Quasifree stochastic cocycles and quantum random walks

The theory of quasifree quantum stochastic calculus for infinite-dimensional noise is developed within the framework of Hudson-Parthasarathy quantum stochastic calculus. The question of uniqueness for the covariance amplitude with respect to which a given unitary quantum stochastic cocycle is quasifree is addressed, and related to the minimality of the corresponding stochastic dilation. The theory is applied to the identification of a wide class of quantum random walks whose limit processes are driven by quasifree noises.


Introduction
Quantum stochastic calculi for gauge-invariant quasifree representations of the canonical commutation and anticommutation relations were originally developed in the 1980s; see [BSW], [HL 1,2 ] and [L 1 ]. The possibilities afforded for semigroup dilation via such calculi were further developed in [App] and [LiM], with the latter treatment using a theory of integral-sum kernel operators. Recently, quasifree stochastic calculus has been extended to the cases of squeezed states and infinite-dimensional noise [LM 1,2 ]. A key ingredient of the latter theory is a partial transpose defined on a class of unbounded operators affiliated to the noise algebra, which defies the failure of complete boundedness for the transpose.
Use of quasifree stochastic calculus may be preferred to the standard theory founded by Hudson and Parthasarathy [HuP, Par] for both physical and mathematical reasons [HL 2 ]. On the one hand, it describes systems which are more physically realistic, at non-zero temperatures for example. On the other hand, the quasifree theory boasts a fully satisfactory martingale representation theorem [HL 1 , LM 1 ], in contrast to the standard theory, whose representation theorem is restricted by regularity assumptions which seem hard to overcome [PS 1 ].
The purpose of this article is twofold. The first is to develop quasifree stochastic calculus in a simplified form within the standard theory, restricting to quasifree states with bounded covariance amplitudes, bounded operator-valued processes and unitary quantum stochastic cocycles with bounded vacuum-expectation semigroups. The second is to give a deeper explanation for an observation of Attal and Joye, who described how the quantum Langevin equation, obtained as limit of a repeated-interactions model with particles in a thermal state, is driven by noises satisfying quasifree Itô product relations [AtJ]. Those parts relating to the first objective are written so as to facilitate the second.
The quasifree CCR representations that we employ are of Araki-Woods type, determined by two maps: the doubling map where k is the Hilbert space conjugate to the quasifree noise-dimension space k, and an operator for which the real-linear map Σ • ι is symplectic. The corresponding Weyl operators W Σ (f ) act on the double Boson Fock space in the following manner: where W (g) denotes the Fock-Weyl operator with test function g, and the operators ι and Σ are extended to act on functions pointwise; for example, (βf )(t) := βf (t) for all t ∈ R + . The symplectic hypothesis ensures that W Σ defines a CCR representation. This class of representations is sufficiently general to include a range of interesting examples, while being concrete enough to render the resulting stochastic calculus straightforward to employ with a minimum of technicalities. Details of this representation theory are given in Section 2.
Section 3 collects the relevant results from standard quantum stochastic analysis, chosen in light of the requirements for the passage to quasifree stochastic calculus in Section 4. We motivate the definition of quasifree stochastic integrals by combining the Itô-type quantum stochastic integration of simple processes with the realisation of quasifree creation and annihilation operators in terms of creation and annihilation operators for the Fock representation. It is notable that quasifree stochastic integrability is unaffected by squeezing the state; indeed, the resulting transformation of quasifree integrands may be viewed as a change-of-variables formula for quasifree stochastic calculus (Theorem 4.4). Our approach demonstrates the central rôle in the theory played by a partial conjugation, which constrains the class of admissible integrands when the noise is infinite dimensional. This corresponds to the partial-transpose operation at the heart of the coordinate-free quasifree stochastic calculus [LM 1,2 ]. Viewing quasifree integrals as particular cases of standard quantum stochastic integrals allows us to employ the existing modern quantum stochastic theory [L 2 ] and, in particular, to avoid any application of Tomita-Takesaki theory. While maintaining strict mathematical rigour, the simplicity of our approach makes it very suitable for applications.
Various uniqueness questions are addressed in Section 5. We first show that the change-ofvariables effect of squeezing on quasifree integrals means that, for present purposes, we may restrict to gauge-invariant quasifree states. Then the stochastic generators of quasifree Hudson-Parthasarathy cocycles on an initial Hilbert space h are parameterised by triples of operators (A, H, Q), where A ∈ B(k) is non-negative, H ∈ B(h) is self adjoint, and Q ∈ B(h; k ⊗ h) is k-conjugatable; see Definition 4.2. The set of triples that generating the same cocycle is parameterised by a class of self-adjoint operators in B(k). Uniqueness for quasifree Hudson-Parthasarathy cocycles inducing a given inner Evans-Hudson flow j (Definition 3.17) is related to the minimality of j, as a stochastic dilation of its vacuum-expectation semigroup, in the sense of [Bha].
The final section, Section 6, concerns quantum random walks and the repeated-interactions model [AtP]. After a brief summary of the relevant results from the standard theory [Be 1 , BGL], we extend the example of Attal and Joye in two directions: to allow infinite-dimensional noise, and to incorporate an enlarged class of interaction Hamiltonians. We show that their example is part of a more general phenomenon: if the particles in the repeated-interactions model are in a faithful normal state whose density matrix ̺ enjoys exponential decay of its eigenvalues, and the interaction Hamiltonian is p-conjugatable and has no diagonal part with respect to the eigenspaces of ̺, then the quantum Langevin equation which governs the limit cocycle U is driven by gauge-invariant quasifree noise. We also give sufficient further conditions on the matrix components of the interaction Hamiltonian for the quasifree noise to be the unique one within the class for which U is quasifree (Theorem 6.8). The GNS space given by the particle state splits naturally into mutually conjugate upper-triangular and lower-triangular parts; this splitting may be viewed as being the origin of the double Fock space arising in the relevant CCR representation.
We expect the results below to be of interest to researchers in quantum optics and related fields; the importance of quantum stochastic calculus to quantum control engineering, for example, is clearly demonstrated in many of the papers contained in [Gou]. In future work, we intend to explore quantum control theory within this quasifree framework. For initial results on quasifree filtering, which show the potential benefit of using squeezed fields for state restoration, see [Bou].
