We take some examples of *-bialgebras and give straightforward classifications of the possible L´evy processes on them and as a result which (LK) type properties each has. We specifically focus on the isometry *-bialgebras.
We make use of the following useful result which tells us that if the maps in our Sch¨urmann triple are well defined with respect to the relations on our algebra they are well defined everywhere.
Lemma 3.2.1 ([FKS16, Lemma 5.8]). Let A be a *-algebra generated by a col- lection of elements, a1, . . . , an, let be a character on A, and let (ρ, η, L) be a Sch¨urmann triple onA. LetB0 be the quotient ofAby the two-sided ideal generated
by the polynomial relations r1(a1, a∗1, . . . , an, a∗n) = 0, . . . , rk(a1, a∗1, . . . , an, a∗n) = 0 and their adjoints.
If ρ, and η vanish on r1, r1∗, . . . , rk, r∗k, then ρ is a representation of B0 andη
is aρ−-cocycle onB0. If, moreover,Lvanishes onr1, r∗1, . . . , rk, rk∗, then(ρ, η, L) is a Sch¨urmann triple on B0.
L´evy Processes on Isometry *-Bialgebras
In [Fra06, Section 2.1] a characterisation of L´evy processes on the unitary *- bialgebras which correspond to the deformed biunitary *-bialgebrasAd(I, I,0,0)
0
is given. A similar characterisation holds for the isometry *-bialgebras that is
I(d)0 :=Ad(I,0,0,0)0 ∼= * (uij)1≤i,j≤d; d X k=1 u∗kiukj =δi,j1 + .
This section is devoted to this characterisation and related results including the characterisation of L´evy processes on the bicyclic semigroup algebra.
The isometry algebras fit into the framework of Propositions 3.1.24 and 3.1.25 so we can use Hilbert spaces instead of pre-Hilbert spaces for our Sch¨urmann triples.
Theorem 3.2.2. Let H be a Hilbert space, V ∈ Md(B(H)) be an isometry, A ∈ Md(H)and λ∈Md(C)be Hermitian. Then there exists a unique Sch¨urmann triple on I(d)0 such that
ρ(uij) =PiV Pj∗, η(uij) =aij, and L(uij −u∗ji) = 2iλij.
For alli, j where Pi :H⊗Cd7→H⊗Cei, is the projection into thei-th copy of H. Furthermore, every Sch¨urmann triple on I(d)0 arises this way.
Proof. If we have a Sch¨urmann triple on I(d)0 we can easily see the existence of (V, A, λ) as above.
Let (V, A, λ) be as above. We can define maps on the free algebra with gener- ators (uij) by the values the linear maps take on the generators:
ρ(uij) = Vij, η(uij) = Aij and L(uij) =iλij − 1
2
X
k=1
hAki, Akji
and applying the necessary product and involutive rules:
ρ(a∗) = ρ(a)∗ and ρ(ab) = ρ(a)ρ(b), η(u∗ji) = − d X k=1 Vki∗Akj and η(ab) = ρ(a)η(b) +η(a)(b), L(a∗) = L(a) and L(ab) = (a)L(b) +hη(a∗), η(b)i+L(a)(b).
These product rules act associatively i.e. η(a(bc)) =η((ab)c). Therefore, these maps are well defined on the free algebra. The maps are constructed to be a Sch¨urmann triple on the free algebra. It is now only a matter to check that the
maps vanish on the relations of the algebra by Lemma 3.2.1 to get a Sch¨urmann triple on I(d)0.
TheVij satisfy the relations of the algebra, thereforeρis a unital *-homomorphism fromI(d)0 toB(H). Fix i and j
η X k (u∗kiukj−δij,1) ! =X k (ρ(u∗ki)η(ukj) +η(u∗ki)(ukj))−0 =X k Vki∗Akj − X l Vli∗Alkδk,j ! = 0 Note thatL(uij +u∗ji) = − P khAki, Akji L X k (u∗kiukj−δi,j) ! =X k
L(u∗ki)(ukj) +(u∗ki)L(ukj) +hη(uki), η(ukj)i
−0
=L(u∗ji+uij) +X
k
hAki, Akji
= 0
for all i, j. Therefore, (ρ, η, L) defines a Sch¨urmann triple on I(d)0.
