Independence and L ́evy Processes in Quantum Probability 97(φ 1 ·φ 2 )(a 1 a 2 ) = c 1 φ 1 (a 1 )φ 2 (a 2 ), (1.8.5)
(φ 1 ·φ 2 )(a 2 a 1 ) = c 2 φ 1 (a 1 )φ 2 (a 2 ), (1.8.6)for all linear functionalsφ 1 :A 1 →C,φ 2 :A 2 →C, and elements
a 1 ∈ A 1 ,a 2 ∈ A 2 with ‘universal scaling constants’c 1 ,c 2 ∈C, that
is, constants that do not depend on the algebras, the functionals, or
the algebra elements. That for every universal independence such
constants have to exist is part of the proof of the classifications in
[BG01, BGS99, Mur03].
Show that if the products of states are again states, then we have
c 1 =c 2 =1. Hint: Take forA 1 andA 2 the algebra of polynomials
onRand forφ 1 andφ 2 evaluation in a point.
Even without symmetry one can show that there always exist
constantsc 1 ,c 2 as in Equations (1.8.5) and (1.8.6). If one imposes
positivity, then it follows thatc 1 =c 2 =1, as in the Exercise above.
In general, if these two constants are equal and not zero, one has
the families
φ 1 • qφ 2 =q(
(q−^1 φ 1 )·(q−^1 φ 2 ))
,parametrized by a complex numberq∈C{ 0 }, for each of the five
products, • ∈ {⊗ ̃,∗,,.,/}, similar to the situation in the
symmetric case described in Equation (1.8.4). If the constants are
not equal, a new family of products, called(r,s)-products, arise,
see [GL14, Ger15, Lac15]. It is conjectured that the degenerate
product in Equation (1.8.3) is the only possibility in the case
c 1 =c 2 =0.
The proof of the classification of universal independences can be
split into three steps.
Using the ‘universality’ or functoriality of the product, one can
show that there exist some ‘universal combinatorial constants’ - not
depending on the algebras - and a formula for evaluating
(φ 1 ·φ 2 )(a 1 a 2 ···am)fora 1 a 2 ···am∈ A 1 ‰A 2 , withak∈ Aek,e 16 =e 26 =··· 6=em, as
a linear combination of productsφ 1 (M 1 ),φ 2 (M 2 ), whereM 1 ,M 2
are ‘sub-monomials’ ofa 1 a 2 ···am. Then, in the second step, it is
shown by associativity that only products withorderedmonomials
M 1 ,M 2 contribute. This is the content of [BGS02, Theorem 5] in the
commutative case and of [Mur03, Theorem 2.1] in the general case.