Notation and conventions. Throughout, the symbol h, sometimes adorned with primes or subscripts, stands for a generic Hilbert space; with this understanding, we usually refrain from saying "let h and h ′ be Hilbert spaces, et cetera". All Hilbert spaces considered are complex and separable, with inner products linear in their second argument. A conjugate Hilbert space of h is a pair (h, K) consisting of a self-adjoint anti-unitary operator K from h to a Hilbert space h; this is unique up to isomorphism in the natural sense. For any x ∈ h and A ∈ B(h), the vector Kx ∈ h and the operator KAK −1 ∈ B(h) are abbreviated to x and A respectively. The closed linear span of a subset S of a Hilbert space is denoted Lin S; the range of a bounded operator T and its closure are denoted Ran T and Ran T respectively. The domain of an unbounded operator T is denoted Dom T . We employ the Dirac-inspired bra and ket notation for any vector x ∈ h.
Algebraic, Hilbert-space and ultraweak tensor products are denoted ⊗ , ⊗ and ⊗ , respectively. The indicator function of a set S is denoted 1 S . The group of complex numbers with unit modulus is denoted T. The integer part of a real number r is denoted ⌊r⌋.

CCR representations
In this section, we collect some key facts on CCR representations and quasifree states. In particular, we introduce the squeezing matrices and AW amplitudes that determine the class of quasifree states that are relevant to us.
Recall that every real-linear operator T : h → h ′ is uniquely decomposable as L + A, where L is complex linear and A is conjugate linear; L and A are referred to as the linear and conjugatelinear parts of T . Explicitly, Im Zx, Zy = Im x, y for all x, y ∈ h.
We denote the space of symplectic operators from h to h ′ by S(h; h ′ ), or S(h) when h ′ = h, and the group of symplectic automorphisms of h by S(h) × .
For T ∈ B(h; h ′ ), it is easily verified that T is isometric if and only if it is symplectic and complex linear. In particular, U (h) is the subgroup of S(h) × consisting of its complex-linear elements.
It is shown in the appendix that symplectic automorphisms of h are automatically bounded. Thus S(h) × is a subgroup of the group of bounded invertible real-linear operators on h.
A parameterisation B = B U,C,P for the elements of S(h) × is also given in the appendix.
For the rest of this section, we fix a Hilbert space H and let (H, K) be its conjugate Hilbert space.
Fock space. As emphasised by Segal [Seg], the Boson Fock space over H has two interpretations, particle and wave: Here H ∨n denotes the nth symmetric tensor power of H, with H ∨0 := C, and ε(x) is the exponential vector corresponding to the test vector x: The normalised exponential vector exp(− 1 2 x 2 )ε(x) is denoted ̟(x), and the distinguished vector ε(0) = ̟(0) is denoted Ω H and called the Fock vacuum vector. For all x, y ∈ H, ε(x), ε(y) = exp x, y , and the map λ → ε(x + λy) is holomorphic from C to Γ + (H). As well as being total in Γ + (H), the exponential vectors are linearly independent.
For any x ∈ H, the Fock-Weyl operator W H (x) is the unique unitary operator on Γ + (H) such that W H (x)̟(y) = exp(−i Im x, y )̟(x + y) for all y ∈ H. (2.2) For all x, y ∈ H, and CCR representations. We let CCR(H) denote the C * -algebra generated by the set unitary elements {w x : x ∈ H} which satisfy the canonical commutation relations in Weyl form: Its existence, uniqueness and simplicity were established in [Sla], and these imply that, for any operator B ∈ S(H) × , there is a unique automorphism α B of CCR(H) such that see [BrR, Pet].
). They are closed and mutually adjoint operators with common domain Dom R H (ix)∩Dom R H (x), on which the following canonical commutation relations hold [BrR]: For any dense subspace D of H, the subspace Lin{ε(z) : z ∈ D} is a common core for all Fock creation and annihilation operators, on which their actions are as follows: for all x, z ∈ H.
for all x ∈ H; (2.5) see [BrR, Pet]. Being non-negative, the form a polarises to a symmetric bilinear form [Kur]; in other words, the following map is real linear in each argument: . In particular, the following regularity property holds: for all x, y ∈ H, the map t → a[x + ty] is continuous on R. If dim H < ∞ then a is bounded and therefore there exists a bounded non-negative real-linear operator T on H such that a[x] = Re x, T x for all x ∈ H.
Definition 2.2. A state ϕ on CCR(H) is said to be quasifree if it satisfies (2.5) for some non-negative real quadratic form a satisfying (2.4); then a is called the covariance of ϕ, and any real-linear operator Z : Remark. Covariances of gauge-invariant quasifree states on CCR(H) are precisely the complex quadratic forms a on H such that x 2 for all x ∈ H.
(2.6) Example 2.3. The Fock vacuum state ϕ H on CCR(H), given by the identity is the basic example of a gauge-invariant quasifree state, in view of (2.6) and the identity (2.3c).
Lemma 2.4. Let Z ∈ S(H; h). Then Z is a covariance amplitude for a quasifree state ϕ on CCR(H). Moreover, if Z is complex linear then ϕ is gauge invariant.
Proof. The first part follows since The second part is immediate.
Remark. Proposition 2.6 below shows that a covariance amplitude of a quasifree state need not be complex linear for the state to be gauge invariant.
Definition 2.5. The doubling map for H is the following bounded real-linear operator: Note that the range of the doubling map is total, since for all x, z ∈ H.
Conversely, let ϕ be a gauge-invariant quasifree state on CCR(H), the covariance of which is a bounded complex quadratic form on CCR(H). Then ϕ has a covariance amplitude of the form Σ • ι for a unique operator Σ ∈ AW 0 (H).
It follows that Σ • ι is symplectic, and is therefore a covariance amplitude of a quasifree state ϕ on CCR(H). The resulting covariance a Σ : and is thereby manifestly gauge invariant.
Conversely, let a be the covariance of a gauge-invariant quasifree state on CCR(H) and suppose that a is bounded. Since a is bounded and such that a[x] x 2 for all x ∈ H, there is a unique operator R ∈ B(H) such that x, Rx = a[x] for all x ∈ H, and R I H . The map A → cosh 2A is a bijection from B(H) + onto {R ∈ B(H) + : R I H }, and therefore, by the identity (2.8), it follows that a = a Σ for a unique operator Σ = Σ A ∈ AW 0 (H).
We now introduce the notion of squeezing, important in quantum optics. For any B ∈ S(H) × , set where L and A are the linear and conjugate-linear parts of B.
Proof. (a) Let L and A be the linear and conjugate-linear parts of an operator B ∈ S(H) × . The block-matrix components of the bounded operator M B are complex linear, so M B ∈ B(H ⊕ H), and Uniqueness follows from the totality of Ran ι.
