Markovian processes on mutually commuting von Neumann algebras

Carlo Cecchini
1998 Banach Center Publications  
1. The aim of this paper is to study markovianity for states on von Neumann algebras generated by the union of (not necessarily commutative) von Neumann subagebras which commute with each other. This study has been already begun in [2] using several a priori different notions of noncommutative markovianity. In this paper we assume to deal with the particular case of states which define odd stochastic couplings (as developed in [3] ) for all couples of von Neumann algebras involved. In this
more » ... tion these definitions are equivalent, and in this case it is possible to get the full noncommutative generalization of the basic classical Markov theory results. In particular we get a correspondence theorem, and an explicit structure theorem for Markov states. 2. Let M be a von Neumann algebra acting on an Hilbert space H. For ξ in H we denote by ω ξ the vector state on B(H) implemented by ξ. In order to simplify our notations we shall often write (ω ξ ) M for ω ξ | M or simply (ω ξ ) α if the von Neumann algebra involved is endowed with an index α. We shall say C is a self-dual positive cone for M in H if there is a separating vector Ω for M in H, such that C is the selfdual positive cone for EM E in EH (in the sense of the modular theory for von Neumann algebras) which contains Ω, with E the orthogonal projection from M to the closure of {aΩ, a ∈ M }. Let γ be an index, M γ be a von Neumann algebra acting on a Hilbert space H, and let Ω be a vector in H which is separating for M γ . We shall denote by H γ the closure of {aΩ, a in M γ }, by E γ the orthogonal projection from H to H γ , and with the usual notations endow with an index γ the objects of the modular theory for the action of
doi:10.4064/-43-1-111-118 fatcat:mkgmk7mg3vc5dgnap4yp4epata