Independence and L ́evy Processes in Quantum Probability 31Definition 1.5.1 Let(A,Φ)be a quantum probability space andB
a∗-algebra. The quantum random variablesj 1 ,... ,jn:B → Aare
calledtensor or Bose-independent(w.r.t. the stateΦ), if
(i)Φ(
j 1 (b 1 )···jn(bn))
=Φ(
j 1 (b 1 ))
···Φ(
jn(bn))
for allb 1 ,... ,bn∈
B, and
(ii)[jl(b 1 ),jk(b 2 )] =0 for allk 6 =land allb 1 ,b 2 ∈B.
Recall that aninvolutive bialgebra(B,∆,ε)is a unital∗-algebraB
with two unital∗-homomorphisms∆ : B → B⊗B,ε :B → C
calledcoproductorcomultiplicationandcounit, satisfying
(id⊗∆)◦∆= (∆⊗id)◦∆ (coassociativity)
(id⊗ε)◦∆=id= (ε⊗id)◦∆ (counit property).Letj 1 ,j 2 :B → Abe two linear maps with values in some algebra
A, then we define theirconvolution j 1 ?j 2 by
j 1 ?j 2 =mA◦(j 1 ⊗j 2 )◦∆.HeremA:A⊗A→Adenotes the multiplication ofA,mA(a⊗b)=
abfora,b∈A.
We shall frequently useSweedler notationfor the coproduct of an
elementb ∈ B, that is, omit the summation and the index in the
formula∆(b) =∑ib( 1 ),i⊗b( 2 ),iand write simply∆(b) =b( 1 )⊗b( 2 ).
With this notation we write(j 1 ?j 2 )(b) = j 1 (b( 1 ))j 2 (b( 2 )). Ifj 1
andj 2 are two independent quantum random variables, thenj 1 ?j 2
is again a quantum random variable, that is, a∗-homomorphism.
The fact that we can compose quantum random variables allows us
to define Levy processes, that is, processes with independent and ́
stationary increments.
Example 1.5.2 Let(G,e)be a semigroup with identitye. We call a
functionf:G→Crepresentative, if there exists a finite-dimensional
representationπ:G→Mnand vectorsu,v∈Cnsuch thatf(g) =
〈v,π(g)u〉for allg∈G. These functions are sometimes also called
coefficient functions. The algebra
R(G) ={f:G→Ca representative function}is an involutive bialgebra with pointwise multiplication and
conjugation, and the coproduct∆ : R(G) → R(G)⊗R(G) ∼=
R(G×G)and counitε:R(G)→Cdefined by