Noncommutative Mathematics for Quantum Systems

Independence and L ́evy Processes in Quantum Probability 37

We will call a Schurmann triple ̈ surjective, if the cocycleη:B →D
is surjective.

Two surjective Schurmann triples ̈ (ρi,ηi,Li),i=1, 2, on a unital
∗-algebraBequipped with a characterε, defined over pre-Hilbert
spacesD 1 andD 2 , are said to beunitarily equivalent, ifL 1 =L 2 and
there exists a bijectionU:D 1 →D 2 such thatρ 2 (a)U=Uρ 1 (a)and
η 2 (a) =Uη 1 (a)for alla∈B. The conditionL 1 =L 2 guarantees that
Uis an isometry.
The functionals of two unitarily equivalent Schurmann triples ̈
have to coincide by definition, and, conversely, if the functionals
of two surjective Schurmann triples agree, then the triples have to ̈
be unitarily equivalent. This can be shown in the same way as the
uniqueness of the GNS construction.
Note that the bijectionU:D 1 →D 2 that implements the unitary
equivalence extends to a unitary operator between the completions
ofD 1 andD 2.

Theorem 1.5.10 LetBbe an involutive bialgebra. We have one-to-one
correspondences between L ́evy processes on B(modulo equivalence),
convolution semigroups of states onB, generating functionals onB, and
surjective Sch ̈urmann triples onB(modulo unitary equivalence).

Proof It only remains to establish the one-to-one correspondence
between generating functionals and Schurmann triples. ̈
Let(ρ,η,L)be a Schurmann triple, then we can show that ̈ Lis
a generating functional, that is, a hermitian, conditionally positive
linear functional withL( 1 ) =0.
The cocycle has to vanish on the unit element 1 , since

η( 1 ) =η( 1 · 1 ) =ρ( 1 )η( 1 ) +η( 1 )ε( 1 ) = 2 η( 1 ).

This implies

L( 1 ) =L( 1 · 1 ) =ε( 1 )L( 1 ) +〈η( 1 ),η( 1 )〉+L( 1 )ε( 1 ) = 2 L( 1 ) =0.

Furthermore, L is hermitian by definition and conditionally
positive, since by (1.5.3) we get

L(b∗b) =〈η(b),η(b)〉=‖η(b)‖^2 ≥ 0

forb∈kerε.