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Communications on Stochastic Analysis

Volume 4 | Number 4 Article 7

12-1-2010 Transformation of quantum Lévy processes on Hopf algebras Michael Schürmann

Michael Skeide

Silvia Volkwardt

Follow this and additional works at: https://digitalcommons.lsu.edu/cosa Part of the Analysis Commons, and the Other Commons

Recommended Citation Schürmann, Michael; Skeide, Michael; and Volkwardt, Silvia (2010) "Transformation of quantum Lévy processes on Hopf algebras," Communications on Stochastic Analysis: Vol. 4 : No. 4 , Article 7. DOI: 10.31390/cosa.4.4.07 Available at: https://digitalcommons.lsu.edu/cosa/vol4/iss4/7 Communications on Stochastic Analysis Serials Publications Vol. 4, No. 4 (2010) 553-577 www.serialspublications.com

TRANSFORMATION OF QUANTUM LEVY´ PROCESSES ON HOPF ALGEBRAS

MICHAEL SCHURMANN,¨ MICHAEL SKEIDE*, AND SILVIA VOLKWARDT

Abstract. A quantum L´evyprocess is given by its generator, a conditionally positive linear functional on the underlying or bialgebra. A transformation between two bialgebras, in the sense of this paper, is a counit preserving algebra . We show that transformation on the level of the corresponding quantum L´evyprocesses is given by product integrals. This general result is applied to a bialgebra and its ‘generator Hopf algebra’ as well as to its ‘Weyl bialgebra’. It follows that a quantum L´evy process can be realized on Bose Fock space as a convolution product integral of its generator process such that the vacuum vector is cyclic. At the same time, it can be reconstructed from its Weyl process. A further application are Trotter product formulae for quantum L´evyprocesses.

1. Introduction

A stochastic process Xt : E → G, t ≥ 0, over some probability space E taking values in a topological G is called a (stationary) L´evyprocess on G if the −1 ≤ ≤ increments Xs,t = Xs Xt, 0 s t, of disjoint intervals are independent, if the distribution of Xs,t only depends on t − s, and if, for t → 0 from the right, we have that Xt converges in law to the Dirac measure concentrated at the unit element of G. This can be generalized to stochastic processes (Xs,t)0≤s≤t taking values in a G if the evolution equations Xr,sXs,t = Xr,t hold. Classical L´evyprocesses are commutative in the following sense. If we replace G and E by suitable ∗–algebras of functions (on G and E; e.g. replace G by L∞(G) and E by ∞ L (E)) then Xs,t : E → G will give the ∗–algebra homomorphism js,t mapping a function f on G to the function js,t(f) = f◦Xs,t on E. The js,t form a commutative process in the sense that they are defined on a commutative ∗–algebra. Replacing the monoid G by a ∗–bialgebra and the classical probability space E by what is called a quantum probability space, the notion of a quantum L´evyprocess (QLP) on a ∗–bialgebra over a quantum probability space has been introduced in L. Accardi, M. Sch¨urmann,and W. von Waldenfels [1].

Received 2010-7-12; Communicated by D. Applebaum. 2000 Mathematics Subject Classification. Primary 46L53, 16T10, 60G51; Secondary 81S25, 60J25, 60B15, 46L55. Key words and phrases. Quantum probability, noncommutative processes with independent increments, L´evyprocesses, Hopf algebras in quantum theory, quantum stochastic differential equations. * Supported by research funds of the Italian MIUR and of the University of Molise. 553 554 MICHAEL SCHURMANN,¨ MICHAEL SKEIDE, AND SILVIA VOLKWARDT

A representation theorem for such processes, M. Sch¨urmann[11, Theorem 2.5.3], says that they can always be realized on a Boson Fock space as solutions to quantum stochastic differential equations in the sense of R.L. Hudson and K.R. Parthasarathy [4]. As pointed out in M. Skeide [14], QLPs can also be viewed as tensor product systems of type I in the sense of W. Arveson [2]. They are (up to stochastic equivalence) uniquely determined by their generators which are precisely the hermitian, normalized conditionally positive linear functionals on the underlying ∗–bialgebra. In this paper we are mainly interested in the following sit- uation. If there are given two bialgebras and an algebra homomorphism between them with the additional property that the homomorphism preserves the counits, then generators are transformed into generators. The question arises how the two QLPs given by the two generators can be transformed into each other. Using infinitesimal convolution products which can be regarded as convolution product integrals, we establish a transformation on the level of the QLPs. We describe very briefly what we do in a simplified setting. (For a precise de- scription of the general situation see Sections 2 and 3.) In this simplified setting the situation is as follows. Suppose (B, ∆, δ) is a ∗–bialgebra. Then the comulti- plication ∆ induces a convolution  ? for algebra-valued linear mappings on B; see Section 2. A QLP j = js,t 0≤s≤t satisfies ··· js,t(b) = jt0,t1 ? ? jtn−1,tn (b) for all s = t0 < t1 < . . . < tn−1 < tn = t. Suppose on B there is a second comultiplication ∆0. We shall show that, in the canonical representation of j on a pre-Hilbert space D with cyclic vector Ω, the expressions 0 ··· 0 jt0,t1 ? ? jtn−1,tn (b)Ω (with the convolution with respect to ∆ replaced by the convolution with respect to ∆0) form a Cauchy net over the partitions of the interval [s, t]. From this it is easy to show that their limits, which we denote by ks,t(b)Ω determine on their linear hull a unique QLP k over (B, ∆0, δ), the tranform of j. Moreover, we shall show that under suitable cyclicity conditions this procedure can be reversed. See Theorems 3.4, 3.6 and 3.7 for a precise formulation in a more general context. The transformation has various applications. For example, there are two QLPs associated with a given QLP in a natural way. One is the QLP’s Weyl operator type process, the other is the generator process of the QLP which is composed of annihilation, preservation and creation processes on Boson Fock space. The Weyl type process can be used to show in a nice way why the result of M. Skeide [3] holds which says that the vacuum vector is always cyclic for the QLP. The generator process allows for a construction of the QLP as a product system by infinitesimal convolution products as a kind of convolution product stochastic in- tegral. Both types of processes admit direct realizations on the Boson Fock space. Writing down the backwards transformations provides two different new proofs of the fact that every QLP may be realized as a (cyclic) process on a Boson Fock space. Other applications are the approximation of the Az´emamartingales by infinitesimal convolution products of the Wiener process (and vice versa), and Trotter product formulae for QLPs. TRANSFORMATION OF QUANTUM LEVY´ PROCESSES ON HOPF ALGEBRAS 555

In Section 2 we repeat the necessary definitions that, in Section 3, are used to formulate the transformation theorems. Section 3 also provides the constructions of several related ∗–bialgebras and applications of the theory. Section 4 presents the proof of the transformation theorems. There is work in progress (for a first step into this direction see M. Sch¨urmann and S. Voss [12]) to generalize the results of this article to QLPs on Dual Groups in the sense of D. Voiculescu [15] and to a more general notion of non-commutative independence. Our method relies on finite-dimensional arguments based on the Fundamental Theorem on (algebraic) , which says that a is the union of its finite-dimensional sub-coalgebras. A natural question is as to whether the theory presented in this paper can be generalized to the topological context of C∗-bialgebras (cf. J.M. Lindsay and A. Skalski [7]). For compact quantum groups there should not be a problem, because these unital C∗–bialgebras always have a dense ∗–subalgebra which is a proper algebraic coalgebra. For the noncompact case, this is an open problem.

2. Preliminaries A ∗– is a vector space V with an involution, i.e. an anti-linear mapping v 7→ v∗ on V satisfying (v∗)∗ = v.A ∗–algebra is an algebra A which is also a ∗–vector space such that (ab)∗ = b∗a∗ for all a, b ∈ A. If A is a ∗–algebra, A ⊗ A ⊗ ∗ ∗ ⊗ ∗ then so is with involution defined by (a1 a2) = a1 a2. A complex vector space C is a coalgebra if there are linear maps ∆: C → C ⊗ C and δ : C → C, called coproduct and counit respectively, satisfying (∆ ⊗ id) ◦ ∆ = (id ⊗ ∆) ◦ ∆ (coassociativity) (δ ⊗ id) ◦ ∆ = id = (id ⊗ δ) ◦ ∆ (counit property).

Following Sweedler we frequently use the notation c(1) ⊗ c(2) for ∆(c) suppressing both summation and indices. Let ∆0 := δ, and for n ≥ 1 define

∆n = (∆n−1 ⊗ id) ◦ ∆.

Sweedler’s notation extends to writing c(1) ⊗ c(2) ⊗ · · · ⊗ c(n) for ∆n(c), n ≥ 1. Sometimes we shall need to equip also the conjugate vector space C with a coalgebra structure. Note that the canonical bijection i = i1 : c 7→ c from C to C is an anti-linear isomorphism. The same is true for the canonical bijections in from the n–fold tensor power of C to the n–fold tensor power of C. We may write

in(c1 ⊗ · · · ⊗ cn) = c1 ⊗ . .. ⊗ cn = c1 ⊗ . .. ⊗ cn.

