Seminars and Workshops

ABSTRACT

ABSTRACT
This seminar proposes a generalized Langevin equation for a classical mechanical system imbedded in a reservoir. Firstly, a finite number n of particles is considered in the reservoir and the action of this reservoir on the small system is described by a memory kernel and a zero-mean gaussian process of a very general nature, based on the gaussian transform introduced in [2]. This gives raise to an integrodifferential equation for the evolution of a generic particle in the main system. The existence of a unique solution X_n of the above equation is proved: it is also a gaussian process. A thermodynamics limit when n goes to infinity is considered and it is proved that X_n converges in distribution to the solution of a non-Markovian evolution equation. The above class of equations includes as a particular case the one studied by Kupferman in [1]. This report is based on a joint research with Carlos Lizama from the University of Santiago. [1] R. Kupferman. Fractional kinetics in Kac--Zwanzig heat bath models. Journal of Statistical Physics, Jan 2004. [2] C. Lizama and R. Rebolledo. Comm. on Stochastic Analysis, 4(4):541--551, 2010.

ABSTRACT

ABSTRACT
A natural characterization of the structure of positive maps will be present. Our arguments will be based on the concept of exposed points, links between tensor products and mapping spaces, and smoothness of unit balls in certain Banach spaces. The emphasize will be put on the geometrical characterization of the set of unital positive maps (given by the norm dual to the certain projective norm) and analytical aspects of smoothness of unit balls in certain Banach spaces. The last will play the crucial role in the characterization of exposed unital positive maps. These results seem to be an answer to old open problems studied both in Operator Algebras and Quantum Information.

ABSTRACT

ABSTRACT
We investigate some contractivity properties for quantum Markov semigroups defined on the algebra B(h) of bounded operators on a separable Hilbert space h. In particular we concentrate on a possible generalization to the non-commutative context of some well-established results for commutative functions, about the relations linking hypercontractivity and log-Sobolev inequalities with a uniform exponentially fast decay of the entropy.