Strict weak mixing of some C*-dynamical systems based on free shifts

We define a stronger property than unique ergodicity with respect to the fixed-point subalgebra previously investigated by Abadie and Dykema. Such a property is denoted as F-strict weak mixing (F stands for the Markov projection onto the fixed-point operator system). Then we show that the free shifts on the reduced C*-algebras of RD-groups, including the free group on infinitely many generators, and amalgamated free product C*-algebras, considered by Abadie and Dykema, are all strictly weak mixing and not merely uniquely ergodic.Comment: 10 page

Quantum Hypothesis Testing and Non-Equilibrium Statistical Mechanics

We extend the mathematical theory of quantum hypothesis testing to the general $W^*$-algebraic setting and explore its relation with recent developments in non-equilibrium quantum statistical mechanics. In particular, we relate the large deviation principle for the full counting statistics of entropy flow to quantum hypothesis testing of the arrow of time.Comment: 60 page

Compact Hypergroups from Discrete Subfactors

Conformal inclusions of chiral conformal field theories, or more generally inclusions of quantum field theories, are described in the von Neumann algebraic setting by nets of subfactors, possibly with infinite Jones index if one takes non-rational theories into account. With this situation in mind, we study in a purely subfactor theoretical context a certain class of braided discrete subfactors with an additional commutativity constraint, that we call locality, and which corresponds to the commutation relations between field operators at space-like distance in quantum field theory. Examples of subfactors of this type come from taking a minimal action of a compact group on a factor and considering the fixed point subalgebra. We show that to every irreducible local discrete subfactor $\mathcal{N}\subset\mathcal{M}$ of type ${I\!I\!I}$ there is an associated canonical compact hypergroup (an invariant for the subfactor) which acts on $\mathcal{M}$ by unital completely positive (ucp) maps and which gives $\mathcal{N}$ as fixed points. To show this, we establish a duality pairing between the set of all $\mathcal{N}$-bimodular ucp maps on $\mathcal{M}$ and a certain commutative unital $C^*$-algebra, whose spectrum we identify with the compact hypergroup. If the subfactor has depth 2, the compact hypergroup turns out to be a compact group. This rules out the occurrence of compact \emph{quantum} groups acting as global gauge symmetries in local conformal field theory.Comment: 58 page