Note that in ⊗im = in+m (where the tensor product of antilinear mappings is well- defined). By i0 we denote complex conjugation of C. It is, then, easy to convince ◦ ◦ −1 ◦ ◦ −1 C oneself that δ := i0 δ i1 and ∆ := i2 ∆ i2 make ( , ∆, δ) a coalgebra. We shall also need the tensor product (C1⊗C2, ∆, δ) of two coalgebras (C1, ∆1, δ1) and (C2, ∆2, δ2) where δ := δ1 ⊗δ2 and ∆ := (id ⊗τ ⊗id)◦(∆1 ⊗∆2) and τ denotes the flip c ⊗ d 7→ d ⊗ c. A ∗–bialgebra (B, ∆, δ) is a coalgebra which is also a unital ∗–algebra, and in such a way that ∆ and δ are ∗–algebra . If A is a unital ∗–algebra 556 MICHAEL SCHURMANN,¨ MICHAEL SKEIDE, AND SILVIA VOLKWARDT with the map m: A ⊗ A → A defined by setting m(a1 ⊗ a2) = a1a2, then we define the convolution of two linear mappings j, k : B → A by j ? k := m ◦ (j ⊗ k) ◦ ∆. In particular, the convolution of two linear functionals ϕ and ψ on B is ϕ ? ψ = (ϕ ⊗ ψ) ◦ ∆. Unitality for a bialgebra (B, ∆, δ) means that it is unital as an algebra, i.e. there exists 1 ∈ B such that m(b ⊗ 1) = m(1 ⊗ b) = b for all b ∈ B and the coproduct and counit are unital, i.e. ∆(1) = 1 ⊗ 1 and δ(1) = 1. We only consider unital algebras. A ∗–bialgebra B with an antipode S (i.e. a mapping S : B → B such that m ◦ (S ⊗ id) ◦ ∆ = δ1 = m ◦ (id ⊗S) ◦ ∆) is called a Hopf ∗–algebra. A linear functional Φ on a ∗–algebra A is called positive if Φ(a∗a) ≥ 0 for all a ∈ A. Let (A, Φ) be a quantum probability space, that is, a unital ∗–algebra with a state (a normalized  positive linear functional Φ: A → C). A quantum stochastic process j = ji i∈I , indexed by some index set I, is a family of quantum random variables ji (that is, of unital ∗–algebra homomorphisms ji : B → A). By ϕi := Φ ◦ ji we denote the distribution of ji. The notion of independence used for quantum L´evyprocesses on ∗–bialgebras in this paper is the tensor independence. A (stationary) quantum L´evyprocess on B over A is a quantum stochastic process j = js,t , indexed by s, t ∈ R+, s ≤ t, satisfying the following four conditions.

(LP1) The increments js,t of disjoint intervals are tensor independent in Φ, that is,  ··· ··· Φ js1,t1 (b1) jsn,tn (bn) = ϕs1,t1 (b1) ϕsn,tn (bn)

for all n ∈ N, bk ∈ B,(s1, t1],..., (sn, tn] mutually disjoint intervals of R+, and

[jsk,tk (b1), jsl,tl (b2)] = 0

for all k =6 l and all b1, b2 ∈ B (LP2) The increments are stationary, that is, ϕs,t = ϕ0,t−s for all 0 ≤ s ≤ t. (LP3) The process is weakly continuous in Φ, that is, limt→0 ϕ0,t(b) = δ(b) for all b ∈ B. (LP4) The js,t are increments under convolution, that is, jr,s ? js,t = jr,t for all 0 ≤ r ≤ s ≤ t and jt,t(b) = δ(b)1 for all 0 ≤ t < ∞.

(For a topological extension of this notion of a QLP to compact quantum groups and operator space coalgebras see [7].) We observe that by (LP1) and (LP4) every L´evyprocess fulfills the condition

(LP4’) ϕr,s ? ϕs,t = ϕr,t for all 0 ≤ r ≤ s ≤ t and ϕt,t = δ.

Therefore, by (LP2) and (LP3) the states ϕt := ϕ0,t form a weakly continuous under convolution. By (LP1), (LP2) and (LP4) this convolution semi- group determines all joint moments (that is exactly all expressions of the form of the left-hand side of the first equation of (LP1), even if we drop the condition that the (sk, tk] are mutually disjoint). In other words, two L´evyprocesses are stochastically equivalent, if and only if they have the same convolution semigroup. tψ We can associate a generator ψ with a convolution semigroup through ϕt = e? for all t ≥ 0. Essentially, this follows from the Fundamental Theorem on Coalgebras; TRANSFORMATION OF QUANTUM LEVY´ PROCESSES ON HOPF ALGEBRAS 557 see [1] and Section 4.1. Then ψ is a linear functional on B, satisfying ψ(1) = 0, and it is conditionally positive and hermitian which means that ψ(b∗b) ≥ 0 for all b in the kernel of the counit δ and that ψ(b∗) = ψ(b) for all b ∈ B. Thus, L´evy processes on ∗–bialgebras can also be characterized (up to equivalence) by their generator. If B is a Hopf ∗–algebra a quantum stochastic process jt on B indexed by time t ∈ R+ is called a QLP if the ‘increments’ js,t = (js ◦ S) ? jt, 0 ≤ s ≤ t, form a QLP in the above sense. Let D be a pre-Hilbert space and denote by La(D) the ∗–algebra of adjointable operators on D, i.e. La(D) consists of all mappings T on D for which there is a ∗ ∗ mapping T on D such thathT x, yi = hx, T yi for all x, y ∈ D. If Ω is a unit vector in D, then La(D), hΩ, · Ωi is a quantum probability space. We call it a concrete quantum probability space and write it as (D, Ω). If a L´evyprocess j takes values in a concrete quantum probability space, then we say j is a concrete L´evypro- cess. By GNS-construction every quantum probability space (A, Φ) gives rise to a concrete quantum probability space (D, Ω), determined uniquely by the properties that there is a ∗–representation π : A → La(D) such that Φ(·) = hΩ, π(·)Ωi and that Ω is cyclic for A, that is π(A)Ω = D. Consequently, every L´evyprocess gives rise to a concrete L´evyprocess over (D, Ω). We will say the L´evyprocess is cyclic, if Ω is cyclic for the ∗–subalgebra n o A ··· ∈ N ∈ B ≤ · · · ≤ j := span jt0,t1 (b1) jtn−1,tn (bn): n , bk , 0 = t0 tn of La(D). Notice that by (LP1) this space does not change, if we allow that the disjoint intervals are not consecutive, and by (LP4) it also does not change if we allow for arbitrary intervals. By restricting to the invariant subspace AjΩ of D that is generated by the process from Ω, we obtain from every L´evyprocess over D a cyclic L´evyprocess on AjΩ = Dj. By a GNS-type construction applied to a generator ψ on B we obtain a pre- Hilbert space K, a surjective mapping η : B → K and a ∗–representation ρ: B → La(K) such that η(ab) = ρ(a)η(b) + η(a)δ(b) and −hη(a∗), η(b)i = δ(a)ψ(b) − ψ(ab) + ψ(a)δ(b) (2.1) for all a, b ∈ B. The specified triple (ρ, η, ψ) is called a surjective L´evytriple. There is a one-to-one correspondence between L´evyprocesses (modulo equivalence) on B, convolution of states on B, generators on B and surjective L´evy triples on B (modulo unitary equivalence).  For every convolution semigroup ϕ = ϕt there is (up to unitary equiv- t∈R+ alence) at most one cyclic L´evyprocess. (Unitary equivalence is much stronger than stochastic equivalence.) Effectively, if j is a cyclic process on (D, Ω) which fulfills (LP1) - (LP3) and (LP4’), then it is not difficult to show that also (LP4) holds. M. Sch¨urmann[11, Proposition 1.9.5] shows that for every convolution semigroup of states on a ∗–bialgebra there is a (unique up to unitary equiva- lence) cyclic L´evyprocess (even without continuity). This construction involves the GNS-construction of all ϕt, their tensor products and an inductive limit over 558 MICHAEL SCHURMANN,¨ MICHAEL SKEIDE, AND SILVIA VOLKWARDT the interval partitions of R+. However, it is completely algebraic and does not involve analytic tools. On the contrary, [11, Theorem 2.5.3] constructs a L´evy process as the solution of quantum stochastic differential equations in the sense of R.L. Hudson and K.R. Parthasarathy [4] of the form

