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A simple proof of the classification theorem for positive natural products

Muraki N.

Probability and Mathematical Statistics

2013

Vol. 33, Fasc. 2

315--326

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.

Diagonalizability of non-homogenous quantum Markov states and associated von Neumann algebras

Fidaleo F., Mukhamedov F.

Probability and Mathematical Statistics

2004

Vol. 24, Fasc. 2

401--418

We give a constructive proof of the fact that any Markov state (even non-homogeneous) on [wzór] is diagonalizable. However, due to the local en-tanglement effects, they are not necessarily of Ising type (Theorem 3.2). In addition,we prove that the underlying classical measure is Markov, and therefore, in the faithful case, it naturally defines a nearest neighbour Hamiltonian. In the translation invariant case, we prove that the spectrum of the two-point block of this Hamiltonian, in some cases, uniquely determines the type of the von Neumann factor generated by the Markov state (Theorem 5.3). In particular, we prove that, if all the quotients of the differences of two such eigenvalues are rational, then this factor is of type IIIλ for some λ ∈ (0,1), and that, if this factor is of type III1, then these quotients cannot be all rational. We conjecture that the converses of these statements are also true.