Neumann Algebra Research Articles

Christensen–Evans theorem and extensions of GNS-symmetric quantum Markov semigroups

In this article we prove a refined version of the Christensen–Evans theorem for generators of uniformly continuous GNS-symmetric quantum Markov semigroups. We use this result to show the existence of GNS-symmetric extensions of GNS-symmetric quantum Markov semigroups. In particular, this implies that the generators of GNS-symmetric quantum Markov semigroups on finite-dimensional von Neumann algebra can be written in the form specified by Alicki's theorem.

Quasi-free Isomorphisms of Second Quantization Algebras and Modular Theory

Using Araki–Yamagami’s characterization of quasi-equivalence for quasi-free representations of the CCRs, we provide an abstract criterion for the existence of isomorphisms of second quantization local von Neumann algebras induced by Bogolubov transformations in terms of the respective one particle modular operators. We discuss possible applications to the problem of local normality of vacua of Klein-Gordon fields with different masses.

The condenser quasicentral modulus

We introduced the quasicentral modulus to study normed ideal perturbations of operators. It is a limit of condenser quasicentral moduli in view of a recently noticed analogy with capacity in nonlinear potential theory. We prove here some basic properties of the condenser quasicentral modulus and compute a simple example. We also discuss some associated noncommutative variational problems. Part of the results are in the more general setting of a semifinite von Neumann algebra.

Schur inequality for Murray–von Neumann algebras and its applications

Schur inequality for Murray–von Neumann algebras and its applications

Entanglement in Interacting Majorana Chains and Transitions of von Neumann Algebras.

We consider Majorana lattices with two-site interactions consisting of a general function of the fermion bilinear. The models are exactly solvable in the limit of a large number of on-site fermions. The four-site chain exhibits a quantum phase transition controlled by the hopping parameters and manifests itself in a discontinuous entanglement entropy, obtained by constraining the one-sided modular Hamiltonian. Inspired by recent work within the AdS/CFT correspondence, we identify transitions between types of von Neumann operator algebras throughout the phase diagram. We find transitions of the form II_{1}↔III↔I_{∞} that reduce to II_{1}↔I_{∞} in the strongly interacting limit, where they connect nonfactorized and factorized ground states. Our results provide novel realizations of such transitions in a controlled many-body model.

Linear isometries of noncommutative L0$L_0$‐spaces

AbstractThe description of (commutative and noncommutative) ‐isometries has been studied thoroughly since the seminal work of Banach. In the present paper, we provide a complete description for the limiting case, isometries on noncommutative ‐spaces, which extends the Banach–Stone theorem and Kadison's theorem for isometries of von Neumann algebras. The result is new even in the commutative setting.

Simplicity and tracial weights on non-unital reduced crossed products

We extend theorems of Breuillard–Kalantar–Kennedy–Ozawa on unital reduced crossed products to the non-unital case under mild assumptions. As a result simplicity of [Formula: see text]-algebras is stable under taking reduced crossed products over discrete [Formula: see text]-simple groups, and a similar result for uniqueness of tracial weight. Interestingly, our analysis on tracial weights involves von Neumann algebra theory. Our generalizations have two applications. The first is to locally compact groups. We establish stability results of (non-discrete) [Formula: see text]-simplicity and the unique trace property under discrete group extensions. The second is to the twisted crossed product. Thanks to the Packer–Raeburn theorem, our results lead to (generalizations of) the results of Bryder–Kennedy by a different method.

