A generalization of the Pistone-Sempi argument, demonstrating the utility of non-commutative Orlicz spaces, is presented. The question of lifting positive maps defined on von Neumann algebra to maps on corresponding noncommutative Orlicz spaces is discussed. In particular, we describe those Jordan * -morphisms on semifinite von Neumann algebras which in a canonical way induce quantum composition operators on noncommutative Orlicz spaces. Consequently, it is proved that the framework of noncommutative Orlicz spaces is well suited for an analysis of a large class of interesting noncommutative dynamical systems.
Abstract. We first use properties of the Fuglede-Kadison determinant on L p (M ), for a finite von Neumann algebra M , to give several useful variants of the noncommutative Szegö theorem for L p (M ), including the one usually attributed to Kolmogorov and Krein. As an application, we solve the longstanding open problem concerning the noncommutative generalization, to Arveson's noncommutative H p spaces, of the famous 'outer factorization' of functions f with log |f | integrable. Using the Fuglede-Kadison determinant, we also generalize many other classical results concerning outer functions.
We present a new rigorous approach based on Orlicz spaces for the description of the statistics of large regular statistical systems, both classical and quantum. This approach has the advantage that statistical mechanics is much better settled. In particular, a new kind of renormalization leading to states having a well defined entropy function is presented.
Abstract. We generalize, to the setting of Arveson's maximal subdiagonal subalgebras of finite von Neumann algebras, the Szegő L p -distance estimate and classical theorems of F. and M. Riesz, Gleason and Whitney, and Kolmogorov. As a byproduct, this completes the noncommutative analog of the famous cycle of theorems characterizing the function algebraic generalizations of H ∞ from the 1960's. A sample of our other results: we prove a Kaplansky density result for a large class of these algebras, and give a necessary condition for every completely contractive homomorphism on a unital subalgebra of a C * -algebra to have a unique completely positive extension.
Abstract. We generalize some facts about function algebras to operator algebras, using the "noncommutative Shilov boundary" or "C * -envelope" first considered by Arveson. In the first part we study and characterize complete isometries between operator algebras. In the second part we introduce and study a notion of logmodularity for operator algebras. We also give a result on conditional expectations. Many miscellaneous applications are provided.
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