On Noncommutative Joinings Bannon, Jon P; Cameron, Jan; Mukherjee, Kunal
International mathematics research notices,
08/2018, Volume:
2018, Issue:
15
Journal Article
Peer reviewed
Open access
Abstract
This article 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 $W^{\ast}$-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.
The modular symmetry of Markov maps Bannon, Jon P.; Cameron, Jan; Mukherjee, Kunal
Journal of mathematical analysis and applications,
07/2016, Volume:
439, Issue:
2
Journal Article
Peer reviewed
Open access
A state-preserving automorphism of a von Neumann algebra induces a canonical unitary operator on the GNS Hilbert space of the state which fixes the vacuum. This unitary commutes with both the modular ...operator of the state and its modular conjugation. We prove an extension of this result for state-preserving unital completely positive maps.
We study the relationship between the dynamics of the action α of a discrete group G on a von Neumann algebra M, and structural properties of the associated crossed product inclusion L(G)⊆M⋊αG, and ...its intermediate subalgebras. This continues a thread of research originating in classical structural results for ergodic actions of discrete, abelian groups on probability spaces. A key tool in the setting of a noncommutative dynamical system is the set of quasinormalizers for an inclusion of von Neumann algebras. We show that the von Neumann algebra generated by the quasinormalizers captures analytical properties of the inclusion L(G)⊆M⋊αG such as the Haagerup Approximation Property, and is essential to capturing “almost periodic” behavior in the underlying dynamical system. Our von Neumann algebraic point of view yields a new description of the Furstenberg-Zimmer distal tower for an ergodic action on a probability space, and we establish new versions of the Furstenberg-Zimmer structure theorem for general, tracial W⁎-dynamical systems. We present a number of examples contrasting the noncommutative and classical settings which also build on previous work concerning singular inclusions of finite von Neumann algebras.
We introduce a notion of transitive family of subspaces relative to a type II_{1} factor, and hence a notion of transitive family of projections in such a factor. We show that whenever \mathcal{M} is ...a factor of type II_{1} and \mathcal{M} is generated by two self-adjoint elements, then \mathcal{M}\otimes M_{2}(\mathbb{C}) contains a transitive family of 5 projections. Finally, we exhibit a free transitive family of 12 projections that generate a factor of type II_{1}.
We introduce a notion of transitive family of subspaces relative to a type II1factor, and hence a notion of transitive family of projections in such a factor. We show that whenever M is a factor of ...type II1and M is generated by two self-adjoint elements, then$\mathcal{M} \otimes M_{2}(\mathbb{C})$contains a transitive family of 5 projections. Finally, we exhibit a free transitive family of 12 projections that generate a factor of type II1.
A finite von Neumann algebra 𝓜 with a faithful normal trace τ has Haagerup's approximation property if there exists a pointwise deformation of the identity in 2-norm by subtracial compact completely ...positive maps. In this paper we prove that the subtraciality condition can be removed. This enables us to provide a description of Haagerup's approximation property in terms of correspondences. We also show that if 𝓝 ⊂ 𝓜 is an amenable inclusion of finite von Neumann algebras and 𝓝 has Haagerup's approximation property, then 𝓜 also has Haagerup's approximation property.
In this article, we introduce an isomorphism invariant for type
II
1
factors using the Connes–Følner condition. We compute bounds of this number for free group factors.
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