(b) By definition, the operator M I H = I H⊕H , and since , 2, and let L and A be the linear and conjugate-linear parts of B. Then As C 2 and K are invertible, this implies that A = 0, so B is complex linear and thus unitary, and C 1 = C 2 B. This implies that C 2 1 = C 2 BB * C 2 = C 2 2 , so C 1 = C 2 and C 1 = C 1 B. As C 1 is invertible, it follows that B = I H and (c) holds. We refer to the elements of M (H), AW (H) and AW 0 (H) respectively as squeezing matrices, AW amplitudes and gauge-invariant AW amplitudes for H.
Remarks. (i) The AW abbreviation is in acknowledgement of Araki and Woods [ArW].
(ii) Each AW amplitude for H is of the form Σ A,B for a unique pair (A, B) ∈ B(H) + × S(H) × , by Proposition 2.7.
(iii) Let Σ = Σ A,B ∈ AW (H). Then Σ • ι is symplectic, since it is the composition of symplectic maps (Σ A •ι)•B, and so is a covariance amplitude of a quasifree state on CCR(H), by Lemma 2.4.
(iv) In terms of the parameterisation B = B U,C,P := U (cosh P − C sinh P ) of B ∈ S(H) × as in Theorem A.2, the squeezing matrices (2.10) Araki-Woods representations. We are interested in the class of representations W Σ of CCR(H) of Araki-Woods type, and the corresponding quasifree states ϕ Σ , determined by AW amplitudes Σ = Σ A,B as follows: Remark. Let Σ = Σ A be gauge invariant. On one hand, if A is injective then Ω H⊕H is a cyclic vector for the representation W Σ . On the other hand, if These AW representations W Σ inherit regularity from the Fock representation W H⊕H . As in the Fock case, given any x ∈ H, setting x) defines creation and annihilation operators via the quasifree field operators {R Σ (z) : z ∈ H}, which are the Stone generators of the corresponding unitary groups (W Σ (tz)) t∈R . We now relate these to Fock creation and annihilation operators.
For convenience, let the AW amplitude Σ ∈ B(H ⊕ H) has the block-matrix form for all x ∈ H.
It follows that R Σ (x) is the closure of the operator by [ReS,Theorem VIII.33], which implies that Thus, in terms of a parameterisation Σ = Σ A,U,C,P , as in (2.10), In particular, for a gauge-invariant AW amplitude Σ = Σ A , Remark. The absence of minus signs in these relations is due to our choice of signs in the definition of the doubling map ι, and the choice of parameterisation of the symplectic automorphism B.

Quantum stochastic calculus
In this section we summarise the relevant elements of standard quantum stochastic calculus [Par,Mey,Fag,L 2 ] in a way which is adapted to the requirements of the quasifree stochastic calculus developed in Section 4. This section ends with discussions of the non-uniqueness of implementing quantum stochastic cocycles for an Evans-Hudson flow, and Bhat's minimality criterion for quantum stochastic dilations.
For the rest of this article, we fix a Hilbert space h, which is referred to as the initial space or system space. For this section, we also fix a Hilbert space K as the multiplicity space or noise dimension space. In later sections, this will vary or have further structure.
Notation. We use the abbreviations Ω, W , a + , a − and F for Ω H , W H , a + H , a − H and Γ + (H), respectively, where the Hilbert space H = L 2 (R + ; K). As is customary, we abbreviate the simple tensor u ⊗ ε(f ) to uε(f ) whenever u ∈ h and f ∈ L 2 (R + ; K).
The space of step functions from R + to K having compact support is denoted S. Although we view S as a subspace of L 2 (R + ; K), we always take the right-continuous version of each step function, thus allowing us to evaluate these functions at any point in R + .
Note that S enjoys the following useful properties: In what follows we restrict our attention, as much as possible, to processes composed of bounded operators.
where I [t is the identity operator on F [t , and measurable, so that the function is weakly measurable for all ξ ∈ h ⊗ F. By separability, weak measurability may be replaced with strong measurability here.
An h-h ′ process X is (i) simple if it is piecewise constant and right continuous, so that there exists a strictly increasing sequence (t n ) n≥1 ⊆ R + such that t 1 = 0 and t n → ∞ as n Notation. It is convenient to augment the multiplicity space, by setting Remark. One may also begin with a Hilbert space K and, by choosing a distinguished unit vector ω ∈ K, obtain K by setting K := K ⊖ Cω. This observation will be useful in Section 6.
Definition 3.2. An K-integrand process on h, or simply an integrand process, is a K⊗h process F such that, in terms of its block-matrix form K M L N , s → K s vε(g) and s → M s g(s) ⊗ vε(g) are locally integrable, and s → L s vε(g) and s → N s g(s) ⊗ vε(g) are locally square-integrable, for all v ∈ h and g ∈ S.
Remark. Suppose F is a K ⊗ h process such that, for all x, y ∈ K, the function is locally integrable. Then F is an integrand process.
Remark. The identity (3.1) is known as the first fundamental formula of quantum stochastic calculus. It follows from it that the family of operators Λ(F ) is unique.
is an integrand process and its adjoint process If the integrand process F is such that the operator Λ(F ) t is bounded, for all t ∈ R + , then taking the closure of each operator defines a continuous h process which, by a slight abuse of notation, we also denote by Λ(F ).
be an integrand process. Then The following proposition, which is readily verified, connects the definition of quantum stochastic integrals of Theorem 3.3 with the classical Itô integration of simple processes.
Remark. The preservation integral A × (N ) has a similar expression (see [Par]) and the time integral is given by the straightforward prescription The following result is the quantum Itô product formula, or second fundamental formula. To state it, we define the quantum Itô projection which is ampliated to 0 0 0 I K⊗h for appropriate choices of h without change of notation.
Theorem 3.6. Let F and G be integrand processes, Definition 3.7. The map Remark. The vacuum expectation is normal, unital and completely positive, and is invariant for the action of σ = (σ t ) t 0 , which is an E 0 semigroup [Arv]: it is a unitary QS cocycle if the process is unitary. The family (E Ω [Y t ]) t 0 is called the vacuum expectation semigroup of Y . A Hudson-Parthasarathy cocycle, or HP cocycle in short, is a unitary QS cocycle whose expectation semigroup is norm continuous.
t 0 is a one-parameter semigroup follows from the adaptedness relations In this case, for all u, v ∈ h and x, y ∈ K. In particular, the vacuum expectation semigroup of U has generator K.  Remark. If F is the stochastic generator of an HP cocycle then Theorem 3.9 implies that F * is also such a generator, since where W = W * , L = −W * L and H = −H. However, it is usually not the case that Y F * and (Y F ) * are equal. An exception is when h = C, described in Example 3.13.
