Φ β maps paths in D tending to a ζ ∈ ∂ D \ {1, −1} from within a sector of opening < π having vertex at ζ , and symmetric about the inner normal to ∂ D in ζ , to paths tending to Φ β (ζ) ∈ ∂ S β in a similar non-tangential way. Therefore, if f ∈ H ∞ (S β ) , the non-tangential limit exists for almost all ζ in ∂ S β by Fatou's Theorem applied to f • Φ β . Similarly we can transcribe the above quoted results concerning H ∞ (D) in the setting of is an isometric algebra homomorphism with weak * closed range in L ∞ (∂ S β ) and f ∈ H ∞ (S β ) is equal to zero whenever f vanishes almost everywhere on a Borel subset of ∂ S β with non-zero arc length measure.
Step 2. We consider the map where, as noticed in Step . The function f ξ,η is uniquely determined by (2.23) due to the uniqueness result quoted in Step 1. Since the right hand side of (2.23) is sesquilinear in ξ and η , the mapping F is also sesquilinear. We shall prove in this step that F is bounded. We first prove that the graph of F is closed. Suppose that ξ n −→ ξ o , η n −→ η o and f ξn,ηn −→ f o with respect to the norm of L ∞ (∂ S β ) , hence also f ξn,ηn −→ f o uniformly. By the continuity of the right hand side of (2.23) in ξ and η , f o has to satisfy (2.23) for ξ = ξ o and η = η o almost everywhere on the set Ξ o . Therefore f o = f ξo,ηo , again by the uniqueness theorem of the Riesz brothers, proving that the graph of F is closed.
Let us consider F (ξ , · ) for a fixed ξ . Denote by H the conjugate of the Hilbert space H and by η the canonical image of η ∈ H in H . By the above proved closedness of the graph of F , the graph of the linear map H ∋ η −→ F (ξ , η) ∈ L ∞ (∂ S β ) is closed and hence, by the Closed Graph Theorem, Step 3. We now take ξ ∈ K c (B) , η ∈ H c (A) . Then is an entire function satisfying the boundary condition (2.23), so that g ξ,η = f ξ,η by the uniqueness theorem of the Riesz brothers. Therefore |g ξ,η (z)| ≤ c ξ η , z ∈ S β and hence the same estimate holds for all z ∈ S β by continuity. This implies because H c (A) is a core of A iz , and Since K c (B) is a core of B −iz , setting z = i β we obtain (4).
Proof of Proposition 2.4.
Let us denote: by λ t the translation operator ξ −→ ξ( · − t) on L 2 (R) , t ∈ R , by α t the * -automorphism Ad (λ t ) of B(H) , t ∈ R , and by b the multiplication operator with e −e π · on L 2 (R) . Clearly, (α t ) t∈R is an so-continuous one-parameter group of * -automorphisms of (3.4) By the Stone Representation Theorem there exists an invertible positive selfadjoint linear operator A on L 2 (R) such that λ t = A it , t ∈ R . Then Let s > 0 be arbitrary. Since 0 ≤ b ≤ 1 L 2 (R) and ℜ (ise −πz ) = s e −πℜz sin (πℑz) , the complex power b ise −πz ∈ B L 2 (R) is defined and b ise −πz ≤ 1 for every z in the closed strip S 1 . Using (3.4), it is easily seen that is an so-continuous extension of R ∋ t −→ α t (b is ) , which is analytic in S 1 and whose value at i is b −is . Taking into account (3.5), Theorem 2.3 yields that (3.7) Indeed, assuming that Dom (A −1 b −is A) is a core of A , (3.6) and Theorem 2.3 imply that R ∋ t −→ α t (b −is ) has a uniformly bounded so-continuous extension which is analytic in S 1 and whose value at i is b is . Then which is analytic in the interior and which takes the same value b −is at i and −i . Then, by (2.26), F is periodic of period 2 i , so it extends to a uniformly bounded entire mapping, which must be constant by the Liouville Theorem. Thus the orbit Since b is is the multiplication operator with R ∋ r −→ e −ise πr on L 2 (R) , this means that the above function is constant, what is plainly not true. By (3.6) and by (3.7) we conclude that, choosing v = b −is with s > 0 , A −1 vA is densely defined and bounded, but Dom (A −1 vA) is not a core for A .

