On Noncommutative Joinings

Abstract

This paper extends the classical theory of joinings of measurable dynamical systems to the noncommutative setting from several interconnected points of view. Among these is a particularly fruitful identification of joinings with equivariant quantum channels between W * -dynamical systems that provides noncommutative generalizations of many fundamental results of classical joining theory. We obtain fully general analogues of the main classical disjointness characterizations of ergodicity, primeness and mixing phenomena.

Balance Between Quantum Markov Semigroups

“…For comparison to the theory of joinings [22,23,24], note that a joining of systems A and B, with α and β * -automorphisms, is a state ω on A ⊙ B such that ω(a ⊗ 1) = µ(a), ω(1 ⊗ b) = ν(b) and ω • (α ⊙ β) = ω. In addition [11] also assumes that ω • (σ µ t ⊙ σ ν t ) = ω, where σ µ t and σ ν t are the modular groups associated to µ and ν. In [11], however, it is formulated in terms of the opposite algebra of B, which is in that sense somewhat closer to the conventions used above for balance.…”

“…The connection to correspondences in the sense of Connes is also discussed. The connection of correspondences to joinings was already pointed out in [11] and [44,Section 5].…”

“…The theory of balance can be viewed as being parallel to the theory joinings for W*-dynamical systems. The latter was developed in [22,23,24], and studied further in [11], for the case where the dynamics are given by * -automorphism groups. Some aspects of noncommutative joinings also appeared in [60] and [44] related to entropy, and in [33] related to certain ergodic theorems.…”

“…maps and diagonal couplings play a key role in developing the theory of balance. This is related to [11,Section 4], although in the latter, certain assumptions involving modular groups are built into the framework, while analogous assumptions are not made in the definition of balance leading to the theory developed in this paper. Section 4 gives a characterization of balance in terms of intertwinement with the u.c.p.…”

Relative weak mixing of W$^*$-dynamical systems via joinings

“…Subsequent work was done in [3], which among other things developed various characterizations of joinings and also obtained a more complete theory for weak mixing, building on an approach to noncommutative joinings outlined in [19,Section 5]. Earlier work related to noncommutative joinings appeared in [22], connected to entropy, and [10], regarding ergodic theorems.…”