Symmetries and ergodic properties in quantum probability
Volume 149 / 2017
Colloquium Mathematicum 149 (2017), 1-20
MSC: Primary 60G10, 46L53, 46L55, 37A30; Secondary 46L30, 46N50.
DOI: 10.4064/cm6863-9-2016
Published online: 27 March 2017
Abstract
We deal with the general structure of (noncommutative) stochastic processes by using the standard techniques of operator algebras. Any stochastic process is associated to a state on a universal object, i.e. the free product $C^*$-algebra, in a natural way. In this setting, one recovers the classical (i.e. commutative) probability scheme and many others, like those associated to the monotone, boolean and $q$-deformed canonical commutation relations including the Bose/Fermi and Boltzmann cases. Natural symmetries like stationarity and exchangeability, as well as the ergodic properties of the stochastic processes are reviewed in detail for many interesting cases arising from quantum physics and probability.