In this article, we are mainly concerned with the following subclass of HP cocycles.
Definition 3.11. An HP cocycle is Gaussian if its stochastic generator lies in B( K ⊗ h) 0 . Equivalently, its parameterisation has the form (H, L, I K⊗h ).
which restricts to a bijection Example 3.13. [Pure-noise cocycles] For any z ∈ K, setting W z := (W (z1 [0,t) )) t 0 defines an HP cocycle on C. An operator F ∈ B( K) is the generator of an HP cocycle on C if and only if The Gaussian pure-noise cocycles are precisely those of the form (e iαt W z t ) t 0 for some α ∈ R and z ∈ K.
Lemma 3.14. Let U be an HP cocycle on h and let u be a pure-noise HP cocycle with the same noise dimension space. Then of U and f ∼ (α, |z , w) of u are related as follows: (3.5) Proof. That U is a unitary QS cocycle follows from the fact that σ r (U t ) and I h ⊗u r commute for all r, t ∈ R + . The quantum Itô product formula, Theorem 3.6, implies that It now follows from the uniqueness part of Theorem 3.9 that U equals the HP cocycle Y F , so that U = U ( H, L, W ) where ( H, L, W ) is given by (3.5).
Definition 3.15. A quantum dynamical semigroup P = (P t ) t 0 is a semigroup of completely positive contractive normal maps on B(h) which is pointwise weak operator continuous. If P t is unital for all t ∈ R + then P is called conservative.
Remark. The generator L of a norm-continuous conservative quantum dynamical semigroup is expressible in Lindblad form [Lin]: there exists a separable Hilbert space K, a self-adjoint operator H ∈ B(h) and an operator L ∈ B(h; K ⊗ h) such that where [ , ] and { , } denote the commutator and anti-commutator, respectively.
Theorem 3.16. Let U be an HP cocycle with stochastic generator (H, L, W ). For all t ∈ R + , let Furthermore, the family j = (j t ) t 0 is the unique mapping process consisting of normal * -homomorphisms that satisfies (3.8). (b) The mapping process j obeys the cocycle relation for all a ∈ B(h) and b ∈ Ran σ K r ⊆ B(F). Moreover, setting P := (E Ω • j t ) t 0 defines a norm-continuous conservative quantum dynamical semigroup on B(h), the vacuum expectation semigroup of j.
(c) For all a ∈ B(h), u, v ∈ h and x, y ∈ K, In particular, the vacuum expectation semigroup of j has generator L.
Proof. That j satisfies (3.8) follows from the quantum Itô product formula. In turn, part (c) follows from (3.8), the first fundamental formula, Theorem 3.3, and the strong continuity of U . For (b) and the uniqueness part of (a), see [L 2 ] and [LW 1 ].
Definition 3.17. An inner Evans-Hudson flow on B(h), or inner EH flow in short, is a mapping process j induced by an HP cocycle on h, as above [Eva]. The map θ is called the stochastic generator of j.
Remark. Let j be an inner EH flow on B(h). Using the ampliations introduced in Theorem 3.16, the prescription J : where U is any HP cocycle inducing j. In turn, we can recover j from J, since j t = J t • ι F for all t ∈ R + , where the ampliation Given an HP cocycle U , Lemma 3.14 provides sufficient conditions for an HP cocycle U ′ to induce the same EH flow as U . In the next result we show that these conditions are also necessary.
Proposition 3.18. Suppose j and j ′ are inner EH flows on B(h) with noise dimension space K, induced by HP cocycles U and U ′ and having stochastic generators (H, L, W ) and (H ′ , L ′ , W ′ ), respectively. The following are equivalent.
(i) The flows j and j ′ are equal.
(ii) The process (U ′ t U * t ) t 0 is the ampliation to h of a pure-noise HP cocycle. (iii) There is a scalar α ∈ R, a vector z ∈ K and an operator w ∈ U (K) such that Proof. If (ii) holds then Lemma 3.14 implies that (iii) holds.
If (iii) holds then it is easily verified that θ ′ , defined from (H ′ , L ′ , W ′ ) rather than (H, L, W ), coincides with θ. Thus (i) holds by the uniqueness part of Theorem 3.16(a).
Finally, suppose that (i) holds, and let X denote the unitary process (U ′ t U * t ) t 0 . For all t ∈ R + , the operator X t commutes with all operators in B(h) ⊗ I F , so X t = I h ⊗ u t for some unitary operator u t ∈ B(F). This implies that X r commutes with σ r (U * t ) for all r, t ∈ R + , and so Hence u = (u t ) t 0 is a unitary QS cocycle on C. Since (U ′ ) * and U * are both strongly continuous and unitary, so u is strongly continuous and therefore its vacuum expectation semigroup P is too. As P is a semigroup on C, this implies that P is norm continuous. Thus u is an HP cocycle and therefore (ii) holds.
Remarks. Given a norm-continuous conservative quantum dynamical semigroup P on B(h), its generator L is expressible in Lindblad form (3.6) for some separable Hilbert space K and operators H = H * ∈ B(h) and L ∈ B(h; K ⊗ h). In turn, Theorem 3.16 implies that the inner EH flow j induced by the HP cocycle with generator (H, L, I K⊗h ) has vacuum expectation semigroup P. In this sense, the flow j is a stochastic dilation of P.
The non-uniqueness of triples (K, H, L) determining the generator L of a norm-continuous quantum dynamical semigroup on B(h) is analysed in [PS 3 ]; this may be compared to the non-uniqueness of triples (H, L, W ) determining the stochastic generator θ of a given inner EH flow j characterised in Proposition 3.18.
The construction of stochastic dilations was a major motivation for the original development of quantum stochastic calculus [HuP, Par].
We end this summary of standard quantum stochastic calculus by connecting it to Bhat's analysis of dilations of the above form, in particular the question of minimality.
Theorem 3.19 ([Bha, Theorem 9.1]). Let j be an inner EH flow. The following are equivalent.
(ii) The stochastic generator (H, L, W ) of any HP cocycle which induces j satisfies Remarks. To see directly that (ii) is independent of the choice of HP cocycle which induces j, note that for two such HP cocycles with stochastic generators (H 1 , L 1 , W 1 ) and (H 2 , L 2 , W 2 ), it holds that by Proposition 3.18. This also gives the following further equivalent condition.