Lebesgue Continuity, Tomita Algebras
In this section we prove Lemmas 2.5 and 2.6, as well as Propositions 2.7 and 2.8. Throughout this section M ⊂ B(H) will stand for a von Neumann algebra, in standard form with respect to a normal semifinite faithful weight ϕ on M .
Proof of Lemma 2.5.
Since M ϕ is a hereditary * -subalgebra of M , there is an increasing approximate unit {b ι } ι for M ϕ (for example, the upward directed set {b ∈ M ϕ ∩ M + ; b < 1} , labeled by itself). Then, by the so-density of M ϕ in M , we have so − lim has an entire extension Clearly, σ ϕ z (a ι ) * = σ ϕ z (a ι ) for all ι and z ∈ C . Since, for every z ∈ C , the function with 0 ≤ f j ∈ L 1 (R) , 1 ≤ j ≤ 4 , using (2.6) we deduce easily that σ ϕ z (a ι ) ∈ M ϕ and σ ϕ z (a ι ) ≤ e (ℑz) 2 for all ι and z ∈ C . On the other hand, so − lim Proof of Lemma 2.6.
First we prove that For let x ∈ A ϕ be arbitrary. Then (2.5) yields On the other hand, since Now, (4.2) and (4.3) imply (1) If x ∈ N ϕ , then by (2.5) Conversely, assume that x ∈ M and c ≥ 0 are such that Let {a ι } ι be a net as in Lemma 2.5. Then we have for every ι and, according to (4.1) and (4.4), −→ x * x in the so-topology and ϕ is lower wo-semi-continuous on the bounded subsets of M + , it follows that ϕ(x * x) ≤ c 2 , that is x ∈ N ϕ and x ϕ ≤ c .
(2) Since the implication "=⇒" is an immediate consequence of (2.5), we have to prove only the converse implication. For let x ∈ M and ξ ∈ H be such that x J ϕ y ϕ = J ϕ y J ϕ ξ , y ∈ M ϕ . (4.5) Then (4.4) holds with c = ξ , so by the above part of the proof we have x ∈ N ϕ . But then (2.5) and (4.5) yield J ϕ y J ϕ x ϕ = x J ϕ y ϕ = J ϕ y J ϕ ξ , so by the so-density of M ϕ in M ∋ 1 H we conclude that x ϕ = ξ .
Proof of Proposition 2.7.
(1) Let D be the linear span of {J ϕ a * J ϕ b ϕ ; a , b ∈ N ϕ } , which is dense in H . Define the linear functional F : D −→ C by where the limit exists due to the convergence Since F is bounded by it extends to a continuous linear functional on H , and hence there exists ξ ∈ H satisfying F (η) = (η | ξ) for all η ∈ D . In particular, by (4.6), This implies that x J ϕ a ϕ = J ϕ a J ϕ ξ for all a ∈ N ϕ and by Lemma 2.6.(2) we get x ∈ N ϕ and x ϕ = ξ . Furthermore, the density of D in H and the boundedness of the net {(x ι ) ϕ } ι yield that (x ι ) ϕ ι −→ x ϕ in the weak topology of H .
(2) Let x ∈ M be any wo-limit point of the bounded net {x ι } ι . Then, for every a ∈ N ϕ , x J ϕ a ϕ is a weak limit point of the net {x ι J ϕ a ϕ } ι a ∈ N ϕ and, using Lemma 2.6. (2), we obtain x ∈ N ϕ and x ϕ = ξ .
By the injectivity of the mapping N ϕ ∋ y −→ y ϕ , the uniqueness of the wo-limit point x of {x ι } ι follows and we conclude that wo − lim Proof of Proposition 2.8.
