Diagonalizability of non homogeneous quantum Markov states and associated von Neumann algebras

2004/11/09 by Francesco Fidaleo, Fidaleo, Francesco, Farruh Mukhamedov +1 · 1 citation
Mathematics · Physics and Astronomy · #46L35 #46L50 #60J99 #82A15 #82B20 #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Operator Algebras (math.OA) #Spectral Theory in Mathematical Physics #math-ph #math.MP #math.OA #msc:46L35 #msc:46L50 #msc:60J99 #msc:82A15 #msc:82B20

paper · pdf · doi:10.48550/arxiv.math/0411200

22 pages

arxiv created 2004/11/09 · openalex publication_date 2004/11/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We clarify the meaning of diagonalizability of quantum Markov states. Then, we prove that each non homogeneous quantum Markov state is diagonalizable. Namely, for each Markov state ϕ on the spin algebra A:=⊗_{j∈ Z}M_d_j^C^* there exists a suitable maximal Abelian subalgebra D⊂ A, a Umegaki conditional expectation E:A↦ D and a Markov measure μ on spec(D) such that ϕ=ϕ_μ∘ E, the Markov state ϕ_μ, being the state on D arising from the measure μ. An analogous result is true for non homogeneous quantum processes based on the forward or the backward chain. Besides, we determine the type of the von Neumann factors generated by GNS representation associated with translation invariant or periodic quantum Markov states.

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• On factors associated with quantum Markov states corresponding to nearest neighbor models on a Cayley tree

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