Contained within the preceding proof is the fact the isometry *-bialgebras have the property (AC) and therefore (LK).
Definition 3.2.3. We call (V, A, λ) from Theorem 3.2.2 a L´evy process triple onI(d)0.
Definition 3.2.4. Given a Sch¨urmann triple (ρ, η, L) we refer to Ran(η) as the associated Hilbert space.
Proposition 3.2.5. Given a surjective Sch¨urmann triple (ρ, η, L) on I(d)0 with L´evy process triple (V, A, λ) the associated Hilbert space is equal to the closure of
K := Lin{Vθk ik,jk. . . V θ1 i1,j1A θ0 i0,j0;k ∈Z+, θl ∈ {1,∗}, il, jl ∈ {1, . . . , d}, l∈ {0, . . . , k}}.
Proof. It is clear to see that η(uij) = Aij and η(u∗ji) = −
P
kV
∗
kiAkj are elements
of K. Using the cocycle property we see that
η uθ1 iju θ0 kl =Vθ1 ij η u θ0 kl +η uθ1 ij =Vθ1 ij A θ0 kl +A θ1 ij ∈K
for i, j, k, l∈ {1, . . . d} and θ0, θ1 ∈ {1,∗}. Continuing inductively on word length and using linearity ofη we find that Ran(η) =K.
Example 3.2.6. Ifd= 1 the associated Hilbert space is given byK = Lin{VkA;k ∈
Z} where V−k := V∗k for all k ∈ N. For example if V = I then K ∼= C for any choice of A.
Proposition 3.2.7. Let (V1, A1, λ1) and (V2, A2, λ2) be L´evy process triples on
I(d)0 with associated Hilbert spaces K1 and K2 respectively. The existence of a
unitary operator U : K2 → K1 such that (V1)ij|K1 = U(V2)ij|K2U
∗, (A 1)ij = U(A2)ij and λ1 = λ2 is an equivalence relation between L´evy process triples on
I(d)0.
If such a unitary operator exists the associated generating functionals are iden- tical.
Proof. The first claim is straightforward, clearly every L´evy process triple onI(d)0 is related to itself this way by the existence of the identity operator. If T1 and T2 are L´evy process triples on I(d)0 related by the unitaryU then T2 is related toT1 by the unitaryU∗. The transitivity relation follows from the fact that products of unitary operators are themselves unitary.
For the second claim it is enough to check that L1(uij ±u∗ij) = L2(uij±u∗ij):
L1(uij −u∗ij) = 2i(λ1)ij L1(uij +u∗ij) =− X k h(A1)ki,(A1)kji = 2i(λ2)ij =− X k hU(A2)ki, U(A2)kji =L2(uij −u∗ij) =− X k h(A2)ki,(A2)kji =L2(uij +u∗ij).
What follows is the classification of certain types of L´evy processes on the isometry *-bialgebras as previously discussed.
Proposition 3.2.8. The triple (V, A, λ) corresponds to a drift L´evy process if and only if the associated Hilbert space is trivial i.e. H ={0}.
Proof. IfH ={0}thenη = 0 andV = id{0} = 0 and thereforeL(ab) =(a)L(b) +
L(a)(b). Conversely ifLis an-derivation thenL((a−(a)1)∗(a−(a)1)) = 0 for alla∈A and the associated Hilbert space is therefore trivial.
Proposition 3.2.9. The triple (V, A, λ) corresponds to a Gaussian L´evy process if and only if V is the identity operator.
Proof. This is straightforward:
(Vij) = (ρ(uij)IH) = IMd(B(H)) ⇐⇒ ρ(uij) = δi,jIH =(uij)IH.
Proposition 3.2.10. The triple (V, A, λ) corresponds to a Poisson L´evy process if and only if Aij =Vijw−δi,jw and λij = w, Vij −Vji∗ w 2i for somew∈H.
Proof. If L(a) = τ(φ(a) − (a)) for some state φ and τ > 0 then by a GNS construction we get a unital *-representationρto some Hilbert space H and some
w ∈ H such that φ(a) = hw, ρ(a)wi. As ρ is a unital *-homomorphism we get
ρ(uij) = Vij is in the necessary form. Finally simple calculations show thatη(uij) =
(Vij −δi,j1)w defines a cocycle that completes the Sch¨urmann triple.
