Independence and L ́evy Processes in Quantum Probability 83It is easy to deduce from Subsection 1.7.1 that the multiplicative
boolean convolution onM 1 (T)is well defined. It is associative,
commutative,∗-weakly continuous in both arguments, but not
affine.
Theorem 1.7.39 [Fra08, Theorem 2.2] Let U and V be two unitary
operators on a Hilbert space H,Ω ∈ H a unit vector and assume,
furthermore, that U− 1 and V− 1 are boolean-independent w.r.t.Ω.
Then the products UV and VU are also unitary and their distribution
w.r.t. Ω is equal to the multiplicative boolean convolution of the
distributions of U and V, that is,
L(UV,Ω) =L(VU,Ω) =L(U,Ω)∪×L(V,Ω).1.8 The Five Universal Independences
In classical probability theory there exists only one canonical
notion of independence. However, in quantum probability many
different notions of independence have been used, e.g., to obtain
central limit theorems or to develop a quantum stochastic calculus.
If one requires that the joint law of two independent random
variables should be determined by their marginals, then an
independence gives rise to a product of algebraic probability
spaces. Imposing certain natural conditions, e.g., that functions of
independent random variables should again be independent and
that the underlying product is associative, it becomes possible to
classify all possible notions of independence. This program has
been carried out by Schurmann [Sch95a], Speicher [Spe97], Ben ̈
Ghorbal and Schurmann ̈ [BGS99][BGS02], and Muraki
[Mur03, Mur02]. See also the recent generalization by Gerhold and
Lachs [GL14]. If one imposes a normalization condition of the
correlations of length two, then there are exactly five universal
independences: tensor independence, freeness, boolean
independence, and monotone and anti-monotone independence.
Without this condition there are five one- or two- parameter
families,cf.[BGS02, GL14].
In this section we will present the results of these classifications.
Universal independences and universal products are of interest
for us, because it turns out that it is possible to define Levy ́
processes for them. However, since bialgebras are built on the
tensor products of algebras, they can only be used for Levy ́