djs,t = js,t ? dIt; jt,t = δ (2.2) where ∗ − ∗ It = At(η(b )) + Λt(ρ(b) δ(b)) + At (η(b)) + ψ(b)t (2.3) ∗ denotes the ‘generator process’ of js,t with At, Λt,At the annihilation, preservation 2 2 and creation processes on Boson Fock space over Γs(L (R+, K)) = Γs(L (R+)⊗K where K denotes the completion of the index space K. For quite a long time it was an open problem, to decide whether Fock space and differential equation can be set in such a way that the Fock vacuum is cyclic for the resulting L´evyprocess. Only quite recently and simultaneously, U. Franz, M. Sch¨urmann,and M. Skeide came up, not with just one, but with a whole bunch of proofs for the affirmative answer. The proof due to M. Skeide (see U. Franz [3, Theorem 1.21]) uses in an essential way the representation on the Fock space and equation (2.2) and shows that for every b ∈ B with δ(b) = 1 the vectors ··· jt0,t1 (b) jtn−1,tn (b)Ω, (2.4) s = t0 ≤ t1 ≤ ... ≤ tn−1 ≤ tn = t, converge over the interval partitions of (s, t] to an exponential vector of the form exp(k1(s,t]) where k ∈ K is a vector depending on b. (Cyclicity is, then, a simply consequence of Skeide’s proof in [13] of a result due to K.R. Parthasarathy and V.S. Sunder [9].) Immediately, from this construction, the idea emerged to construct an explicit isomorphism from the space of the abstract L´evyprocess of [11, Proposition 1.9.5] to the Fock space of the L´evyprocess obtained via [11, Theorem 2.5.3]. Namely, if in (2.4) we replace j and Ω with the abstract process ej and its cyclic vector Ω,e we know from [3, Theorem 1.21] that they converge. Sending the limit to exp(k1(s,t]) establishes a unitary from the abstract representation space De to the Fock space. If we can manage to do this without using [3, Theorem 1.21], then we will obtain a direct proof of representability of the L´evyprocess as a cyclic process on the Fock space. The idea for a transformation of a (cyclic) L´evyprocess originates in the fol- lowing observation. Let us put B1 := {b ∈ B : δ(b) = 1}. Suppose the element b ∈ B1 is group-like, that is, ∆(b) = b ⊗ b. (Note that b ∈ B being group-like, the counit property forces b = 0 or b ∈ B1.) Then ··· ··· jt0,t1 (b) jtn−1,tn (b) = jt0,t1 ? ? jtn−1,tn (b) = js,t(b) so that the limit is over a constant and gives back what js,t(b) does to the cyclic vector. In general, there need not be group-like elements in B1, and if, then they need not generate B. However, if we were able to define a different comultiplication on B for which all elements in B1 are group-like, then ··· ks,t(b)Ω = lim jt0,t1 (b) jtn−1,tn (b)Ω TRANSFORMATION OF QUANTUM LEVY´ PROCESSES ON HOPF ALGEBRAS 559 would define a family of homomorphisms ks,t that form a L´evyprocess with re- spect to the group-like comultiplication. In other words, we transformed one L´evy process into another. It is easy to give a direct realization of such a group-like process on a suitable Fock space; see Section 4.1. Thus, provided that the process k acts cyclicly on Ω, we would find the representation theorem. The easiest way to establish cyclicity is to reconstruct j from k by a reverse transformation. Recall that the construction of k involved replacing the original comultiplication with one that makes all b ∈ B1 ··· 0 ··· 0 into group-like elements so that jt0,t1 (b) jtn−1,tn (b) is nothing but jt0,t1 ? ? jtn−1,tn with respect to the new comultiplication. Now we do just the opposite and look at the limit of ··· kt0,t1 ? ? ktn−1,tn (b)Ω (2.5) for the original comultiplication. If this reverse transformation gives back j, then, knowing that the representation space of the intermediate group-like process k is isomorphic to a Fock space, we will know that also the representation space of j is a Fock space. Technically, in general, it is not possible to equip B directly with a comultiplication that makes the elements of B1 group-like. However, it is possible to associate with every ∗–bialgebra B its Weyl bialgebra CB1. The vector space CB1 contains the set B1 as a basis consisting entirely of group-like elements. And the ks,t(b)Ω defined on elements of B1 determine a unique L´evyprocess on CB1. But now the kst do no longer define a linear mapping a a e e B → L (D). (They do define a linear mapping CB1 → L (D) where D is the linear span in D of what the ks,t(b) generate from Ω.) So the in (2.5) with respect to the comultiplication of B do no longer have a meaning. The problem is solved if we associate again with B a special kind of ∗–bialgebra; see example 3.2. We will equip this tensor ∗–bialgebra with a certain comultiplication, so that the convolutions in (2.5) are defined with respect to this comultiplication.

3. Statement of Results and Applications In Section 3.1 we state the main result of this paper (Theorem 3.4) on the transformation of QLPs. We introduce two ∗–bialgebra structures on the tensor ∗–algebra over the kernel of the counit of a ∗–bialgebra. Moreover, we show that lifted generators give rise to QLPs on the tensor ∗–bialgebra which by restriction lead back to a version of the original QLP (Proposition 3.3). Section 3.2 is on the reversion of the transformation which is always possible if the transformation is surjective. In Section 3.3 we treat applications of our results to classical L´evy processes, to realizations of QLPs by their Weyl process and by their generator process, to the passage from the Wiener process to Az´emamartingales, and to Trotter product formulae for QLPs.

3.1. Transformation of QLPs. Let (B, ∆, δ) and (C, Λ, λ) be two ∗–bialgebras. A transformation of B is a unital ∗–algebra homomorphism κ : C → B satisfying δ ◦ κ = λ. (3.1) This means that κ preserves the counit. Since κ(1) = 1 it is easy to see that (3.1) is equivalent to the condition κ(C0) ⊂ B0 where C0 = ker λ, B0 = ker δ. In the 560 MICHAEL SCHURMANN,¨ MICHAEL SKEIDE, AND SILVIA VOLKWARDT sequel, if we have such a situation κ : C → B, we should warn the reader that we call B the first and C the second ∗–bialgebra. Example 3.1. (Generator Hopf algebra associated with a ∗–bialgebra) For a vector space V the (unital) T(V ) over V is the vector space M T(V ) = V ⊗n n∈N where V ⊗n denotes the n-fold tensor product of V with itself, V ⊗0 = C, with unit element (1, 0,... ) and the multiplication given by

(v1 ⊗ · · · ⊗ vn)(w1 ⊗ · · · ⊗ wm) = v1 ⊗ · · · ⊗ vn ⊗ w1 ⊗ · · · ⊗ wm for n, m ∈ N, v1, . . . , vn, w1, . . . , wm ∈ V . The tensor algebra satisfies the following universal property. There exists a vector space embedding ι: V → T(V ) of V into T(V ) such that for any linear mapping f from V into an algebra A there is a unique algebra homomorphism T(f): T(V ) → A such that T(f)◦ι(v) = f(v) for all v ∈ V . Then, any algebra homomorphisms g : T(V ) → A is uniquely determined by its restriction to V . In a similar way, an involution on V gives rise to a unique extension as an involution on T(V ). Thus, for a ∗–vector space V we can form the tensor ∗–algebra T(V ) over V . This ∗–algebra becomes a ∗–bialgebra if we extend the mappings Λ: V → T(V ) ⊗ T(V ), v 7→ v ⊗ 1 + 1 ⊗ v and T(0): V → C, v 7→ 0 as ∗–algebra homomorphisms to define the comultiplication and the counit on T(V ). The elements v ∈ V are so-called primitive elements of the coalgebra T(V ). By extending v 7→ −v as an algebra anti-homomorphism, we obtain an antipode so that T(V ) becomes a Hopf ∗–algebra. We call T(V ) the tensor Hopf ∗–algebra over V . Let (B, ∆, δ) be any ∗–bialgebra. The set B0 = {b ∈ B : δ(b) = 0} is a ∗–ideal of B. The tensor Hopf ∗–algebra (T(B0), Λ, T(0)) is called the generator Hopf algebra of B. We obtain a pair of ∗–bialgebras by taking for the first ∗–bialgebra B itself, and for the second one the generator Hopf algebra of B. The role of κ is played by the counit preserving ∗–algebra homomorphism κ defined by κ(b1⊗· · ·⊗bn) = b1 ··· bn for b1, . . . , bn ∈ B0. We call κ the multiplication map and denote it by M. Example 3.2. (Induced tensor ∗–bialgebra associated with a ∗–bialgebra) Let (B, ∆, δ) and (T(B0), Λ, T(0)) be as in example 3.1. We can define another coalgebra structure on T(B0). Denote by

E : B0 ⊕ B0 ⊕ (B0 ⊗ B0) → T(B0) ⊗ T(B0) the canonical embedding coming from the identifications of B0 with B0 ⊗ 1 and 1 ⊗ B0 respectively and of B0 ⊗ B0 ⊂ T(B0) ⊗ T(B0). Moreover, consider the restriction ∆0 of ∆ to B0. Then

∆0 : B0 → B0 ⊕ B0 ⊕ (B0 ⊗ B0) and (T(B0), T(E ◦ ∆0), T(0)) is a ∗–bialgebra. We can understand this ∗–bialgebra as a kind of big version of B and (T(B0), T(∆0), T(0)) is called the induced ten- sor ∗–bialgebra associated with B. In the context of the algebraic set-up the first ∗–bialgebra is (T(B0), Λ, T(0)) and the second ∗–bialgebra is (T(B0), T(∆0), T(0)). TRANSFORMATION OF QUANTUM LEVY´ PROCESSES ON HOPF ALGEBRAS 561

The identity on T(B0) is an example of a counit preserving ∗–algebra homomor- phism κ. The following result will frequently be needed. If ψ ∈ B0, B0 the algebraic of the vector space B, is a generator on B we have that ψ ◦ M is a generator on the induced tensor ∗–bialgebra T(B0). By the bialgebra property of B we have ∆ ◦ m = (m ⊗ m) ◦ (∆ ⊗ ∆) where m: B ⊗ B → B again denotes the multiplication of B. This implies

(ϕ1 ? ϕ2) ◦ M = (ϕ1 ◦ M) ? (ϕ2 ◦ M) (3.2) 0 for all ϕ1, ϕ2 ∈ B where the first ? in (3.2) is with respect to ∆ and the second with respect to T(∆0).

