We introduce and study a class of quadratic Wasserstein distances on spaces consisting of generalized dynamical systems on a von Neumann algebra. We emphasize how symmetry of such a Wasserstein ...distance arises, but also study the asymmetric case. This setup is illustrated in the context of reduced dynamics, and a number of simple examples are also presented.
The concept of balance between two state-preserving quantum Markov semigroups on von Neumann algebras is introduced and studied as an extension of conditions appearing in the theory of quantum ...detailed balance. This is partly motivated by the theory of joinings. Balance is defined in terms of certain correlated states (couplings), with entangled states as a specific case. Basic properties of balance are derived, and the connection to correspondences in the sense of Connes is discussed. Some applications and possible applications, including to non-equilibrium statistical mechanics, are briefly explored.
A definition of relative discrete spectrum of noncommutative W*-dynamical systems is given in terms of the basic construction of von Neumann algebras, motivated from three perspectives: First, as a ...complementary concept to relative weak mixing of W*-dynamical systems. Second, by comparison with the classical (i.e., commutative) case. And, third, by noncommutative examples.
Joinings of W ∗ -dynamical systems Duvenhage, Rocco
Journal of mathematical analysis and applications,
07/2008, Volume:
343, Issue:
1
Journal Article
Peer reviewed
Open access
We study the notion of joinings of
W
∗
-dynamical systems, building on ideas from measure theoretic ergodic theory. In particular we prove sufficient and necessary conditions for ergodicity in terms ...of joinings, and also briefly look at conditional expectation operators associated with joinings.
C∗-fermi systems and detailed balance Crismale, Vitonofrio; Duvenhage, Rocco; Fidaleo, Francesco
Analysis and mathematical physics,
03/2021, Volume:
11, Issue:
1
Journal Article
Peer reviewed
Open access
A systematic theory of product and diagonal states is developed for tensor products of
Z
2
-graded
∗
-algebras, as well as
Z
2
-graded
C
∗
-algebras. As a preliminary step to achieve this goal, we ...provide the construction of a
fermionic
C
∗
-
tensor product
of
Z
2
-graded
C
∗
-algebras. Twisted duals of positive linear maps between von Neumann algebras are then studied, and applied to solve a positivity problem on the infinite Fermi lattice. Lastly, these results are used to define fermionic detailed balance (which includes the definition for the usual tensor product as a particular case) in general
C
∗
-systems with gradation of type
Z
2
, by viewing such a system as part of a compound system and making use of a diagonal state.
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