DYNAMICAL MCDUFF-TYPE PROPERTIES FOR GROUP ACTIONS ON VON NEUMANN ALGEBRAS

Abstract We consider the notion of strong self-absorption for continuous actions of locally compact groups on the hyperfinite II $_1$ factor and characterize when such an action is tensorially absorbed by another given action on any separably acting von Neumann algebra. This extends the well-known McDuff property for von Neumann algebras and is analogous to the core theorems around strongly self-absorbing C $^*$ -dynamics. Given a countable discrete group G and an amenable action $G\curvearrowright M$ on any separably acting semifinite von Neumann algebra, we establish a type of measurable local-to-global principle: If a given strongly self-absorbing G-action is suitably absorbed at the level of each fibre in the direct integral decomposition of M, then it is tensorially absorbed by the action on M. As a direct application of Ocneanu’s theorem, we deduce that if M has the McDuff property, then every amenable G-action on M has the equivariant McDuff property, regardless whether M is assumed to be injective or not. By employing Tomita–Takesaki theory, we can extend the latter result to the general case, where M is not assumed to be semifinite.

Dixmier-type traces on symmetric spaces associated with semifinite von Neumann algebras

Dixmier-type traces on symmetric spaces associated with semifinite von Neumann algebras

On α-z-Rényi divergence in the von Neumann algebra setting

We will investigate the α-z-Rényi divergence in the general von Neumann algebra setting based on Haagerup non-commutative Lp-spaces. In particular, we establish almost all its expected properties when 0 < α < 1 and some of them when α > 1. In an Appendix we also give an equality condition for generalized Hölder’s inequality in Haagerup non-commutative Lp-spaces.

A short proof of Tomita's theorem

Tomita-Takesaki theory associates a positive operator called the “modular operator” with a von Neumann algebra and a cyclic-separating vector. Tomita's theorem says that the unitary flow generated by the modular operator leaves the algebra invariant. I give a new, short proof of this theorem which only uses the analytic structure of unitary flows, and which avoids operator-valued Fourier transforms (as in van Daele's proof) and operator-valued Mellin transforms (as in Zsidó's and Woronowicz's proofs). The proof is similar to one given by Bratteli and Robinson in the special case that the modular operator is bounded.

2-positive contractive projections on noncommutative Lp-spaces

We prove the first theorem on projections on general noncommutative Lp-spaces associated with non-type I von Neumann algebras where 1⩽p<∞. This is the first progress on this topic since the seminal work of Arazy and Friedman (1992) [6] where the problem of the description of contractively complemented subspaces of noncommutative Lp-spaces is explicitly raised. We show that the range of a 2-positive contractive projection on an arbitrary noncommutative Lp-space is completely order isometrically isomorphic to some noncommutative Lp-space. This result is sharp and is even new for Schatten spaces Sp. Our approach relies on non-tracial Haagerup's noncommutative Lp-spaces in an essential way, even in the case of a projection acting on a Schatten space and is unrelated to the methods of Arazy and Friedman.

Hartle-Hawking state and its factorization in 3d gravity

We study 3d quantum gravity with two asymptotically anti-de Sitter regions, in particular, using its relation with coupled Alekseev-Shatashvili theories and Liouville theory. Expressions for the Hartle-Hawking state, thermal 2n-point functions, torus wormhole correlators and Wheeler-DeWitt wavefunctions in different bases are obtained using the ZZ boundary states in Liouville theory. Exact results in 2d Jackiw-Teitelboim (JT) gravity are uplifted to 3d gravity, with two copies of Liouville theory in 3d gravity playing a similar role as Schwarzian theory in JT gravity. The connection between 3d gravity and the Liouville ZZ boundary states are manifested by viewing BTZ black holes as Maldacena-Maoz wormholes, with the two wormhole boundaries glued along the ZZ boundaries. In this work, we also study the factorization problem of the Hartle-Hawking state in 3d gravity. With the relevant defect operator that imposes the necessary topological constraint for contractibility, the trace formula in gravity is modified in computing the entanglement entropy. This trace matches with the one from von Neumann algebra considerations, further reproducing the Bekenstein-Hawking area formula from entanglement entropy. Lastly, we propose a calculation for off-shell geometrical quantities that are responsible for the ramp behavior in the late time two-point functions, which follows from the understanding of the Liouville FZZT boundary states in the context of 3d gravity, and the identification between Verlinde loop operators in Liouville theory and “baby universe” operators in 3d gravity.

