A SIMPLE PROOF OF THE CLASSIFICATION THEOREM FOR POSITIVE
NATURAL PRODUCTS
Abstract: A simplification of the proof of the classification theorem for natural notions of
stochastic independence is given. This simplification is made possible after adding the
positivity condition to the algebraic axioms for a (non-symmetric) universal product (i.e. a
natural product). Indeed, this simplification is nothing but a simplification, under the
positivity, of the proof of the claim that, for any natural product, the ‘wrong-ordered’
coefficients all vanish in the expansion form. The known proof of this claim involves a
cumbersome process of solving a system of quadratic equations in 102 unknowns, but in our
new proof under the positivity we can avoid such a process.
2000 AMS Mathematics Subject Classification: Primary: 46L53; Secondary: 60A05,
81S25.
Keywords and phrases: Non-commutative probability, quantum probability,
independence, universal products, natural products.