First we show that every y ∈ A ϕ can be approximated by a sequence {y n } n≥1 in T ϕ as required in the statement and such that (2.29) holds for x = y n , n ≥ 1 . Set Then every orbit has the entire extension and by (2.27) we have σ ϕ z (y n ) ∈ N ϕ for every z ∈ C . Similarly, has an entire extension C ∋ z −→ σ ϕ z (y * n ) and σ ϕ z (y * n ) ∈ N ϕ for every z ∈ C . Since σ ϕ z (y n ) * = σ ϕ z (y * n ) , we have that is y n ∈ T ϕ for all n ≥ 1 . By the so-continuity of R ∋ t −→ σ ϕ t (y) ∈ M and R ∋ t −→ σ ϕ t (y * ) ∈ M we get y n so −→ y and y * n so −→ y * , while using e −n(t−z) 2 = e −n(t−ℜz) 2 e n(ℑz) 2 e 2 n i(t−ℜz) ℑz = e −n(t−ℜz) 2 e n(ℑz) 2 (4.8) it is easily seen that σ ϕ z (y n ) ≤ e n(ℑz) 2 y for all n ≥ 1 and z ∈ C . On the other hand, by (2.27) we have and by the norm-continuity of R ∋ t −→ ∆ it ϕ y ϕ ∈ H and R ∋ t −→ ∆ it ϕ (y * ) ϕ ∈ H we get the convergences (y n ) ϕ −→ y ϕ and (y * n ) ϕ −→ (y * ) ϕ in the norm-topology. Furthermore, every orbit has the entire extension and thus (see [28], Lemma 3.2 and [12], Theorem 6.1) (y n ) ϕ ∈ z∈C Dom ∆ iz ϕ and Moreover, using (4.8), we get for every n ≥ 1 and z ∈ C ∆ iz ϕ (y n ) ϕ ≤ e n(ℑz) 2 y ϕ , ∆ iz ϕ (y * n ) ϕ ≤ e n(ℑz) 2 (y * ) ϕ . Finally, for every n ≥ 1 , (4.7), (2.27) and (4.9) yield hence (2.29) holds for x = y n . It remains to prove that (2.29) holds in full generality. First we show that By Lemma 2.6 this is equivalent to the implication what we now are going to prove. Choose a sequence {y n } n≥1 in T ϕ as in the above part of the proof. For each n ≥ 1 , (y n ) ϕ ∈ Dom (∆ −i z ϕ ) implies by (2.1) that so, according to Theorem 2.3, Since, by (4.10) and by (2.5), (4.13) Taking into account that x ϕ ∈ Dom (∆ iz ϕ ) and, by Theorem 2.3 and by (2.1), 13) implies the equality σ ϕ z (x) J ϕ (y n ) ϕ = J ϕ y n J ϕ ∆ iz ϕ x ϕ . Passing now to the limit for n → ∞ , we conclude that holds for every integer k ≥ 1 . Indeed, ϕ is clear and if (4.14) holds for some k ≥ 1 and x ∈ T ϕ , then we have by (4.11) On the other hand, for every x ∈ T ϕ and k ≥ 1 we have σ ϕ k 2 i (x) ∈ T ϕ , so (4.14) yields σ ϕ k 2 i (x) ϕ ∈ Dom ∆ k 2 ϕ and, using (4.11), we deduce Therefore (4.14) holds also for every integer k ≤ −1 that is This last inclusion together with (4.11) imply (2.29).

Hermitian Maps
In this section we analyse the notion introduced in Definition 2.10 by proving Lemma 2.9, Lemma 2.11 and Theorem 2.12.
Proof of Lemma 2.9.
Let ξ ∈ Dom S ψ be arbitrary. Then there is a sequence (x n ) n≥1 in A ψ such that
For (2) and (3) we first notice that, for every ξ ∈ K S ψ and x ∈ A ψ , Conversely, let us assume that ξ ∈ K is such that hence, by our assumption on ξ , Proof of Lemma 2.11.
If T ∈ B(K, H) implements ψ in ϕ and x * = x ∈ N ψ , then T x ψ = (T x T * ) ϕ with (T x T * ) * = T x T * ∈ N ϕ and hence Lemma 2.9.(1) implies T K S ψ ⊂ H Sϕ , proving (1). In this case the inclusion T N ψ T * ⊂ N ϕ and the wo-density of N ψ in N imply T N T * ⊂ M , proving (2).
