Noncommutative Mathematics for Quantum Systems

54 Noncommutative Mathematics for Quantum Systems

Then (Tt)t≥ 0 is the quantum Markov semigroup of a (uniquely
determined) L ́evy process onPol(G)if and only if Tt is translation
invariant for all t≥ 0.

Proof If(Tt)t≥ 0 comes from a Levy process on Pol ́ (G), then there
exists a generating functionalφon Pol(G)such that the generator
of the semigroup(Tt)t≥ 0 isLφ(a) =φ?a(a∈Pol(G)) andTt =
exp(−tLφ)on Pol(G). ThenLφis translation invariant on Pol(G):

(id⊗Lφ)◦∆(a) =a( 1 )⊗(φ?a( 2 )) =a( 1 )⊗a( 2 )φ(a( 3 ))

=∆(a( 1 ))φ(a( 2 )) =∆◦Lφ(a).

Next, we observe that the powers of a translation invariant operator
are again translation invariant: if(id⊗Lφ)◦∆=∆◦Lφ, then, by
induction, we have

(id⊗Lnφ)◦∆= (id⊗Lφ)◦∆◦Lnφ−^1 =∆◦Lnφ.

Therefore,Tt = exp(−tLφ) = ∑n≥ 0 (−t)

n

n! L

n
φ is also translation-
invariant on Pol(G)for eacht≥0. By continuityTtis translation-
invariant onA.
Reciprocally, ifTtis translation invariant, then, by the previous
Lemma, it induces a linear mapT ̃tonCr(G) =A/Nhthat maps
Pol(G)to itself, and soφt:=ε◦T ̃tis well defined on Pol(G)(εis
defined on Pol(G), but may not extend toA). From the Markov
semigroup properties of (Tt)t≥ 0 we deduce that (φt)t≥ 0 is a
convolution semigroup of states on Pol(G). The generating
functional of this semigroup defines uniquely a Levy process on ́
Pol(G), and we have

Tt(a) = (id⊗φt)(a) +Nh, (1.6.11)

that is, the Markov semigroup(Tt)t≥ 0 is equal to convolution with
states from the convolution semigroup(φt)t≥ 0 , modulo elements of
the null space of the Haar state.

Remark 1.6.7 Equation (1.6.11) can be interpreted as saying that
the restriction of(Tt)t≥ 0 to Pol(G)coincidesh-almost everywhere
with the semigroup(Lφt)t≥ 0 of the associated L ́evy process. We do
not know if we can removeNhfrom Equation (1.6.11)

A closely related result for C∗-bialgebras with the counit,
satisfying the residual vanishing at infinity condition, was proved