(iii) The stochastic generator (H, L, W ) of any HP cocycle which induces j is such that the degeneracy space K L = {0}; for the definition of K L , see (5.1).
Bhat actually deals with the associated E 0 semigroup J := (  t • σ t ) t 0 on B(h ⊗ F) which, in view of the remark following Definition 3.17, is equivalent.

Quasifree stochastic calculus
In this section we produce a simplified form of the coordinate-free multidimensional quasifree stochastic calculus [LM 1,2 ] with respect to a fixed AW amplitude Σ = Σ A,B for a Hilbert space k, the quasifree noise dimension space.
The conjugate Hilbert space of L 2 (R + ; k) is identified with L 2 (R + ; k), and L 2 (R + ; k)⊕L 2 (R + ; k) is identified with L 2 (R + ; k ⊕ k). Note that we are here working with the Boson Fock space F over L 2 (R + ; k ⊕ k).
Suppose the noise dimensions space k is finite dimensional, with orthonormal basis (e i ) i∈I , let R be a simple (k ⊗ h)-h process, let t > 0 and suppose the partition {0 = t 0 < · · · < t n = t} contains the points of discontinuity of R on [0, t). Then with H denoting L 2 (R + ; k). Note that, for any u ∈ h, f , g ∈ S k and x, y ∈ k, Thus ⊗ uε(f, g) ds for all u ∈ h and f, g ∈ S, and therefore I 1 (t) ⊇ A − R (Σ * 1 ⊗ I h⊗F ) t . Applying this reasoning to I 2 (t) * , and exploiting adaptedness to commute the terms R j (t j ) * and I h ⊗ a − H⊕H (γe i 1 [t j ,t j+1 ) , δe i 1 [t j ,t j+1 ) ), where i ∈ I and j = 0, . . . , n − 1, yields the relation for all i ∈ I and s ∈ R + ; we say that R T is partially transpose to R. It follows that . Moreover, this also shows, for a suitable h-(k ⊗ h) process Q, that The preceding discussion shows clearly the need for a partial transpose operation for infinitedimensional k. A comprehensive theory is developed in [LM 1,2 ]. Here we specialise to our context of AW amplitudes, and it is convenient to concentrate on the composition of the partial transpose and adjoint operations.
First note that, for any Y ∈ B(h 1 ; h ⊗ h 2 ), the quantity c(Y ) := sup i∈I Y * (e i ⊗ u) 2 1/2 : u ∈ h 2 , u = 1 ∈ [0, ∞] is independent of the choice of orthonormal basis (e i ) i∈I for h. When it is finite, where · 2 denotes the Hilbert-Schmidt norm. Let B 2 (h; h ′ ) denote the space of Hilbert-Schmidt operators from h to h ′ .
(a) The following are equivalent.
(i) There is an operator Y c ∈ B(h 2 ; h ⊗ h 1 ) such that In this case, the operator Y c is unique and c The following statements hold.
Proof. Let (e i ) i∈I be an orthonormal basis for h and note the trivial identity For (a), note first that if c(Y ) < ∞ then the prescription u → i∈I e i ⊗ Y * (e i ⊗ u) defines an operator Y c from h 2 to h ⊗ h 1 which is bounded with norm c(Y ) and such that for all y ∈ h and u ∈ h 2 , so that (4.1) holds. Conversely, suppose that an operator Y c ∈ B(h 2 ; h ⊗ h 1 ) satisfies (4.1). Then (4.2) implies that so (ii) holds. Uniqueness of the operator Y c is immediate, and the fact that now c(Y c ) = Y and Y cc = Y follows from taking the adjoint of identity (4.1).

Parts (b) and (d) are readily verified, and part (c) follows from the identity
which is valid for all u ∈ h and any unit vector u ′ ∈ h ′ .
Definition 4.2. We let and note that it is a subspace of B(h 1 ; h ⊗ h 2 ) on which c defines a norm. The elements of this space are h-conjugatable or partially conjugatable operators, and partial conjugation is the conjugate-linear isomorphism In this case, the quasifree stochastic integral of V is the process Λ Σ (V ) := Λ(V Σ ).
Remarks. If V is a Σ-integrand process on h, with block-matrix form K R Q 0 , then A sufficient condition for a k ⊗ h process K R Q 0 to be a Σ-integrand process is that the function is locally integrable on R + . If dim k < ∞ then this reduces to the local integrability of the function t → K t + Q t 2 + R t 2 .
We will now show that Σ-integrability is unaffected by squeezing. The transformation of integrands resulting from squeezing the AW amplitude may be viewed as a change-of-variables formula.
Theorem 4.4. Let Σ = Σ M , where Σ and M are an AW amplitude and squeezing matrix for k, respectively, and let W be a Σ-integrand process. Then there is a Σ-integrand process W such that Λ Σ ( W ) = Λ Σ (W ).
Proof. Let W have block-matrix form K R Q 0 , let M = M U,C,P as in (2.9), and let for all t 0, where c := cosh P , s := sinh P and I := I h⊗F . To show that W := K R Q 0 is as desired, it now suffices to verify the following.
(a) The processes Q and R are conjugatable.
(b) For all t ∈ R + , it holds that equivalently, Now, Theorem 4.1 gives (a), and the following identities: and so (b) holds as required.
The following identity is the first fundamental formula for quasifree stochastic integrals. In view of Theorem 3.3, it holds by definition.
Proposition 4.5. Let V be a Σ-integrand process on h. With the notation given in (4.3), The following is readily verified from the definitions. Let F H = Γ + L 2 (R + ; H) for any choice of H.
Corollary 4.6. Suppose that the AW amplitude Σ is gauge invariant, so has the form Σ A , and let k 0 := ker A. Then any Σ-integrand process V on h compresses to a k 0 -integrand process V 0 Remark. Here k 0 is being viewed as a subspace of K := k ⊕ k as well as of k, and F k 0 is being identified with the subspace This observation shows the quasifree stochastic calculus constructed here incorporates standard quantum stochastic integrals as well as purely quasifree stochastic integrals, making them useful for the investigation of repeated interaction systems with particles in a non-faithful state; see Section 6 and [Be 3 ].
The following result is the second fundamental formula for quasifree stochastic integrals, and should be compared with Theorem 3.6. The final term on the right-hand side is the quasifree Itô correction term.
Theorem 4.7. Let X : and W = J T S 0 are Σ-integrand processes and X 0 , Y 0 ∈ B(h) ⊗ I F . In the notation of (4.3), Proof. This follows immediately from Theorem 3.6, Definition 4.3 and the identity which holds for all x, y ∈ K, u, v ∈ h, f , g ∈ S K and s ∈ R + .