Proposition 3.3. Let ψ be a generator on the ∗–bialgebra B and let Js,t be a cyclic QLP on the induced ∗–bialgebra T(B0), over (DJ , Ω) with generator ψ ◦ M. Then the restriction of Js,t to B ⊂ T(B0) is a QLP on B with generator ψ.

Proof. We show that Js,t satisfies

Js,t(b1) ··· Js,t(bn) = Js,t(b1 ··· bn) (3.3) for all n ∈ N, b1, . . . , bn ∈ B0, 0 ≤ s ≤ t. By equation (3.2) we have for Φt(·) := hΩ,J0,t(·)Ωi

hΩ,Js,t(b1) ··· Js,t(bn)Ωi = Φt−s(b1 ⊗ · · · ⊗ bn) (t−s)ψ = e? (b1 ··· bn)

= hΩ,Js,t(b1 ··· bn)Ωi Using the properties of a QLP, we obtain from that

hζ, Js,t(b1) ··· Js,t(bn)ξi = hζ, Js,t(b1 ··· bn)ξi for all ζ, ξ ∈ DJ which proves (3.3).  A generator ψ of a L´evyprocess on B is lifted via κ to a generator ψ ◦ κ of a L´evyprocess on C. The question arises, what is the relationship between the two L´evyprocesses? We will show how the second process can be computed from the first one and vice versa. In the sequel, Zst denotes the set of all partitions of an interval [s, t] ⊂ R+. Let α = {s = t0 < t1 < ··· < tn−1 < tn = t} be a partition of [s, t] and define

kαk = max{tj+1 − tj | 0 ≤ j ≤ n − 1}.

We turn Zst into a directed set by writing α1 ≺ α2 :⇔ α1 ⊂ α2.

Theorem 3.4. Let (B, ∆, δ) be a ∗–bialgebra and let (js,t)0≤s≤t be the unique cyclic L´evyprocess over (Dj, Ω) whose convolution semigroup is given by a gener- ator ψ. Let (C, Λ, λ) be another ∗–bialgebra and let κ : C → B be a transformation of B. Denote by Hk the Hilbert subspace of Dj defined by n ◦ κ · ·· ◦ κ Hk := span (jt ,t )(c1) (jt − ,t )(cn)Ω: 0 1 n 1 n o n ∈ N, c1, . . . , cn ∈ C, 0 ≤ s ≤ t, s = t0 ≤ t1 ≤ · · · ≤ tn−1 ≤ tn = t . 562 MICHAEL SCHURMANN,¨ MICHAEL SKEIDE, AND SILVIA VOLKWARDT  Then for every c ∈ C and 0 ≤ s ≤ t the net ϑα(c) converges in norm to α∈Zst an element in Hk where ◦ κ ··· ◦ κ ϑα(c) = (jt0,t1 ) ? ? (jtn−1,tn )(c)Ω. (3.4) Moreover, setting

ks,t(c)Ω := lim ϑα (c) α  C determines a unique cyclic L´evyprocess k = ks,t 0≤s≤t on over a dense subspace (Dk, Ω) of Hk. The convolution semigroup of this process has generator ψ ◦ κ. The proof will be given in Section 4. We call k the transform of j. We formally will describe the construction of ks,t out of js,t in the above theorem by the short hand writing → Y? (js,t ◦ κ) = ks,t. (3.5) Λ We may call ks,t the infinitesimal convolution product of js,t ◦ κ. Remark 3.5. In more detail, Theorem 3.4 says that, under the assumptions of 2 e the theorem, the following L -type construction holds. If ks,t is a concrete cyclic L´evyprocess over (D,e Ω),e De a pre-Hilbert space with cyclic vector Ω,e on C with generator ψ ◦ κ, then e e ks,t(c)Ω 7→ ks,t(c)Ω defines a unitary mapping e U : Dk → D such that e ks,t(c) U = U ks,t(c) (3.6) for all c ∈ C. Notice that the formal writing (3.5) can always be given a mathe- e matical meaning by (3.6). Of course, when the process ks,t consists of bounded e e operators, equations (3.6) make sense on Hk and H where H denotes the Hilbert space which is the completion of De. However, boundedness does not always hold in the applications; cf. Examples 3.10, 3.12, 3.13. 3.2. Reversion of the transformation. The reverse transformation of a L´evy process on (C, Λ, λ) into a L´evyprocess on (B, ∆, δ) requires a counit preserving ∗–algebra homomorphism κe which, roughly speaking, is the inverse of κ. The construction of κe assumes the surjectivity of κ. This is equivalent to κ(C0) = B0 and the existence of an injective linear ∗–mapping

υ : B0 → C0 such that κ ◦ υ = idB. The linear ∗–mapping  υ is not unique. Its existence follows from the existence of a ∗ B ∈ C self-adjoint basis bi i∈I of the –vector space 0, I some index set. Choose ci self-adjoint such that κ(ci) = bi. This is possible since κ is surjective. Define the linear ∗–map υ by υ(bi) = ci. In view of the universal property of tensor algebras we extend the linear ∗–map υ to a ∗–algebra homomorphism

κe = T(υ): T(B0) → C TRANSFORMATION OF QUANTUM LEVY´ PROCESSES ON HOPF ALGEBRAS 563 to the induced tensor ∗–bialgebra T(B0). The coalgebra structure on T(B0) is that of the induced tensor ∗–algebra of Example 3.2. Indeed, the ∗–algebra homomor- phism κe preserves the counits. It is sufficient to show this for the generators of T(B0). For all b ∈ B0 we have ◦ ◦ κ ◦ ◦ λ υ(b) = δ υ(b) = δ idB0 (b) = 0 = δ(b). The above situation is described by κe κ (T(B0), T(∆0), T(0)) −→ (C, Λ, λ) −→ (B, ∆, δ). An application of Theorem 3.4 to this setting gives

Theorem 3.6. Let κ be a surjective transformation of the ∗–bialgebra B. For a B C generator ψ on let ks,t 0≤s≤t be a cyclic L´evyprocess on ( , Λ, λ) over (Dk, Ω) with generator ψ ◦ κ.  For every b ∈ B and 0 ≤ s ≤ t the net ζα converges in norm to an α∈Zst element in Dk where

ζ (b) := (k ◦ κe) ? ··· ? (k − ◦ κe)(b)Ω. α t0,t1 T(∆0) T(∆0) tn 1,tn Moreover, setting js,t(b)Ω := lim ζα (b) α  B determines a unique cyclic L´evyprocess j = js,t 0≤s≤t on over a dense subspace (Dj, Ω) of Dk. The convolution semigroup of this process has generator ψ. The following result says that the transform j in Theorem 3.6 is the reversion of the transform k in Theorem 3.4.

Theorem 3.7. Let κ be a surjective transformation of the ∗–bialgebraB. and let B ψ be a generator on . An application of Theorem 3.6 to the process ks,t 0≤s≤t of Theorem 3.4 gives back the original L´evyprocess on (B, ∆, δ). Moreover, we have Hk = Dj = Dk. For a proof of Theorems 3.6 and 3.7 see Section 4. 3.3. Applications. Example 3.8. (Group-like ∗–bialgebras) For a set S the vector space generated by S is the vector space CS consisting of all functions f : S → C with finite support. Assume in addition that S is a monoid with identity e ∈ S. Since S is a basis, the multiplication map S × S → S induces a map m: CS ⊗ CS → CS that turns CS into an algebra with identity element e ∈ S ⊂ CS. Since S is a basis of CS the mapping m induces an algebra structure on CS with unit element e. The vector space generated by a set satisfies the following universal property. There exists an embedding ι: S → CS such that any mapping φ from S to some vector space V can be uniquely extended to a linear mapping φe: CS → V such that φ = φe◦ι. This can be used to define a coalgebra structure on CS. We understand S as a set of group-like elements. We extend the mappings Λ: S → CS ⊗CS, Λ(s) = s ⊗ s and λ: S → C, λ(s) = 1 to linear mappings on CS. We will denote the 564 MICHAEL SCHURMANN,¨ MICHAEL SKEIDE, AND SILVIA VOLKWARDT comultiplication and the counit on CS again by Λ and λ. If S is a monoid Λ and λ are algebra homomorphism since Λ(xy) = xy ⊗ xy = (x ⊗ x)(y ⊗ y) = Λ(x)Λ(y) and λ(xy) = 1 = λ(x)λ(y) for all x, y ∈ S. An involution on S can also be uniquely extended to an involution on CS. Thus, for a ∗–monoid S we can form the group-like ∗–bialgebra (CS, Λ, λ) over S. Let (B, ∆, δ) be a ∗–bialgebra. The set B1 = {b ∈ B : δ(b) = 1} is a ∗–monoid with multiplication and involution of the ∗–algebra B. Hence, (CB1, Λ, λ) is a ∗–bialgebra which will be called the Weyl bialgebra of B; see also the next example. b In the sequel, we write b for the element b in B1 ⊂ CB1. The comultiplication b b b b Λ and the counit λ on CB1 are defined by Λ( b ) = b ⊗ b and λ( b ) = 1 for b b b ∈ C(B1). B1 is equal to the set of group-like elements in CB1, i.e. B1 = {0 =6 b ∈ b b b CB1 : Λ( b ) = b ⊗ b}. The situation is described by κe κ (T(B0), T(∆0), T(0)) −→ (CB1, Λ, λ) −→ (B, ∆, δ) where the counit preserving ∗–algebra homomorphisms κ and κe are defined by b [ b κ( b ) = b for b ∈ B1 and κe(b) = b + 1 − 1 for b ∈ B0. Now we are able to express the reverse transformation (2.5) by (k ◦ κe) ? ··· ? (k − ◦ κe)(b)Ω t0,t1 T(∆0) T(∆0) tn 1,tn for b ∈ B0. Example 3.9. (Construction of quantum L´evyprocesses I) We apply Theorem 3.4 to Example 3.8. Let (B, ∆, δ) be some ∗–bialgebra and let B js,t 0≤s≤t be a cyclic L´evyprocess on over (Dj, Ω) with generator ψ. In view of Theorem 3.4 we have that ˆ ks,t(b)Ω := lim jt0,t1 (b) . . . jtn−1,tn (b)Ω, α  ∈ B CB b 1, defines a cyclic L´evyprocess ks,t 0≤s≤t on the Weyl bialgebra ( 1, Λ, λ) of B over (Dk, Ω) where Dk is a linear subspace of Dj. Thus, for each pair ˆ ks,t(b)Ω, ks,t(ˆc)Ω for b, c ∈ B1 and 0 ≤ s ≤ t < ∞ we have ˆ (t−s)ψ(b∗c) hks,t(b)Ω, ks,t(ˆc)Ωi = e . The generator ψ defines a coboundary by (2.1). Thus, we compute −(t−s)ψ(b) ˆ −(t−s)ψ(c) (t−s)(−ψ(b∗)−ψ(c)+ψ(b∗c)) he ks,t(b)Ω, e ks,t(ˆc)Ωi = e = e(t−s)hη(b),η(c)i

