Noncommutative Mathematics for Quantum Systems

Independence and L ́evy Processes in Quantum Probability 47

(i) fz( 1 ) =1 forz∈C,

(ii) the mappingC 3 z7→ fz(a) ∈Cis an entire holomorphic

function for alla∈Pol(G),

(iii) f 0 (z) =εandfz 1 ?fz 2 =fz 1 +z 2 for anyz 1 ,z 2 ∈C,

(iv) fz(S(a)) = f−z(a)andfz ̄(a∗) = f−z(a)for anyz∈C,a ∈

Pol(G),

(v)S^2 (a) =f− 1 ?a?f 1 fora∈Pol(G),

(vi) the Haar statehsatisfies:

h(ab) =h(b(f 1 ?a?f 1 )), a,b∈Pol(G).

In this case the formula

σt(a) =fit?a?fit, t≥0, (1.6.4)

defines a one-parameter group of modular automorphisms of
Pol(G) and h is the (σ,− 1 )-KMS state, which means that it
satisfies

h(ab) =h(bσ−i(a)), a,b∈Pol(G), (1.6.5)

cf. [BR97, Definition 5.3.1]. Such a state isσ-invariant, that is,
h(σt(a)) =h(a)fora∈Pol(G)andt≥0 (see [BR97, Proposition
5.3.3]).
The matrix elements of the irreducible unitary corepresentations
satisfy the famous generalized Peter–Weyl orthogonality relations

h

((

u(ijs)

)∗

u(kt`)

)

=

δstδj`f− 1

(

u(kis)

)

Ds

, h

(

u(ijs)

(

u(kt`)

)∗)

=

δstδikf 1

(

u(`sj)

)

Ds

,

(1.6.6)

wheref 1 : Pol(G)→Cis the Woronowicz character and

Ds=

ns

∑

`= 1

f 1

(

u(``s)

)

is the quantum dimension ofu(s),cf.[Wor87a, Theorem 5.7.4]. Note
that unitarity implies that the matrix
(
f 1

(

(u(jks))∗

))

1 ≤j,k≤ns

∈Mns(C)