Additive mappings on von Neumann algebras acting as Lie triple centralizer via local actions and related mappings

Additive mappings on von Neumann algebras acting as Lie triple centralizer via local actions and related mappings

Characterization of nonlinear Lie type derivations on von Neumann algebras by local action

Let R be a unital ring containing a nontrivial idempotent. In this article, under a mild condition on R , we prove that if a map δ : R → R satisfies δ ( P n ( A 1 , A 2 , A 3 , … , A n ) ) = ∑ i = 1 n P n ( A 1 , … , A i − 1 , δ ( A i ) , A i + 1 , … , A n ) for any A 1 , A 2 , A 3 , … , A n ∈ R with A 1 A 2 A 3 = 0 , then δ ( A + B ) − δ ( A ) − δ ( B ) ∈ Z ( R ) for any A , B ∈ R . In particular, if R is a von Neumann algebra with no central summands of type I 1 or a factor, then δ ( x ) = d ( x ) + τ ( x ) for all x ∈ R , where d : R → R is an additive derivation and τ : R → Z ( R ) is a map vanishing on each ( n − 1 ) -th commutator p n ( A 1 , A 2 , A 3 , … , A n ) with A 1 A 2 A 3 = 0 .

A 𝐶*-algebra of entire functions

We study the maximal abelian von Neumann algebra corresponding to L ∞ ( R ) L^\infty (\mathbb {R}) via the Bargmann transform. It is naturally an algebra of operators on the Fock space F 2 F^2 , but it can also be realized as a function algebra contained in F 2 F^2 . This provides an interesting example of a C ∗ C^* algebra whose elements are analytic functions.

Nonassociative Lp-spaces and embeddings in noncommutative Lp-spaces

We define a notion of nonassociative Lp-space associated to a JBW⁎-algebra (Jordan von Neumann algebra) equipped with a normal faithful state φ. In the particular case of JW⁎-algebras underlying von Neumann algebras, we connect these spaces to a complex interpolation theorem of Ricard and Xu on noncommutative Lp-spaces. We also make the link with the nonassociative Lp-spaces of Iochum associated to JBW-algebras and the investigation of contractively complemented subspaces of noncommutative Lp-spaces. More precisely, we show that our nonassociative Lp-spaces contain isometrically the Lp-spaces of Iochum and that all tracial nonassociative Lp-spaces from JW⁎-factors arise as positively contractively complemented subspaces of noncommutative Lp-spaces.

The Clifford Algebra Bundle on Loop Space

We construct a Clifford algebra bundle formed from the tangent bundle of the smooth loop space of a Riemannian manifold, which is a bundle of super von Neumann algebras on the loop space. We show that this bundle is in general non-trivial, more precisely that its triviality is obstructed by the transgressions of the second Stiefel-Whitney class and the first (fractional) Pontrjagin class of the manifold.

Operator algebra generalization of a theorem of Watrous and mixed unitary quantum channels

We establish an operator algebra generalization of Watrous’ theorem (Watrous 2009 Quantum Inf. Comput. 9 403–413) on mixing unital quantum channels (completely positive trace-preserving maps) with the completely depolarizing channel, wherein the more general objects of focus become (finite-dimensional) von Neumann algebras, the unique trace preserving conditional expectation onto the algebra, the group of unitary operators in the commutant of the algebra, and the fixed point algebra of the channel. As an application, we obtain a result on the asymptotic theory of quantum channels, showing that all unital channels are eventually mixed unitary. We also discuss the special case of the diagonal algebra in detail, and draw connections to the theory of correlation matrices and Schur product maps.

The second nonlinear mixed bi-skew Lie triple derivations on factor von Neumann algebras

Let be a factor von Neumann algebra with dim . In this paper, it is shown that if a map satisfies for all , then ϕ is an additive ∗-derivation, where .