= hE(η(b) ⊗ 1[s,t]),E(η(c) ⊗ 1[s,t])i where η : B1 → K is the canonical mapping to a dense linear subspace K of a Hilbert space K and E(η(·) ⊗ 1[s,t]) denotes the exponential vector of η(·) ⊗ 1[s,t] 2 in the Boson Fock space Γs(L ([s, t], K)). Here η(·) ⊗ 1[s,t] denotes the function in L2([s, t], K) which is a constant equal to η(·) on the interval [s, t] and 0 elsewhere. The space K is obtained from ψ by the GNS construction mentioned in Section 2. Hence, ∼ (t−s)ψ(b) 2 ks,t(b)Ω = e E(η(b) ⊗ 1[s,t]) ∈ Γs(L ([s, t], K)) where b ∈ B1, ψ(b) ∈ C and η(b) ∈ K. In other words, the vectors ks,t(b)Ω behave 2 like exponential vectors in the Boson Fock space Γs(L ([s, t], K)) and the ks,t(b) act TRANSFORMATION OF QUANTUM LEVY´ PROCESSES ON HOPF ALGEBRAS 565 on these exponential vectors like Weyl operators. Moreover, the vectors ks,t(b)Ω ‘generate’ the Hilbert subspace Dk of Dk where n s,t · ·· Dk = span kt ,t (c1) kt − ,t (cn)Ω: s,t 0 1 n 1 n o n ∈ N, s = t0 ≤ t1 ≤ · · · ≤ tn−1 ≤ tn = t, c1, . . . , cn ∈ C . ∼ 2 ∼ 2 R+ Therefore, we have Dks,t = Γs(L ([s, t], K)) and thus Dk = Γs(L ( , K)). The- ∈ B ⊂ orem 3.6 states that the vectors ks,t(b)Ω, b 1, are total in Djs,t Dj as well, i.e. ∼ 2 + Dj = Dk = Γs(L (R , K)). We proved that each cyclic quantum L´evyprocess on a ∗–bialgebra can be realized 2 + on a Boson Fock space Γs(L (R , K)). Example 3.10. (Construction of quantum L´evyprocesses II) In the situation of Example 3.9, an application of Theorem 3.6 allows to reconstruct js,t from the process ks,t on the Weyl bialgebra of B. The realization of the latter on Fock space can simply be written down. In the present example we describe a realization of a L´evyprocess on a Bose Fock space that parallels the construction in [4] and [11] with the help of quantum stochastic calculus in the sense of R.L. Hudson and K.R. Parthasarathy. Applying our result to the situation of Example 3.1 and 3.2 with κ = id there are two possibilities. If we put the first bialgebra B equal to the induced ∗–bialgebra and C equal to the generator Hopf algebra, then for b ∈ B0 we have Xn

ϑα(b) = jti−1,ti (b)Ω (3.7) i=1 and Theorem 3.4 tells us that 3.7 converges to  ∗ ∗ − Is,t(b)Ω = As,t(η(b )) + Λs,t(ρ(b)) + As,t(η(b)) + ψ(b)(t s) Ω ∗ in norm where As,t, Λs,t,As,t denote the annihilation, preservation and creation 2 operators of the interval [s, t] on Boson Fock space Γs(L (R+, K)); see the preced- ing section. For arbitrary b ∈ B we find ∗ − ∗ − Is,t(b) = δ(b)1 + As,t(η(b )) + Λs,t(ρ(b) δ(b)) + As,t(η(b)) + ψ(b)(t s).

Is,t is called the generator process of the L´evyprocess js,t. We may construct js,t out of Is,t if we take the generator Hopf algebra for the first bialgebra B and the induced one for the second bialgebra C. Then by Theorem 3.4 we obtain a QLP Js,t on T(B0) as the limit → Y? Js,t = T(Is,t − δ) T(∆0) of the convolution products of the generator process where now, of course, convo- lution is with respect to the original comultiplication ∆ of B. The limit is to be understood in the sense of the remark after Theorem 3.4; see equations (3.5) and (3.6). The QLP Js,t has the generator ψ ◦ M and by Proposition 3.3 the restriction of its cyclic version to B is a version of a QLP with generator ψ. So our procedure 566 MICHAEL SCHURMANN,¨ MICHAEL SKEIDE, AND SILVIA VOLKWARDT allows, like quantum stochastic calculus (see equation (2.2)), a construction of ∗ the L´evyprocess js,t from the elementary processes As,t, Λs,t,As,t on Boson Fock space. In fact, if dt is ‘small’, then in all relevant formula one may substitute jt,t+dt with It,t+dt. We find, in a heuristic sense,

js,t+dt − js,t = js,t ? jt,t+dt − js,t ' js,t ?It,t+dt − js,t = js,t ? (It,t+dt − δ1).

If we put dIt = Is,t+dt −Is,t (independent of s < t), this gives a heuristical meaning to Z t js,t = δ1 + js,r ? dIr s as a quantum stochastic integral. We remark that this interpretation as an integral is not limited to the above choice. Whenever k is a transformed process obtained from j via (3.5), we formally may write Z t ks,t = δ1 + ks,r ? (djt ◦ κ), s where djt := jt,t+dt − δ1. Example 3.11. (Classical L´evyprocesses and unitary evolutions) Let G be a topological group and denote by R(G) the space of all coefficient functions of continuous finite-dimensional representations of G. Then f ∈ R(G) iff there are n ∈ N and continuous complex-valued functions f1, . . . fn, g1, . . . gn on G such that Xn f(xy) = fi(x) gi(y) ∀x, y ∈ G. i=1 R(G) is a commutative ∗–algebra. By setting Xn ∆f = fi ⊗ gi, δf = f(e) i=1 R(G) becomes a commutative Hopf ∗–algebra. In various cases (e.g., when G is compact or locally compact abelian) the group G is uniquely determined by R(G). Let us assume that G is compact. Then R(G) is the Kreˆınalgebra of G. A classical L´evyprocess Xt on G gives rise to a quantum L´evyprocess jt on R(G) by putting jt(f) = f ◦ Xt. Here jt = j0.t and js,t = (js ◦ S) ? jt where S is the antipode of R(G). Let us specialize further to the case when G is the group Ud of unitary d × d-matrices. Then R(G) equals the Hopf ∗–algebra C ∗ ∗ [xkl, xkl; k, l = 1, . . . d] divided by the –ideal generated by the elements which are ∗− ∗ − the entries of the matrices x x 1 andPx x 1 where we put x = (xkl)k,l=1,...d. The d ⊗ comultiplication is given by ∆xkl = i=1 xki xil and the counit by δxkl = δkl. ∗ The antipode is given by S(xkl) = xlk. By replacing the commuting indeterminates xkl by non-commuting indeterminates, we define a non-commutative ∗–bialgebra Ch ∗ i ∗ ∗ xkl, xkl; k, l = 1, . . . d /x x = 1, x x = 1 which we denote by Uhdi; cf. [11]. (It is easy to see that Uhdi is not a Hopf algebra.) TRANSFORMATION OF QUANTUM LEVY´ PROCESSES ON HOPF ALGEBRAS 567