The following are equivalent.
(i) The operator W has block-matrix form , where Q is conjugatable and If either condition holds then U is the unique h process satisfying (a) and (b) of (ii).

Proof. Suppose that (i) holds and set
Then F ∈ B( K ⊗ h) 0 and F * + F + F * ∆F = 0. Appealing to Theorem 3.9 and Definition 3.10, there exists a unitary process U := Y F . Since (W · U ) Σ = F · U , so W · U is a Σ-integrand process and Λ Σ (W · U ) t = Λ(F · U ) t = U t − I h⊗F for all t ∈ R + , hence (ii) holds.
Conversely, suppose that (ii) holds for a unitary h process U , and let K R Q 0 be the block-matrix form of W . Theorem 4.1 implies that the operators Q and R * are conjugatable, and Assumption (b) gives that U t = I h⊗F + Λ(F · U ) t for all t ∈ R + , and so, by Theorem 3.9, it holds that F * + F + F * ∆F = 0 and U = Y F . In particular, the uniqueness claim is established. The condition F * + F + F * ∆F = 0 is equivalent to so it remains to prove that X := Q + R * = 0. Note that (a) is equivalent to (Σ ⊗ I h ) X X c = 0 and, in terms of the parameterisation Σ A,U,C,P of the AW amplitude Σ given in (2.10), this is equivalent to It follows from (4.5) that X = −(tanh P · CK −1 ⊗ I h )X c , and so, by Theorem 4.1 and the fact that C commutes with P and C 2 = I k , thus 0 = (I k − tanh 2 P ) ⊗ I h X = (cosh 2 P ⊗ I h ) −1 X and so X = 0.
Remark. From the preceding proof, we see that the unique unitary h process U determined by an operator W ∈ B( k ⊗ h) 0 satisfying Theorem 4.8(i) equals Y F , where F = W Σ as defined in (4.4). In particular, U is an HP cocycle. Cocycle aspects of quasifree processes are further investigated in [LM 2 ].
Definition 4.9. An HP cocycle U on h with noise dimension space k ⊕ k is Σ-quasifree and has Σ-generator , with Q k-conjugatable and Remark. Thus Σ-quasifree HP cocycles form a subclass of the collection of Gaussian HP cocycles with noise dimension space having a decomposition K = k ⊕ k. be the block matrix form of L, and let Σ = Σ A be a gauge-invariant AW amplitude for k. The following are equivalent.
When these hold, the cocycle U has Σ-generator (4.7) Proof. By Theorem 4.8 and Definition 4.9, (i) is equivalent to (ii), and these imply that U has Σ-generator . Properties of the partial conjugation, Theorem 4.1, now imply that (ii) is equivalent to (iii); they also imply that (iii) is equivalent to (iv). When these conditions hold, since the identity (4.7) follows from the identity cosh 2 A − sinh 2 A = I k . Then (j k t • ψ)(a) t 0 is a Σ-integrand process for all a ∈ B(h), where j k t := id B( k) ⊗ j t , and j t (a) = a ⊗ I F + Λ Σ (j k • ψ)(a) t for all a ∈ B(h) and t ∈ R + .
Proof. It is straightforward to verify that for all a ∈ B(h) and s ∈ R + , where j K s := id B( K) ⊗ j s for K = k ⊕ k, and θ is the map from B(h) to B( K ⊗ h) defined in (3.7). It therefore follows from Theorem 3.16 that j t (a) − a ⊗ I F = Λ (j K • θ)(a) t = Λ Σ (j k • ψ)(a) t for all a ∈ B(h) and t ∈ R + , as claimed.

Uniqueness questions
In this section, issues of uniqueness are considered. We begin with the question of uniqueness of AW amplitudes for quasifree HP cocycles. Given an HP cocycle with noise dimension space K and stochastic generator F = K −L * V L V −I K⊗h , we examine the class of pairs (Σ, Q) such that Σ is an AW amplitude, Q is a k-conjugatable operator and ( is a Σ-quasifree generator and F = W Σ . Immediate necessary conditions for this class to be non-empty are that U is Gaussian, so that F ∈ B( K ⊗ h) 0 , and K has a decomposition k ⊕ k, so K must not have finite odd dimension.
We also consider the uniqueness of quasifree HP cocycles implementing a given EH flow j and relate this to the minimality of j as a stochastic dilation of its expectation semigroup.
For the remainder of this section, we fix a quasifree noise dimension space k, and set K = k ⊕ k. Theorem 4.4 has the following consequence.
Corollary 5.1. Let Σ = Σ M , where Σ and M are an AW amplitude and squeezing matrix for k, respectively. Then every Σ-quasifree HP cocycle is also Σ-quasifree.
In light of the above corollary, we restrict to gauge-invariant AW amplitudes for the rest of this section. For an operator X ∈ B(h; k ⊗ h), let the k-degeneracy space of X be Proposition 5.2. Let Σ = Σ A be a gauge-invariant AW amplitude for k, and suppose U is a Σquasifree HP cocycle with stochastic generator K −L * L 0 and Σ-generator Then Furthermore, if Σ = Σ A is another gauge-invariant AW amplitude for k, then the following are equivalent.
Proof. Corollary 4.11 implies that L 2 is k-conjugatable and Q is k-conjugatable, with Thus (5.2) follows from the invertibility of cosh A. Corollary 4.11 also implies that (i) holds if and only if L c 2 = −(tanh A ⊗ I h )L 1 . Therefore (i) and (ii) are equivalent, by (5.3).
For an HP cocycle U with noise dimension space k ⊕ k, let Ξ(U ) := Σ ∈ AW 0 (k) : U is Σ-quasifree be the set of gauge-invariant AW amplitudes for k for which U is Σ-quasifree.
Corollary 5.3. Let U be an HP cocycle with stochastic generator K −L * L 0 . If U is quasifree with respect to a gauge-invariant AW amplitude Σ A then In particular, if k L 1 = {0} then U is quasifree with respect to at most one gauge-invariant AW amplitude.
We now turn to the question of implementability of inner EH flows by quasifree HP cocycles. (i) The cocycles U and U induce the same inner EH flow.
(ii) There exist x ∈ k and α ∈ R such that Proof. Let C and T denote cosh A and tanh A, respectively,where Σ = Σ A , and let If z = (z 1 , z 2 ) ∈ k ⊕ k and α ∈ R are such that (5.4) holds then so z 2 = −T z 1 , and therefore z = Σι(x), where x = C −1 z 1 . It follows from (4.7) that (ii) holds.