LetAij =Vijw−δi,jw for somew∈H. Clearly η(uij) = (ρ(uij)−(uij)wand
η(u∗ij) =−X k Vkj∗η(uki) =− X k Vkj∗(Vkiw−δkiw) =Vij∗w−δi,jw.
If we assume η(a) = (ρ(a) −(a))w and η(b) = (ρ(b)−(b))w we can use the product rule onη to see that
η(ab) =ρ(a)(ρ(b)−(b)1)w+ (ρ(a)−(a)1)w(b) = (ρ(ab)−(ab)1)w.
From here we can use induction on word length and use linearity to show that
η(a) = (ρ(a)−(a))w for all a∈ I(d)0. Fix i and j then
L(uij) = w,(Vij −Vji∗)w−P khVkiw−δkiw, Vkjw−δkjwi 2 = w,(Vij −Vji∗)w−
w, δi,jw−Vji∗w−Vijw+δi,jw
2 =hw, Vijwi −δi,jkwk2.
In other wordsL(uij) =kwk2(φ(uij)−(uij)) whereφ is the state given byφ(a) =
hw, ρ(a)wi/kwk2. Similarly to before let a, b∈ A such that L(a) = kwk2(φ(a)−
(a)) andL(b) =kwk2(φ(b)−(b)). Note that
hη(a∗), η(b)i =hw,(ρ(a)−(a)I)(ρ(b)−(b)I)wi =hw, ρ(ab)wi −(a)hw, ρ(b)wi − hw, ρ(a)wi(b) +(ab)kwk2 =kwk2(φ(ab)−(a)φ(b)−φ(a)(b) +(ab)). Therefore L(ab) = L(a)(b) +(a)L(b) +hη(a∗), η(b)i =kwk2((a)φ(b)−(ab) +φ(a)(b)−(ab)) +hη(a∗), η(b)i =kwk2(φ(ab)−(ab)).
Gaussian Processes on the Universal Rotation Algebra
We consider the universal rotation *-bialgebra and characterise the Gaussian L´evy processes on it.
Example 3.2.11. LetA:=hU, V, Z unitary ;U V =ZV U, U Z =ZU, V Z =ZVi
where T ∈ A is unitary if T∗T =T T∗ = 1. This is referred to as the polynomial algebra of the universal rotation algebra. This algebra has a basis of the form (UnVmZp)
n,m,p∈Z where T
−k=T∗k for k >0 forT unitary.
Therefore, we can giveA the structure of a *-bialgebra by extending the maps
∆(U) =U ⊗U, ∆(V) =V ⊗V, ∆(Z) =Z ⊗Z
and (U) = (V) =(Z) = 1.
We apply Lemma 3.2.1 to characterise the Gaussian L´evy processes on this *-bialgebra.
Proposition 3.2.12. Let H be a Hilbert space, ηU, ηV ∈ H and λU, λV ∈R then there exists a unique Gaussian Sch¨urmann triple on A such that
η(U) =ηU η(V) = ηV
L(U −U∗) = 2iλU L(V −V∗) = 2iλV.
Furthermore, every Gaussian Sch¨urmann triple on A arises this way.
Proof. Let (ρ, η, L) be a Gaussian Sch¨urmann triple on A. It is clear that the relevant elements exist.
LetηU, ηV ∈H andλU, λV ∈R. Letρ(UnVmZp) = idH for alln, m, p∈Z, this
is trivially a unital *-homomorphism. Let η(U) =ηU and η(U∗) =−ηU, similarly forV and let η(Z) = η(Z∗) = 0. Then using the product rule we see that
and
η(U V) = η(U) +η(V) = η(Z) +η(V) +η(U) = η(ZV U).
The commutative relationsη(U Z) =η(ZU) andη(V Z) =η(ZV) are trivial in the Gaussian case. Similarly let L(U) = iλU− 1 2kηUk 2, L(V) =iλ V− 1 2kηVk 2 and L(Z) = hη V, ηUi−hηU, ηVi
Then using the product rule we see that
L(U∗U) =L(U∗) +L(U) +kηUk= 0 and L(V∗V) =L(V∗) +L(V) +kηVk= 0
and
L(Z∗Z) =L(Z) +L(Z∗) = 0
and similarly for the coisometry relations. The commutativity relations are satis- fied as follows L(U V) =L(U) +L(V)− hηU, ηVi =L(U) +L(V) +hηV, ηUi − hηU, ηVi − hηV, ηUi =L(U) +L(V) +L(Z) +hη(V∗), η(U)i+hη(Z∗), η(V)i+hη(Z∗), η(U)i =L(U) +L(V) +L(Z) +hη(Z∗), η(V U)i =L(ZV U).