L´evytriples on Uhdi are given by a Hilbert space K, a unitary operator W on d C ⊗ K, a matrix L ∈ Md(C) ⊗ K and a self-adjoint matrix H ∈ Md(C) via the equations

ρ(xkl) = Wkl ∈ B(K)

η(xkl) = Lkl 1 ψ(x ) = − (LL∗) + i H ; kl 2 kl kl cf. [11] and [3]. The generator process (cf. the previous example) is given by 2 matrices Is,t ∈ Md(C) ⊗ Γ(L (R+, K)) with 1 (I ) = −A ((W ∗L) ) + Λ ((W − 1) ) + A∗ (L ) + (i H − (LL∗)) (t − s) s,t ij s,t ji s,t ij s,t ij 2 ij Theorem 3.4 says that

It0,t1 It1,t2 ...Itn−1,tn converges, again in the sense of the remark after Theorem 3.4, to the L´evyprocess d 2 Us,t which is the unitary process on C ⊗Γ(L (R+, K)) given by (Us,t)ij = js,t(xij). This is a generalization of a construction already given by W. von Waldenfels [16]. A classical L´evyprocess on Ud is a special case of a QLP on Uhdi. Example 3.12. (Az´emamartingales) Consider the ∗–algebra Chx, x∗, yi generated by x and a self-adjoint y. For q ∈ R divide Chx, x∗, yi by the ∗–ideal generated by the element xy − q yx to obtain a ∗–algebra A. On A we consider two ∗–bialgebra structures. The first is the one with x (and x∗) primitive and with y group-like, the second is given by ∆x = x ⊗ y + 1 ⊗ x and δx = 0 ∆y = y ⊗ y and δy = 1 and is called the Az´ema ∗–bialgebra of parameter q. Again we apply our results to these two ∗–bialgebras with κ = id. If we choose for generator  1 if M(x, x∗) = xx∗ ψ(M(x, x∗) yk) = 0 otherwise ∗ ∗ ∗ M(x, x ) ∈ A a monomial in x and x , k ∈ N0, then K = C, η(x ) = 1, η(x) = 0, ρ(x) = 0 and ρ(y) = q. The linear functional ψ is the generator of the quantum ∗ ∗ q-Az´emamartingale (Xt,Xt ,Yt) if we consider the Az´ema –bialgebra, and it ∗ generates the process (At,At ,Yt) in the case of the primitive/group-like structure of A where Yt is the second quantization of multiplication by q 1[0,t]. The process Xt satisfies the quantum stochastic differential equation

dXt = (q − 1)Xt dΛt + dAt,X0 = 0; see [8, 10]. An application of Theorem 3.4 yields the formula nX−1

Wt = lim Ztj ,tj+1 j=0 568 MICHAEL SCHURMANN,¨ MICHAEL SKEIDE, AND SILVIA VOLKWARDT with Z = lim W Y ...Y +W Y ...Y + ... t t0,t1 t1,t2 tn−1,tn t1t2 t2,t3 tn−1,tn  ··· + Wtn−2,tn−1 Ytn−1,tn + Wtn−1,tn where Wt and Zt denote the Wiener process and the q-Az´emamartingale on Boson Fock space respectively. Again convergence is in the sense of the remark after 3.4. Example 3.13. (Trotter formulae for QLPs) This example can be regarded as a motivation of the whole paper. Trotter formulae were considered by V. Liebscher and M. Skeide [5] and initiated the theory of transformations as presented in this paper. In the case when the initial space is finite dimensional, the following is a gener- alization to arbitrary ∗–bialgebras of a formula by J.M. Lindsay and K.B. Sinha [6] for unitary quantum stochastic processes on a Hilbert space which satisfy a quantum stochastic differential equation of the type (2.2) with constant bounded coefficients (see [4] and cf. Example 3.11). Let B be a ∗–bialgebra and let ψ1, ψ2 be two generators on B. Then it is immediate that ψ1 + ψ2 is again a generator on B. One would like to construct (1) (2) the process js,t given by ψ1 + ψ2 from the processes js,t and js,t given by ψ1 and ψ2 respectively. This can be done in the framework of transformation in at least three different ways. 1. For two ∗–bialgebras B1 and B2 we can form the tensor product B1 ⊗ B2 as the ∗–bialgebra with comultiplication coming from the coalgebra tensor product (cf. Section 2) and the usual ∗–algebra structure on B1 ⊗B2 (i.e. (b1 ⊗b2)(c1 ⊗c2) = ∗ ∗ ∗ b1c1 ⊗ b2c2 and (b ⊗ c) = b ⊗ c ). For generators ψ1, ψ2 on B1, B2 we form the generator e ψ := ψ1 ⊗ δ2 + δ1 ⊗ ψ2 ˜ ˜ on B1⊗B2. Indeed, ψ is hermitian with ψ(1) = 0, and if b ∈ B1⊗B2,(δ1⊗δ2)(b) = 0 ψe(˜b∗˜b) = (ψ ⊗ δ )(˜b∗˜b) + (δ ⊗ ψ )(˜b∗˜b) 1 2 1 2   ˜∗ ˜ ˜∗ ˜ = ψ1 (id ⊗δ2)(b )(id ⊗δ2)(b) + ψ2 (δ1 ⊗ id)(b )(δ1 ⊗ id)(b) ≥ 0 since ˜ ˜ δ1(id ⊗δ2)(b) = 0 = δ2(δ1 ⊗ id)(b) so that ψe is conditionally positve. (1) (2) B B For QLPs js,t , js,t on 1, 2 over (D1, Ω1), (D2, Ω2) with generators ψ1, ψ2 we e form the process js,t on B1 ⊗ B2 e ⊗ (1) ⊗ (2) js,t(b1 b2) = js,t (b1) js,t (b2). e Then js,t is a QLP on B1 ⊗ B2 over (D1 ⊗ D2, Ω1 ⊗ Ω2) and consists of two (1) (2) independent components js,t and js,t . Moreover, the convolution semigroup of e js,t is given by ⊗ (1) (2) ϕt(b1 b2) = ϕ1 (b1)ϕ2 (b2) TRANSFORMATION OF QUANTUM LEVY´ PROCESSES ON HOPF ALGEBRAS 569

∈ B ∈ B (1) (2) (1) (2) b1 1, b2 2, if ϕt , ϕt denote the convolution semigroups of js,t , js,t . e e e The generator of js,t is given by ψ, which shows once more that ψ is conditionally positive. In the case B1 = B2 = B we consider the induced tensor ∗–bialgebra (T(B0), T(∆0), T(0)) of B and define the ∗–algebra homomorphism κ : T(B0) → B ⊗ B by setting

κ(b) = b ⊗ 1 + 1 ⊗ b, b ∈ B0.

Then κ is a transformation of B ⊗ B and we have for b1, . . . , bn ∈ B0  (ψe ◦ κ)(b ⊗ · · · ⊗ b ) = ψe (b ⊗ 1 + 1 ⊗ b ) ··· (b ⊗ 1 + 1 ⊗ b ) 1 n X1 1 n n e (3.8) = ψ(bA ⊗ bAc ) A⊂{1,...,n} { ··· } { } ··· where for a subset i1 < < il of 1, . . . , n we put bA = bi1 bil , b∅ = 1. Since b1, . . . , bn are in the kernel of the counit δ we have that (3.8) is equal to e ψ1(b1 ··· bn) + ψ2(b1 ··· bn) and ψ ◦ κ = (ψ1 + ψ2) ◦ M. Denote by Js,t the QLP on T(B0) with generator (ψ1 + ψ2) ◦ M. By Proposition 3.3 the restriction js,t of Js,t to B ⊂ T(B0) is a QLP on B with generator ψ1 + ψ2. Equation (3.5) becomes (b ∈ B0)  (1) ⊗ ⊗ (2) − ⊗ ··· js,t(b) = lim jt0,t1 (b(1)) 1 + 1 jt0,t1 (b(1)) δ(b(1))1 1 α  ··· (1) ⊗ ⊗ (2) − ⊗ jt − ,t (b(n)) 1 + 1 jt − ,t (b(n)) δ(b(1))1 1 n 1 n n 1 n  (1) − ⊗ ⊗ (2) ··· = lim (jt0,t1 δ)(b(1)) 1 + 1 jt0,t1 (b(1)) α  ··· (1) − ⊗ ⊗ (2) (jt − ,t δ)(b(n)) 1 + 1 jt − ,t (b(n)) n 1 n n 1 n (1) ⊗ ⊗ (2) − ··· = lim jt0,t1 (b(1)) 1 + 1 (jt0,t1 δ)(b(1)) α  ··· (1) ⊗ ⊗ (2) − jtn−1,tn (b(n)) 1 + 1 (jtn−1,tn δ)(b(n)) . If we take equidistant partitions we obtain  h i (1) (2) − ?n Ω, js,t(b)Ω = lim ϕ t−s + ϕ t−s δ (b) n→∞ n n