If T is isometric in addition, then shows the injectivity of the map N ∋ x −→ T x T * ∈ M , which is clearly also a * -homomorphism. Furthermore, 0 ≤ a ∈ M ψ implies a 1/2 ∈ N ψ and Therefore (3) holds. Let us finally assume that ψ and ϕ are bounded and ξ ψ = (1 K ) ψ , η ϕ = (1 H ) ϕ . If T ∈ B(K, H) is injective and implements ψ in ϕ , then T N T * ⊂ M by the above proved (2) and H) is injective and satisfies T N T * ⊂ M and T * η ϕ = ξ ψ , then for x ∈ N (T x T * ) ϕ = T x T * η ϕ = T x ξ ψ = T x ψ .

This equation, in turn, is equivalent to the validity of
We have already seen that (1) ⇔ (3). Applying this equivalence to J ψ T * J ϕ , it follows that J ψ T * J ϕ is Hermitian with respect to (ϕ, ψ) if and only if is real for all x * = x ∈ N ψ and y * = y ∈ N ϕ . But this means exactly (3), so (3) ⇔ (7).
By the equivalence of statements (1) and (5) we get also (2.31) by the Three Line Theorem. Finally, since K S ψ and H Sϕ are invariant under ∆ −is ψ and ∆ is ϕ , respectively, for every s ∈ R , due to (2.3) and Lemma 2.9.(1), we obtain the Hermiticity of T (s) = ∆ is ϕ T ∆ −is ψ from (1).

Generalization of the Structure Theorem of Borchers
We prove Theorem 2.13 in two steps: first we prove it for the case where Ξ o and Ξ 1 are empty, and then we reduce the proof of the general case to the above special case.

By our assumptions in this step,
is a bounded wo-continuous map which is analytic in S β and satisfies the boundary conditions (i) T (s) is Hermitian with respect to (ψ, ϕ) for all s ∈ R , (ii) J ϕ T (s + iβ) J ψ is Hermitian with respect to (ψ, ϕ) for all s ∈ R . Let x ∈ T ψ and y ∈ T ϕ be arbitrary (for the Tomita algebras T ψ and T ϕ see the comments before Proposition 2.8) such that for some constant c ≥ 0 . Consider the functions They are continuous and, according to (6.1), bounded on any set of the form Furthermore, the partial functions are analytic. Now, by (2.29) and Theorem 2.12.(4), (i) implies, for every z 1 ∈ C and s ∈ R , Therefore is a well defined continuous function, bounded on every set of the form For each fixed z 1 ∈ C , the function S β ∪ S −β ∋ z 2 −→ f (z 1 , z 2 ) is analytic. Hence, by the Morera Theorem (the one-dimensional edge-of-the-wedge theorem, see for example [3], 2.1.9. (2) or [11], II.2.7), it can be analytically extended across R , that is the partial functions are analytic. Thus we can apply to f the Osgood Lemma (the Hartogs Theorem for continuous functions, see for example [18], Theorem I.A.2) and deduce that it is analytic, as function of two complex variables, on C × z 2 ∈ C ; |ℑz 2 | < |β| .
For every z 1 ∈ C and s ∈ R , (ii) implies by (2.29) and Theorem 2.12.(4), Therefore, for each s ∈ R , the bounded, continuous function which is analytic in the interior, satisfies By the Morera Theorem, g s extends to a periodic entire function with period i , still denoted by g s , which is bounded. By the Liouville Theorem it follows that g s is constant, hence we get successively By the density property of T ϕ stated in Proposition 2.8, the above equalities imply that hence (2.36) holds with T = T (0) . From (2.32) and (2.33) in Theorem 2.12, we obtain (2.37) and (2.38).
Step 2. Proof in the general case.
Consequently, the mapping S β ∋ z −→ T n (z) defined in (6.2) is a restriction of the entire mapping T ζ,n ( · ) . In particular, it is so-continuous (the role of T ζ,n ( · ) is just to prove this statement) and its restriction to S β is analytic. We recall that its boundedness was already noticed in (6.3).
Since R ∋ t −→ f n (t) is a real function, (1) ⇔ (3) in Theorem 2.12 implies that the Hermiticity of T (s) and J ϕ T (s + iβ) J ψ for s ∈ R \ Ξ o respectively s ∈ R \ Ξ 1 is inherited by T n (s) and J ϕ T n (s + iβ) J ψ for all s ∈ R .