Theorem 5.5. Let j be an inner EH flow which is a minimal dilation of its vacuum expectation semigroup. Then there is at most one gauge-invariant AW amplitude Σ such that j is induced by a Σ-quasifree HP cocycle.
Proof. Suppose that j is induced by a Σ-quasifree HP cocycle U and a Σ-quasifree HP cocycle U , where Σ = Σ A and Σ = Σ A are gauge-invariant AW amplitudes for k. Then U and U are Gaussian and so have stochastic generators of the form K −L * L 0 be their respective quasifree generators, it follows that and Proposition 3.18 implies that L = L + |z ⊗ I h for some z = (z 1 , z 2 ) in k ⊕ k. If T := tanh A and T := tanh A then Therefore, by Theorem 3.19, the minimality of j implies that Ran(T − T ) * = {0}, so T = T , A = A and Σ = Σ.

Quantum random walks
In this section we first review the basic theory of unitary quantum random walks for particles in a vector state and their convergence to quantum stochastic cocycles [Be 1 ]; for an elementary treatment via the semigroup decomposition of quantum stochastic cocycles, see [BGL]. Stronger theorems for more general walks may be found in [Be 2 ], for particles in a faithful normal state, and in [Be 3 ], for particles in a general normal state. We then construct quantum random walks in the repeated-interactions model for particles in a faithful normal state ρ. Under the assumption that the interaction Hamiltonian H I has no diagonal component with respect to the eigenspaces of the density matrix of ρ, we demonstrate convergence to HP cocycles of the form U ⊗ I where I is the identity operator of the Fock space over L 2 (R + ; K 0 ) for a subspace K 0 of the GNS space of ρ. The construction yields a quasifree noise dimension space k together with natural conjugate space k and, under the assumption of exponential decay of the eigenvalues of the density matrix corresponding to ρ, a gauge-invariant AW amplitude Σ(ρ) for k. We then show that U is Σ(ρ)-quasifree, assuming only that H I is p-conjugatable. We also show that if the lower-triangular matrix components of H I are strongly linearly independent then Σ(ρ) is the unique gauge-invariant AW amplitude with respect to which U is quasifree.
Particles in a vector state. For this subsection, we fix a noise dimension space K.
Definition 6.1. The toy Fock space Υ over K is the tensor product of a sequence of copies of K := C ⊕ K with respect to the constant stabilising sequence given by ω := 1 0 : We also set Υ [m := As is readily verified [Be 1 ], toy Fock space over K approximates Boson Fock space over K in the following sense. Let F J = Γ + L 2 (J; K) for any subinterval J ⊆ R + , with Ω J its vacuum vector, and, for all τ > 0, let for any finitely-supported sequence (x n ) ⊆ K. Then D τ D * τ → I F in the strong operator topology as τ → 0+.
Definition 6.2. For any U ∈ U ( K ⊗ h), the quantum random walk generated by U is the sequence (U n ) n 0 ⊆ B(h ⊗ Υ) defined recursively as follows: where the normal * -monomorphism and σ n := id B(h) ⊗ σ Υ n is the ampliation of the right shift * -endomorphism of B(Υ) with range I K ⊗n ⊗ B(Υ [n ).
Scaling maps on B( K ⊗ h) are defined by setting Remark. If the generator U has the form A ⊗ X then U n = X n ⊗ A ⊗n ⊗ I [n for all n 0.
Henceforth we focus on the repeated-interactions model of [AtP].
and let Then F * + F + F * ∆F = 0 and, as τ → 0+, Proof. That F is as claimed is readily verified, and the final claim holds by [Be 1 , Theorem 7.6 and Remarks 4.8 and 5.10], since Particles in a faithful state. We now fix a non-zero Hilbert space p, referred to as the particle space, and a faithful normal state ρ on B(p). Let (γ α ) α∈I be the eigenvalues of its density matrix ̺, ordered to be strictly decreasing, and suppose the index set I is either {0, 1, · · · , N } for some non-negative integer N , or Z + . For any α ∈ I, let P α ∈ B(p) be the orthogonal projection with range k α , the eigenspace of ̺ corresponding to the eigenvalue γ α . Thus ̺ = α∈I γ α P α and α∈I γ α d α = 1, where d α := dim k α = tr(P α ).
Let K be the anti-unitary operator from k → k obtained by restricting the adjoint operation on K = B 2 (p). Then K = Cω ⊕ K 1 ⊕ K 0 , and (k, K) is a realisation of the conjugate Hilbert space of k. Note also that (6.1) We now identify the one-dimensional subspace Cω of K with C, so that Theorem 6.4. Let the operators H S ∈ B(h), H P ∈ B(p) and H I ∈ B(p ⊗ h) be self adjoint, and assume that (P α ⊗ I h )H I (P α ⊗ I h ) = 0 for all α ∈ I. Then we have the following.
and H T (τ ) := I p ⊗ H S + H P ⊗ I h + τ −1/2 H I ∈ B(p ⊗ h), Then Proof. (a) It must be shown that If u, v ∈ h, α ∈ I and T ∈ k αα , and H u v := (I p ⊗ u|)H I (I p ⊗ |v ), then and so (6.2) follows from (6.1).
Furthermore, it is straightforward to verify that ω, π(H P )ω I h = ρ(H P )I h and and this last identity implies that F is as claimed. The conclusion now follows from Theorem 6.3, since ω is identified with Remarks. Under the identification h ⊗ F K = h ⊗ F K 1 ⊗ F K 0 , where F H = Γ + L 2 (R + ; H) , the limit process decomposes as The condition on H I has the following physical interpretation: there is no contribution from the interaction Hamiltonian unless the particle undergoes a transition.
This ensures that the following lemma yields an AW amplitude for k. To avoid it would require more of the general theory developed in [LM 1,2 ].
For all α, β ∈ I, let P αβ denote the orthogonal projection with range k αβ .
Thus, under Assumption 6.5, with S(ρ) and C(ρ) as in the preceding lemma, defines a gauge-invariant AW amplitude for k.
Our goal now is to prove that the HP cocycle generated by F in Theorem 6.4 is Σ(ρ)-quasifree, provided that the interaction Hamiltonian H I is p-conjugatable. To this end, note first that, for all T ∈ B(p) and α, β, α ′ , β ′ ∈ I, the vectors P α T P β and P α ′ T P β ′ are orthogonal in B 2 (p) unless α ′ = α and β ′ = β, and therefore so the following prescriptions define bounded operators: For the next proposition we adopt the notation Recall that Theorem 4.1 gives the inclusion , respectively, and that the resulting maps are related via partial conjugation.
