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Markovian processes on mutually commuting von Neumann algebras

100%
Cecchini C.
Banach Center Publications
|
1998
|
tom 43
|
nr 1
111-118
EN
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 situation 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
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Canonical functional extensions on von Neumann algebras

100%
Cecchini C.
Studia Mathematica
|
1999
|
tom 135
|
nr 1
13-24
EN
The topology and the structure of the set of the canonical extensions of positive, weakly continuous functionals from a von Neumann subalgebra $M_0$ to a von Neumann algebra M are described.
3
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Chain rules for canonical state extensions on von Neumann algebras

63%
Cecchini C., Petz D.
Colloquium Mathematicum
|
1993
|
tom 64
|
nr 1
115-119
EN
In previous papers we introduced and studied the extension of a state defined on a von Neumann subalgebra to the whole of the von Neumann algebra with respect to a given state. This was done by using the standard form of von Neumann algebras. In the case of the existence of a norm one projection from the algebra to the subalgebra preserving the given state our construction is simply equivalent to taking the composition with the norm one projection. In this paper we study couples of von Neumann subalgebras in connection with the state extension. We establish some results on the ω-conditional expectation and give a necessary and sufficient condition for the chain rule of our state extension to be true.
4
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Operators on VN(G) commuting with A(G)

32%
Cecchini C.
Colloquium Mathematicum
|
1980
|
tom 43
|
nr 1
137-142
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