Thus T n ( · ) fulfils all the assumptions made in Step 1. Consequently there exists T n ∈ B(K, H) satisfying It follows that which yields by analytic extension On the other hand, from (6.2) we obtain due to the boundedness and norm-continuity of T ( · ) in S β . Thus we get by (6.8) Now choose some s o ∈ R \ Ξ o and denote Then T is Hermitian with respect to (ψ, ϕ) and we have for every that is (2.36) holds. Using (1) ⇒ (8) in Theorem 2.12, (2.32) and (2.33), as well as the uniqueness theorem of the Riesz brothers (see, in Section 3, Step 1 of the proof of (3) ⇒ (4) in Theorem 2.3), we obtain that T ( · ) extends to an so-continuous map on S β , for which (2.37) and (2.38) hold.

Proof of Theorem 2.1
We recall the setting: -N ⊂ M ⊂ B(H) are von Neumann algebras, ϕ is a normal semifinite faithful weight on M such that its restriction ψ to N is semifinite, -we assume that M is in standard form with respect to ϕ and {y ϕ ; y ∈ N ψ } is dense in H , hence N is in standard form with respect to ψ and y ψ = y ϕ for all y ∈ N ψ . We shall use the notations ∆ M = ∆ ϕ , J M = J ϕ and ∆ N = ∆ ψ , J N = J ψ .
The proof of Theorem 2.1 will be performed in nine steps.
Step 1. Application of the Modular Extension Theorem.
Since S ψ ⊂ S ϕ , hence H S ψ ⊂ H Sϕ , the identity map I on H is Hermitian with respect to (ψ, ϕ) . By (1) ⇔ (8) in Theorem 2.12, (2.32) and (2.33), we obtain an so-continuous map which is analytic in S −1/2 and satisfies the conditions is so-continuous, analytic in S −1/2 , and such that Step 2. Hermiticity on the boundary.
Step 3. Change of variable.
We shall use the analytic logarithm branches log + : r e iθ ; r > 0 , − π 2 < θ < 3π 2 ∋ r e iθ −→ log r + i θ , For any β ∈ R , β = 0 , we consider (like in Section 3, in the proof of (3) ⇒ (4) in Theorem 2.3) the one point compactification of the right half and the left half of S β and denote each added point by +∞ and −∞ , respectively.
We notice an additional continuity property of V ( · ) : since Ψ −1 (−∞) = 0 , the limit so − lim exists. Thus the mapping (7.6) has actually an so-continuous extension Step 5. Further change of variable.
Step 8. Invariance properties of U ( · ) with respect to ϕ .
We show in the following that R ∋ s −→ U (s) ∈ B(H) satisfies property (8) in Theorem 2.1, hence also property (9), which is an immediate consequence of (8).
For any y ∈ N ψ and t ∈ R , (2.3) yields Setting s = 1 − e 2πt and using (7.21), we obtain Therefore we have for all s ∈ R , s < 1 , U (s) * y U (s) ∈ N ϕ and U (s) * y U (s) ϕ = U (s) * y ψ . (7.32) Moreover, according to the Lebesgue continuity result Proposition 2.7, (7.32) holds also for s = 1 . According to (7.31), The modular automorphism groups of the normal semifinite faithful weights ψ and ϕ • γ −1 1 are equal. Indeed, by (7.14) we have for every y ∈ N and t ∈ R : . On the other hand, for every y ∈ N ψ , using (7.32) with s = 1 , we obtain and ψ coincide on M ψ . Thus [28], Proposition 5.9 yields ψ = ϕ In particular, for every x ∈ M and n ≥ 0 , Let x ∈ N ϕ be arbitrary. By (7.34) and (7.31) we have γ 1 (x) ∈ N ψ , so (7.32) holds with y = γ 1 (x) and any 0 ≤ s ≤ 1 . Using the group property of U ( · ) , we deduce that, for every 0 ≤ s ≤ 1 , In particular, for s = 0 we get On the other hand, for x ∈ M and s ≥ 0 , denoting by n the integer part of s , that is the integer n ≥ 0 with n ≤ s < n + 1 , we have Consequently property (8) in Theorem 2.1 is satisfied.
Step 9. Description of the generator P .
First we verify that statement (10) in Theorem 2.1 holds with P defined by (7.26).