Proposition 6.7. There are unique operators for all A ∈ B c (p ⊗ h) * , u, v ∈ h and ζ ∈ k α , η ∈ k β with α > β. Furthermore, we have that and the following properties hold.
; it also satisfies (6.8a) since, for all u, v ∈ h, ζ ∈ k α and η ∈ k β , where α > β, In particular, the operator φ h ρ (A) does not depend on the choice of orthonormal bases made above. Similarly, there is an operator φ c(A * ), the identity (6.8b) holds and, for any choice of orthonormal bases e i Recall that a countable family of bounded operators C is said to be strongly linearly independent if there is no non-zero function α : C → C such that T ∈C α(T )T converges to zero in the strong sense.
Theorem 6.8. Let F ∈ B K 1 ⊗ h 0 be as in Theorem 6.4, so that where the operators H S ∈ B(h), H P ∈ B(p) and H I ∈ B(p ⊗ h) are self adjoint, J is the natural isometry from k ⊕ k to C ⊕ (k ⊕ k) ⊕ K 0 and (P α ⊗ I h )H I (P α ⊗ I h ) = 0 for all α ∈ I.
In other words k V 1 = {0} and therefore, by Corollary 5.3, there is no other gauge-invariant AW amplitude Σ with respect to which the HP cocycle U is Σ-quasifree.
Remark. Theorems 6.4 and 6.8 comprise a significant generalisation of the main result of [AtJ], Theorem 7. The restriction to finite-dimensional noise or particle space, is removed, and the interaction Hamiltonian is of a more general form. In [AtJ], the operator H I is taken to have the form 0 V * V 0 , and this assumption corresponds to the Σ(ρ)-quasifree generator K −Q * Q 0 satisfying e j,k | ⊗ I h Q = 0 for all j > k > 0. In conclusion, a large class of unitary quantum random walks, with particles in a faithful normal state, converge to HP cocycles governed by a quasifree quantum Langevin equation.

Appendix
In this appendix, we prove that symplectic automorphisms of a Hilbert space h are necessarily bounded, and give a parameterisation for the elements of the group S(h) × . For the convenience of the reader, this is a streamlined version of the proof given in [HoR], which also covers the case of unbounded symplectic automorphisms of separable pre-Hilbert spaces. Thus L has everywhere-defined adjoint x → 1 2 B −1 x − iB −1 (ix) , and so is closed and bounded, by the closed graph theorem. Similarly, the conjugate-linear operator A has everywhere-defined adjoint x → − 1 2 B −1 x + iB −1 (ix) , and so is also bounded. Thus B is bounded.
For a triple (U, C, P ) consisting of a unitary operator U on h, a bounded non-negative operator P on h and a conjugation (a self-adjoint anti-unitary operator) C on h, such that P and C commute, we define the following bounded real-linear operator on h: B U,C,P := U (cosh P − C sinh P ) (A.1) Remark. Since, with (U, C, P ) as above, the map −C is also a conjugation on h that commutes with P , a deliberate choice is being made here. The reason for this particular choice is that it eliminates minus signs elsewhere.
(a) Let (U, C, P ) be a triple as above.
(ii) Suppose that B U,C,P = B U ′ ,C ′ ,P ′ for another such triple (U ′ , C ′ , P ′ ). Then U ′ = U, P ′ = P and C ′ agrees with C on Ran P .
(b) Conversely, let B ∈ S(h) × . Then there is a triple (U, C, P ) as above, such that B = B U,C,P .
Proof. (a) (i) This is a straightforward matter of verification.
(ii) Set B = B U,C,P , and let L and A be its linear and conjugate-linear parts. Then U cosh P = L = U ′ cosh P ′ and − U C sinh P = A = −U ′ C ′ sinh P ′ .
Since the bounded operators cosh P and cosh P ′ are non-negative and invertible, and U and U ′ are unitary, the uniqueness of polar decompositions implies that U ′ = U and cosh P ′ = cosh P . Hence, by the non-negativity of P ′ and P , so P ′ = P , and thus also C ′ sinh P ′ = C sinh P . It follows that C ′ f (P ′ ) = Cf (P ) for all continuous functions f : R + → C satisfying f (0) = 0; in particular C ′ P = CP , so C ′ and C agree on Ran P . Applying the first of these identities to the symplectic automorphism B −1 , we see that Let U |L| and V |A| be the polar decompositions of L and A, respectively, and set K := ker |A| and K * := ker |A * |. The conjugate-linear partial isometry V has initial space K ⊥ and final space K * ⊥ , and the identities (A.2) and (A.3) imply that L is invertible, so U is unitary, and |L| I h . Thus there exists a unique non-negative operator P ∈ B(h) such that |L| = cosh P and |A| = (|L| 2 − I h ) 1/2 = sinh P . Now |L * | = U |L|U * and |L| = U * |L * |U so, for all x ∈ K and z ∈ K * , |L * |U x = U |L|x = U x and |L|U * z = U * |L * |z = U * z, which implies that U K ⊆ K * and U * K * ⊆ K. Hence U K = K * , and therefore also U K ⊥ = K * ⊥ . It follows that, on h = K ⊕ K ⊥ , U * V has the form {0} ⊕ D 1 for an anti-unitary operator D 1 on K ⊥ . Therefore, setting D := D 0 ⊕ D 1 for an arbitrary conjugation D 0 on K, B = L + A = U |L| + V |A| = U (cosh P + D sinh P ), |A|D = |A|U * V and U * V |A| = D|A|. Thus, using the identities (A.2) and (A.3) once more, |A|U * V = (|L| 2 − I) 1/2 U * V = U * (|L * | 2 − I) 1/2 V = U * |A * |V = U * V |A|.
Therefore D commutes with |A| = sinh P and so commutes with all continuous functions of sinh P such as P itself and |L| = cosh P . The second identity in (A.2) now implies that |L|x, D|A|y = Lx, Ay = Ly, Ax = |L|y, D|A|x = |A|y, D|L|x = |L|x, D * |A|y for all x, y ∈ h, so D and D * agree on Ran |A| = K ⊥ , and thus D * 1 = D 1 . But D * 0 = D 0 , since D 0 is a conjugation on K, therefore D * = D and so the anti-unitary operator D is a conjugation on h. The proof is now completed by letting C be the conjugation −D.