(t−s)(ψ1+ψ2) = e? (b) which, of course, also follows from Theorem 3.4. 2. There is a ‘multiplicative’ version of the above construction. Define the ∗–algebra homomorphism β : CB1 → B ⊗ B by β(b) = b ⊗ b for b ∈ B1. We use the notation of Example 3.8. One proves that κ = κe ◦ e β : T(B0) → B ⊗ B is a transformation of B ⊗ B, and that ψ ◦ κ = ψ ◦ M. This time equation (3.5) can be written (b ∈ B1) X  (1) ⊗ (2) ··· (1) ⊗ (2) js,t(b) = lim λi jt ,t (b1,i) jt ,t (b1,i) jt − ,t (bn,i) jt − ,t (bn,i) α 0 1 0 1 n 1 n n 1 n P i ⊗ · · · ⊗ ∈ B ∈ C with ∆n b = i λi b1,i bn,i, bl,i 1, λi . 570 MICHAEL SCHURMANN,¨ MICHAEL SKEIDE, AND SILVIA VOLKWARDT

3. Formula (3.7) of [6] can also be obtained by transformation. In the trans- formation theorem 3.4 take for the second ∗–bialgebra C the original ∗–bialgebra B itself and for the ∗–bialgebra B in Theorem 3.4 take B ⊗ B. Then the comul- tiplication ∆: B → B ⊗ B is a transformation (notice that ∆ in general is not a coalgebra homomorphism!). Of course, ψe ◦ ∆ = ψ, and we obtain a realisation of our QLP on B with generator ψ as (1) ⊗ (2) ··· (1) ⊗ (2) js,t(b) = lim jt ,t (b(1)) jt ,t (b(2)) jt − ,t (b(2n−1)) jt − ,t (b(2n)). α 0 1 0 1 n 1 n n 1 n In the case of Example 3.11 this gives exactly the formula of [6] for a finite di- mensional initial space. If B is cocommutative (in which case ∆ is a coalgebra (1) ⊗ (2) ◦ homomorphism!) we obtain js,t = (js,t js,t ) ∆, and the Trotter formula be- comes trivial. The three constructions of a quantum L´evyprocess with generator ψ1 + ψ2 depend on the choice of the first bialgebra and the transformation κ. The most natural seems to be the one of 3. where κ is the comultiplication ∆ itself whereas in cases 1. and 2. the transformation κ does not depend on ∆. In 1. and 2. the two original processes are first put together in an additive and group-like way respectively. Case 3. is a ‘real’ Trotter formula for exponentials given by the comultiplication ∆.

4. Proof of Theorems In principle, Theorem 3.4 is proved if we show that the nets in (3.4) are Cauchy. To show this, in Section 4.1 we prove a lemma about infinitesimal products in Banach algebras (an extension of ideas in V. Liebscher and M. Skeide [5]) and a coalgebra version, appealing to the Fundamental Theorem on Coalgebras; see L. Accardi, M. Sch¨urmann,and W. von Waldenfels [1]. These lemmas plus the algebraic Proposition 4.3 allow to prove Proposition 4.4, which is the analytic heart of the proof of Theorem 3.4. 4.1. Preparatory lemmas. We start with a lemma that imitates, like in [5], proofs of the Trotter product formula.

Lemma 4.1. Let A be a Banach algebra. Suppose we have a constant R > 0 and (µ) a family A µ∈M of functions (M being some index set) r 7−→ A(µ) = I + rG + S(µ) ∈ A r r 2 R ∈ A (µ) (µ) ≤ 2 C on + where G and Sr satisfies Sr r 2 for some constant C not depending on µ ∈ M and all r ≤ R. Then for all intervals [s, t] ⊂ R+, all partitions α = {s = t0 < t1 < ··· < tn−1 < tn = t} (n ∈ N) of [s, t] with kαk ≤ R, and an arbitrary choice of elements µ1, . . . , µn of M, we have

2 k k2 kαk kGk (µ1) (µn) (t−s)G (t−s) max(kGk,C) C + G e A − ··· A − − e ≤ kαk (t−s)e . t1 t0 tn tn−1 2

2 (µk) ≤ k k 2 C ≤ r max(kGk,C) Proof. By assumption Ar 1 + r G + r 2 e , and thus

(µ ) (µ ) − k k A ` ··· A k ≤ e(tk t`−1) max( G ,C) t`−t`−1 tk−tk−1 TRANSFORMATION OF QUANTUM LEVY´ PROCESSES ON HOPF ALGEBRAS 571 for all intervals [s, t] ⊂ R+, all partitions αn of [s, t], and all 1 ≤ ` < k ≤ n. The next calculation (cf. [5] proof of Proposition 3.3) is essential for the proof. We compute (µ ) (µ ) − (µ ) (µ ) − − A 1 ··· A n − e(t s)G = A 1 ··· A n − e(t1 t0)G ··· e(tn tn−1)G t1−t0 tn−tn−1 t1−t0 tn−tn−1 Xn   (µ ) (µ − ) (µ ) − = A 1 ··· A j 1 A j − e(tj tj−1)G t1−t0 tj−1−tj−2 tj −tj−1 j=1 − − · e(tj+1 tj )G ··· e(tn tn−1)G. We have

(µj ) (tj −tj−1)G (µj ) A − − e ≤ A − − I − (tj − tj−1)G tj tj−1 tj tj−1 (tj −tj−1)G + I + (tj − tj−1)G − e 2 − k k 2 k k (tj tj−1) G 2 C + G e ≤ (t − t − ) . j j 1 2 From this estimate, from the estimate preceding it, and from Xn Xn 2 (tj − tj−1) ≤ kαk (tj − tj−1) = kαk (t − s) j=1 j=1 the statement follows.  There is a coalgebra version of Lemma 4.1 deduced from the Fundamental Theorem on Coalgebras which yields that the coalgebra generated by a finite subset of a coalgebra is finite dimensional. In the sequel, L(V,W ) denotes the vector space of linear maps between vector spaces V and W . We put L(V,V ) = L(V ). Let (C, ∆, δ) be a coalgebra and let ψ ∈ L(C, C) = C∗ be a linear functional on C. The map T : ψ 7→ (id ⊗ ψ) ◦ ∆ defines an injective unital algebra homomorphism from (L(C, C),?) to (L(C), ◦) with left inverse δ ◦ 1. Moreover, each T (ψ) leaves every sub-coalgebra of C invariant. On an arbitrary finite-dimensional subcoalgebra P∞ C C 3 C T (ψ)  C T (ψ) c c c of the series e c := n=0 n! converges in any norm. By the Fundamental Theorem on Coalgebras for every c ∈ C such a Cc exists. We deduce that the series ∞ X ψ?n eψ(c) := (c) = δ ◦ eT (ψ)(c) (4.1) ? n! n=0 converges for all ψ ∈ C∗ and all c ∈ C. Clearly, this limit of complex numbers cannot depend on the choice of Cc; see [1]. We now prove the coalgebra version of Lemma 4.1.

Lemma 4.2. Let C be a coalgebra. Suppose we have a constant R > 0 and a (µ) family f µ∈M of functions r 7−→ f (µ) = δ + rψ + R(µ) ∈ L(C, C) r r (µ) (µ) 2 on R+ where ψ ∈ L(C, C) and Rr (c) satisfies Rr (c) ≤ r Dc for some constant

Dc > 0, depending on c ∈ C but not on µ, and all r ≤ R. Then there exist constants Cc > 0 and Ψc > 0 such that for all intervals [s, t] ⊂ R+, all partitions 572 MICHAEL SCHURMANN,¨ MICHAEL SKEIDE, AND SILVIA VOLKWARDT

αn = {s = t0 < t1 < ··· < tn−1 < tn = t} (n ∈ N) of [s, t] with kαk ≤ R, and an arbitrary choice of elements µ , . . . , µ of M, we have 1 n − f (µ1) ? ··· ? f (µn) (c) − e(t s)ψ(c) t1−t0 tn−tn−1 ? 2 2 kαk Ψ − C + Ψ e c ≤ kαk (t − s)e(t s) max(Ψc,Cc) c c . 2

Proof. Choose b ∈ C and fix a finite-dimensional sub-coalgebra Cb of C containing b. Fix a norm on C . From the weak estimates R(µ)(c) ≤ r2D we easily con- b r c (µ) 2 clude the strong estimate Rr ≤ r D for a suitable constant D for the linear (µ) functionals Rr on Cb. (Just take your favorite elementary proof of the Uniform Boundedness Principle for finite-dimensional Banach spaces.) Consider the linear operator (µ) (µ)  C Ar := T (fr ) b on Cb, so (µ) (µ) Ar = I + rG + Sr (µ) (µ) where G := T (ψ)  Cb and Sr = T (Rr )  Cb. L(Cb) is a Banach algebra with respect to the operator norm. Since T is a C∗ C∗ ⊂ C bijection from b onto T ( b ) L( b), and since all norms on finite-dimensional 2 spaces are equivalent, S(µ) satisfies S(µ) ≤ r2 C for some constant C. In view r r 2 p k k k k of lemma 4.1p we obtain the claimed statement if we choose Cc = C δ c and Ψc = kGk kδk kck. 