We recall from Subsection (b) of Section 2 that, if we endow R 2 with the Lie group structure defined by the composition law (s 1 , t 1 ) · (s 2 , t 2 ) = (s 1 + e −2πt1 s 2 , t 1 + t 2 ) , then R 2 ∋ (s, t) −→ T s L t ∈ P ↑ + (1) is a Lie group isomorphism. Hence, by (7.27), is an so-continuous unitary representation on H and, according to (7.21), the group π P ↑ + (1) contains {∆ it M , ∆ is N ; t , s ∈ R} and is generated by this set.
Let us consider the elements X 1 , X 2 , X 3 of the Lie algebra p ↑ + (1) ≡ g defined in (2.13). By (2.15) and by the definition of π , taking into account how P ↑ + (1) was identified in Subsection (b) of Section 2 with G , we obtain for every t ∈ R : Therefore, according to (2.16), Since any two of X 1 , X 2 , X 3 is a basis for g ↑ + (1) and P ↑ + (1) is connected and simply connected, the representation π is uniquely determined by any two of the relations (7.38) (see e.g. [2], Ch. 11, § 5 or [30], Proposition 10.5.2). Now, by (2.14), (2.18) and (7.38), we conclude that 8. Complements to the implementation theorem of Borchers and the proof of Theorem 2.2 First we prove Theorem 2.14, which will then be used to prove Theorem 2.2.
Proof of Theorem 2.14.
Step 1. The existence and the uniqueness of b (it is essentially the proof of [1], Theorem 3.1 and [44], Corollary 5.7).
By the lower boundedness of P we have d o = exp(−P ) ∈ B(H) . Moreover, d o is clearly positive and injective. Denoting β s = α −s , (β s ) s∈R is an so-continuous group of * -automorphisms of M such that We recall that, according to [44], Theorem 1.4, we have for every λ ∈ R : Dom (β z ) ; β z (x) ≤ e −λℑz x for all z ∈ C , ℑz ≤ 0 .
(iii) is clear from the construction of b in Step 1.
In order to verify (iv), let σ be a * -automorphism of M such that, for some Then holds clearly There exists a faithful unital normal * -representation π : M −→ B(K) , which is covariant with respect to σ , that is π σ(x) = U π(x) U * , x ∈ M , where U is an appropriate unitary on K : for example, we can choose K = l 2 (Z; H) , the space of all square-summable two-sided sequences in H , π(x) (ξ k ) k∈Z = σ k (x) ξ k k∈Z for x ∈ M , (ξ k ) k∈Z ∈ l 2 (Z; H) , U (ξ k ) k∈Z = (ξ k+1 ) k∈Z for (ξ k ) k∈Z ∈ l 2 (Z; H) .
Proof of (1). Let us consider γ s = Ad U (s) for all s ∈ R (not only for s ≥ 0) and let z be an arbitrary selfadjoint element of the center Z(M ) of M . By (6) in Theorem 2.1 we have γ s (z) ∈ M for all s ≥ 0 , hence z and γ s (z) commute for any s ≥ 0 . But then the elements of the set {γ s (z) ; s ∈ R} are mutually commuting, so the von Neumann algebra C generated by this set is commutative. Since γ s = Ad U (s) | M leaves C invariant for every s ∈ R and U (s) = exp(isP ) , s ∈ R , for some positive selfadjoint operator P in H , according to the implementation theorem of Borchers [5] (see also Theorem 2.14) there exists an element b ∈ C , 0 ≤ b ≤ 1 H , such that Consequently, γ s (z) = z for all s ∈ R .
Proof of (2). By (5)  Since π is faithful, we obtain that U (s) a U (−s) = γ s (a) = a for all s ≥ 0 and all a ∈ eM e . In other words, every U (s) commutes with every operator in eM e . Consequently, for every s ∈ R , the unitary U (s) | eH : eH → eH belongs to the commutant of the reduced von Neumann algebra x | eH : eH → eH ; x ∈ eM e , hence to the induced von Neumann algebra x ′ | eH : eH → eH ; x ′ ∈ M ′ . Since the kernel of the induction * -homomorphism Ad U (s) leaves M −∞ invariant for every s ∈ R , defining thus an so-continuous one-parameter group (α s ) s∈R of * -automorphisms of M −∞ . Using Theorem 2.14, we get an injective b ∈ M −∞ , 0 ≤ b ≤ 1 H , such that