Chain rules for canonical state extensions on von Neumann algebras

• Autorzy: Carlo Cecchini , Dénes Petz
• Języki publikacji: EN

Abstrakty

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.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 1993
• Tom: 64
• Numer: 1
• Strony: 115-119

Daty

wydano

1993

otrzymano

1992-01-20

Twórcy

autor

Carlo Cecchini

• Dipartimento di Matematica e Informatica, Università di Udine, via Zanon 6, 33100 Udine, Italy

autor

Dénes Petz

• Mathematical Institute, Hungarian Academy of Sciences, 1364 Budapest, PF. 127, Hungary

Bibliografia

• [1] L. Accardi and C. Cecchini, Conditional expectations in von Neumann algebras and a theorem of Takesaki, J. Funct. Anal. 45 (1982), 245-273.
• [2] C. Cecchini and D. Petz, State extension and a Radon-Nikodym theorem for conditional expectations on von Neumann algebras, Pacific J. Math. 138 (1989), 9-24.
• [3] C. Cecchini and D. Petz, Classes of conditional expectations over von Neumann algebras, J. Funct. Anal. 92 (1990), 8-29.
• [4] A. Connes, Sur le théorème de Radon-Nikodym pour les poids normaux fidèles semifinis, Bull. Sci. Math. Sect. II 97 (1973), 253-258.
• [5] A. Connes, On a spatial theory of von Neumann algebras, J. Funct. Anal. 35 (1980), 153-164.
• [6] D. Petz, Sufficient subalgebras and the relative entropy of states of a von Neumann algebra, Comm. Math. Phys. 105 (1986), 123-131.