The remaining commutativity relations are straightforward.
Contained within the preceding proof is the fact that the polynomial algebra of the universal rotation algebra has the property (GC) and therefore (LK).
of the universal rotation algebra are of the form L(UnVmZp) = i(nλU+mλV+p(2 imhηV, ηUi))− 1 2(n 2kη Uk+2nmhηU, ηVi+m2kηVk2)
where H is a Hilbert space with ηU, ηV ∈H and λU, λV ∈R.
Proof. First, let us show that
L(Un) = inλU −
1 2n
2kη
Uk
for all n∈N by induction. The base case is trivial. Assume the statement is true for somen =r then using the product rule of L
L(Ur+1) =L(U) +L(Ur) +hη(U∗), η(Ur)i =iλU− 1 2kηUk+irλU − 1 2r 2k
ηUk+h−ηU, rηUi
=i(r+ 1)λU −
1
2(r+ 1) 2kη
Uk.
Using the *-linear property ofL we then get L(U−n) = −inλ
U − 12n2kηUk for all n∈N. Therefore, L(Un) =inλ
U − 12n2kηUk for all n ∈Z. Similarly we can show L(Vn) =inλ
V −12n2kηVk andL(Zn) = 2inimhηV, ηUi. The result then follows by
a final application of the product rule on L.
It is easy to see that if hηU, ηVi ∈Rthen the presentation is simpler still:
L(UnVmZp) = i(nλU +mλV)−
1
2hnηU +mηV, nηU +mηVi.
L´evy Processes on the One Generator Free Inverse Monoid
Example 3.2.14. Consider the unital *-algebra with generatorpsuch thatpp∗p=
p. We refer to this as the semigroup algebra of the free inverse semigroup with one generator and identity. This has the *-bialgebra structure as given in Example 3.1.4.
Proposition 3.2.15. Let H be a Hilbert space, V ∈ B(H), h1, h2 ∈ H such that
V h2 =−V V∗h1 andλ∈R. There exists a unique Sch¨urmann triple(ρ, η, L)such
that
ρ(p) =V, η(p) = h1, η(p∗) =h2 and L(p−p∗) = 2iλ.
Furthermore every Sch¨urmann triple arises this way.
Proof. Let (ρ, η, L) be Sch¨urmann triple. The existence of appropriate V and λ
be as above is trivial. Let h1 =η(p) and h2 =η(p∗) then
0 =η(pp∗p)−η(p) = (V V∗h1 +V h2 +h1)−h1 =⇒ V h2 =−V V∗h1.
Let V, h1, h2 and λ as above. The mapping ρ(p) = V extends to a unital *-homomorphism. Using the product rule we have that
η(pp∗p) = V V∗h1+V h2+h1 =h1 =η(p)
and by Lemma 3.2.1 η is a cocycle on A0. Let L(p) = iλ − kh1k2+kh2k2−kV V∗h1k2
2 then note that V h2 = −V V ∗h 1 and (V V∗)2 =V V∗ and hV h2, h1i =− hV V∗h1, h1i =−kV V∗h1k2. L(pp∗p) = L(p) +L(p∗) +L(p) +hh2, h2i+hh1, h1i+hh2, V∗h1i =iλ− kh1k 2+kh 2k2− kV V∗h1k2 2 + (−iλ− kh1k2+kh2k2− kV V∗h1k2 2 ) +iλ− kh1k 2+kh 2k2− kV V∗h1k2 2 +kh2k 2+kh 1k2+hh2, V∗h1i =iλ− kh1k 2+kh 2k2−3kV V∗h1k2 + 2hV h2, h1i 2 =iλ− kh1k 2+kh 2k2− kV V∗h1k2 2 =L(p).
Therefore,L is well defined and completes the Sch¨urmann triple.
and h2 =−V∗h1. Contained within the preceding proof is the fact the semigroup of the free inverse semigroup with one generator and identity has the property (AC) and therefore (LK).