Noncommutative Mathematics for Quantum Systems

Independence and L ́evy Processes in Quantum Probability 51

where φbt(x) := φt(b∗xb). Moreover, by the antipode property
(1.6.2) we have

UU∗(λ(a)ξ⊗ρt(b)ξt) = U(λ(a( 1 ))ξ⊗ρt(S(a( 2 ))b)ξt)

= λ(a( 1 ))ξ⊗ρt(a( 2 )S(a( 3 ))b)ξt

= λ(a( 1 )ε(a( 2 )))ξ⊗ρt(b)ξt

= λ(a)ξ⊗ρt(b)ξt,

which implies thatUis an isometry with dense image and therefore
extends to a unique unitary operator denoted again byU.
Now the fact that the Markov semigroup(Tt)tis bounded on
Cr(G), that is,

‖Tt(a)‖r=‖λ(Tt(a))‖B(H)≤‖λ(a)‖B(H)=‖a‖r,

follows immediately from the relation

λ(Tt(a)) =Et(U(λ(a)⊗idHt)U∗), (1.6.10)

since

‖λ(Tt(a))‖ = ‖Et(U(λ(a)⊗idHt)U∗)‖≤‖U(λ(a)⊗idHt)U∗‖

= ‖λ(a)⊗idHt‖=‖λ(a)‖.

To see that (1.6.10) holds, let us fixv∈ Handb∈Pol(G)such
thatv=λ(b)ξ. Then,

Et(U(λ(a)⊗idHt)U∗)v

= (πt◦U◦(λ(a)⊗idHt)◦U∗◦it)(λ(b)ξ)

= (πt◦U◦(λ(a)⊗idHt)◦U∗) (λ(b)ξ⊗ξt)

= πt◦U◦(λ(a)⊗idHt)

(

λ(b( 1 ))ξ⊗ρt(S(b( 2 )))ξt

)

= πt◦U

(

λ(ab( 1 ))ξ⊗ρt(S(b( 2 )))ξt

)

= πt

(

λ(a( 1 )b( 1 ))ξ⊗ρt(a( 2 )b( 2 )S(b( 3 )))ξt

)

= πt

(

λ(a( 1 )b)ξ⊗ρt(a( 2 ))ξt

)

= 〈ξt,ρt(a( 2 ))ξt〉λ(a( 1 )b)ξ

= λ(a( 1 )φt(a( 2 )))λ(b)ξ=λ(Tt(a))v.

This way we showed that eachTtextends to a contraction onCr(G).
The extensions form again a semigroup and since both∆andφtare