4.2. Proof of Theorem 3.4. Consider the Hilbert subspaces (0 ≤ s ≤ t) n · ·· Hs,t = span jt ,t (b1) jt − ,t (bn)Ω: 0 1 n 1 n o n ∈ N, s = t0 ≤ t1 ≤ · · · ≤ tn = t, b1, . . . , bn ∈ B of Dj where H0 = C. Put Ht = H0,t. Using the shift and the unit vector Ω, we define mappings Us,t : Hs ⊗ Ht → Hs+t by ··· ⊗ ··· Us,t(js0,s1 (b1) jsn−1,sn (bn)Ω jt0,t1 (c1) jtm−1,tm (cm)Ω) ··· ··· = js0,s1 (b1) jsn−1,sn (bn)jt0+s,t1+s(c1) jtm−1+s,tm+s(cm)Ω where Us,t(Ω ⊗ Ω) = Ω and b1, . . . , bn, c1, . . . , cm ∈ B, n, m ∈ N. Indeed, the mappings Us,t are unitary. The shift is isometric and the unit vector Ω is cyclic which ensures surjectivity. Therefore, we may think of the family of Hilbert spaces (Ht)t≥0 as a tensor product system in the sense of W. Arveson [2]; see M. Skeide [14]. In fact, it is of type I which means that it comes from a Boson Fock space; see Examples 3.10 and 3.9. Let 0 ≤ s = t0 ≤ t1 ≤ · · · ≤ tn−1 ≤ tn = t. Using the unitary isomorphism ⊗ ⊗ · · · ⊗ ∼ Ht0,t1 Ht1,t2 Htn−1,tn = Hs,t, in the sequel, we identify ··· ⊗ · · · ⊗ jt0,t1 (b1) jtn−1,tn (bn)Ω = jt0,t1 (b1)Ω jtn−1,tn (bn)Ω. (4.2) TRANSFORMATION OF QUANTUM LEVY´ PROCESSES ON HOPF ALGEBRAS 573

In what follows we will often exploit in an essential way the coalgebra structure of B ⊗ C (see Section 2) and its interplay with expressions like (4.2). The following proposition expresses the core of all such computations. It’s proof is an easy verification and we omit it.

Proposition 4.3. Let (B, ∆, δ) and (C, Λ, λ) be coalgebras. Let Di (i = 1, 2) be two pre-Hilbert spaces and suppose we have linear mappings Ji : B → Di and Ki : C → Di. Define the linear functionals Li on the coalgebra B ⊗ C by setting

Li(b ⊗ c) := hJi(b),Ki(c)i and denote

J1 ?J2 := (J1 ⊗ J2) ◦ ∆: B −→ D1 ⊗ D2,

K1 ?K2 := (K1 ⊗ K2) ◦ Λ: C −→ D1 ⊗ D2. Then L1 ?L2(b ⊗ c) = hJ1 ?J2(b),K1 ?K2(c)i. Like in Proposition 4.3, in all what follows it is important to pay careful atten- tion to the different comultiplications of the coalgebras B, B, C, C, B ⊗C and C ⊗C which lead to different convolutions. Proposition 4.4. For c, d ∈ C and T > 0 there exists a C > 0 such that the following holds. For each [s, t] ⊂ [0,T ] and α ∈ Zst and for each β ∈ Zst finer than α we have (t−s)ψ◦κ ∗ hϑα(c), ϑβ(d)i − e? (c d) < kαk (t − s)C. (4.3)

Proof. Let the partitions α and β be given by

α = {s = s0 < s1 < ··· < sl = t} and β = {s = s = t(1) < t(1) < ··· < t(1) < t(1) = s 0 0 1 k1−1 k1 1 = t(2) < t(2) < ··· < t(2) < t(2) = s 0 1 k2−1 k2 2 . . = t(l) < t(l) < ··· < t(l) < t(l) = s = t}. 0 1 kl−1 kl l Denote further (n) (n) (n) (n) (n) α = {s − = t < t < ··· < t < t = s } n 1 0 1 kn−1 kn n for n = 1, . . . , l. For a pair of partitions γ, ζ of an interval [s, t] define the linear functionals Lγ,ζ on C ⊗ C by setting Lγ,ζ (c ⊗ d) := hϑγ (c), ϑζ (d)i. Then, by Proposition 4.3,

(1) (l) Lα,β = L{s0,s1},α ?...?L{sl−1,sl},α .

(n) In the expression for L{sn−1,sn},α we have ◦ κ ◦ κ jsn−1,sn (c) = (jt(n),t(n) ? . . . ? jt(n) ,t(n) ) (c) 0 1 kn−1 kn 574 MICHAEL SCHURMANN,¨ MICHAEL SKEIDE, AND SILVIA VOLKWARDT since, by assumption, (js,t)0≤s≤t is a L´evyprocess with respect to the comultipli- cation of B. If for a partition

γ = {s = t0 < t1 < ··· < tm = t} of an interval [s, t] we define the linear functionals Mγ on B ⊗ C by ⊗ h i Mγ (b c) := jt0,t1 ? . . . ? jtm−1,tm (b)Ω, ϑγ (c) then, again by Proposition 4.3,

L (n) (c ⊗ d) = M (n) (n) ?...?M (n) (n) (κ(c) ⊗ d). {sn−1,sn},α {t ,t } {t ,t } 0 1 kn kn−1

(n) ∈ k k (n) For ρ [0, α ] we define Lρ := L{sn−1,sn−1+ρ},α (ρ), where    (n) (n) α (ρ) := [sn−1, sn−1 + ρ] ∩ α ∪ sn−1 + ρ .

(n) (n) (Roughly speaking, if ρ ≤ sn − sn−1, then α (ρ) concides with the part of α up to sn−1 + ρ, and otherwise it adds another interval to the partition.) We define the linear functionals Mr := M{τ,τ+r} on B ⊗ C. Note that these do not depend on τ ≥ 0. We find

M (b ⊗ c) = M{ }(b ⊗ c) r τ,τ+r  ∗ = hjτ,τ+r(b)Ω, jτ,τ+r ◦ κ(c)Ωi = ϕr(b κ(c)) = (δ ⊗ λ) + rG + Rr (b ⊗ c), ∗ where G(b ⊗ c) := ψ(b κ(c)) and Rr fulfills the condition of Lemma 4.2. For fixed [s, t], it follows that for every c ⊗ d ∈ C ⊗ C there exists a constant C such that c,d

(n) ⊗ − ρG κ ⊗ ≤ k (n) k ≤ 2 Lρ (c d) e? ( (c) d) α (ρ) ρCc,d ρ Cc,d

(n) for all partitions α of [sn−1, sn]. (The constant Cc,d might depend on [s, t].) (n) From this it is routine to conclude that the Lρ fulfill the condition of Lemma 4.2 at least for all c⊗d ∈ C ⊗C with the linear first order functional c⊗d 7→ ψ◦κ(c∗d). By taking (finite!) linear combinations, we obtain suitable constants Dγ for every γ ∈ C ⊗ C. From this the statement follows.   Corollary 4.5. The net ϑα(c) is a Cauchy net. α∈Zst

Proof. We have to show that for ε > 0 there is a γ such that α, β ∈ Zst, α γ and β γ, implies kϑα (c) − ϑβ (c)k < ε . By Proposition 4.4 there is a γ such that for η ∈ Zst with η γ, we have 2 − ◦κ ε hϑ (c), ϑ (c)i − e(t s)ψ (c∗c) < . (4.4) γ η ? 16 So, for α γ we have 2 kϑη (c) − ϑα (c)k = hϑη (c), ϑη (c)i + hϑα (c), ϑα (c)i

− hϑη (c), ϑα (c)i − hϑα (c), ϑη (c)i ε2 ≤ . 4 TRANSFORMATION OF QUANTUM LEVY´ PROCESSES ON HOPF ALGEBRAS 575

Thus, for α γ and β γ

kϑα (c) − ϑβ (c)k ≤ kϑα (c) − ϑη (c)k + kϑβ (c) − ϑη (c)k ≤ ε which finishes the proof.   The limit of the Cauchy net ϑα(c) in Dj will be denoted by ϑs,t(c). α∈Zst Remark 4.6. Taking the limit of (4.3) over β α for fixed α, we find the same estimate for hϑα(c), ϑs,t(c)i. The fact that (4.3) does not depend on the precise form of α but only on its width kαk and computations similar to the proof of the corollary, show that kϑα(c) − ϑs,t(c)k is small, whenever kαk is sufficiently small. In particular, it follows that

lim ϑα (c) = ϑs,t(c) n→∞ n for each sequence αn in Zst with limn→∞ ||αn|| = 0. To conclude the proof of Theorem 3.4, we start by observing that

ϑs,t(c) = ϑt0,t1 ? . . . ? ϑtn−1,tn . (4.5)

(To see this, simply take the limit of ϑβ over the subnet of partions β α.) For α = (s = t < t < . . . < t − < t ) ∈ Z (0 ≤ s < t) we define 0 1 n n 1 n st o ⊗ ⊗ ∈ C Dkα := span ϑt0,t1 (c1) ... ϑtn−1,tn (cn): c1, . . . , cn . ⇒ ⊃ By (4.5), ϑs,t(Sc) = ϑt0,t1 ? . . . ? ϑtn−1,tn it follows β α = Dkβ Dkα . We 0 0 put D := D . Of course, [s , t ] ⊃ [s, t] =⇒ D 0 0 ⊃ D . We put ks,t S α kα ks ,t ks,t 3 Dkt,∞ := t≤r