Narrow Results

This paper gives applications of local nets of GW${}^{\ast }$-algebras, as developed in [27]. These local nets do not only contain all local observables (as is the case of local nets of von Neumann algebras), but also the field operators appearing in Wightman’s theory.

We consider an infinite spin chain as a bipartite system consisting of the left and right half-chains and analyze entanglement properties of pure states with respect to this splitting. In this context, we show that the amount of entanglement contained in a given state is deeply related to the von Neumann type of the observable algebras associated to the half-chains. Only the type I case belongs to the usual entanglement theory which deals with density operators on tensor product Hilbert spaces, and only in this situation separable normal states exist. In all other cases, the corresponding state is infinitely entangled in the sense that one copy of the system in such a state is sufficient to distill an infinite amount of maximally entangled qubit pairs. We apply this results to the critical XY model and show that its unique ground state φ_S provides a particular example for this type of entanglement.

As earlier conjectured by several authors and much later established by Solèr (relying on partial results by Piron, Maeda–Maeda and other authors), from the lattice theory point of view, Quantum Mechanics may be formulated in real, complex or quaternionic Hilbert spaces only. Stückelberg provided some physical, but not mathematically rigorous, reasons for ruling out the real Hilbert space formulation, assuming that any formulation should encompass a statement of Heisenberg principle. Focusing on this issue from another — in our opinion, deeper — viewpoint, we argue that there is a general fundamental reason why elementary quantum systems are not described in real Hilbert spaces. It is their basic symmetry group. In the first part of the paper, we consider an elementary relativistic system within Wigner’s approach defined as a locally-faithful irreducible strongly-continuous unitary representation of the Poincaré group in a real Hilbert space. We prove that, if the squared-mass operator is non-negative, the system admits a natural, Poincaré invariant and unique up to sign, complex structure which commutes with the whole algebra of observables generated by the representation itself. This complex structure leads to a physically equivalent reformulation of the theory in a complex Hilbert space. Within this complex formulation, differently from what happens in the real one, all selfadjoint operators represent observables in accordance with Solèr’s thesis, and the standard quantum version of Noether theorem may be formulated. In the second part of this work, we focus on the physical hypotheses adopted to define a quantum elementary relativistic system relaxing them on the one hand, and making our model physically more general on the other hand. We use a physically more accurate notion of irreducibility regarding the algebra of observables only, we describe the symmetries in terms of automorphisms of the restricted lattice of elementary propositions of the quantum system and we adopt a notion of continuity referred to the states viewed as probability measures on the elementary propositions. Also in this case, the final result proves that there exists a unique (up to sign) Poincaré invariant complex structure making the theory complex and completely fitting into Solèr’s picture. This complex structure reveals a nice interplay of Poincaré symmetry and the classification of the commutant of irreducible real von Neumann algebras.

Let M be a von Neumann algebra on a Hilbert space ℋ with a cyclic and separating unit vector Ω and let ω be the faithful normal state on M given by ω(·) = (Ω,·Ω). Moreover, let {N_i : i ∈ I} be a family of von Neumann subalgebras of M with faithful normal conditional expectations E_i of M onto N_i satisfying ω = ω ◦ E_i for all i ∈ I and let N = ∩_i∈I N_i. We show that the projections e_i, e of ℋ onto the closed subspaces and respectively satisfy e = ∧_i∈I e_i. This proves a conjecture of Jones and Xu in [1].

A quadrilateral of factors is an irreducible inclusion of factors N ⊂ M with intermediate subfactors P and Q such that P and Q generate M and the intersection of P and Q is N. We investigate the structure of a noncommuting quadrilateral of factors with all the elementary inclusions P ⊂ M, Q ⊂ M, N ⊂ P, and N ⊂ Q 2-supertransitive. In particular, we classify noncommuting quadrilaterals with the indices of the elementary subfactors less than or equal to 4. We also compute the angles between P and Q for quadrilaterals coming from α-induction and asymptotic inclusions.

For finite dimensional abelian subalgebras of a finite von Neumann algebra, we obtain the value of conditional relative entropy defined by Choda. We also consider the modified invariant defined by Pimsner and Popa.

The definition of Choi matrices for linear maps on the n × n matrices is extended to factors, and the basic theorems for Choi matrices are proved in this more general context.

Let $E$ and $F$ be two symmetric quasi-Banach spaces and let ${ℳ}$ be a semifinite von Neumann algebra. The purpose of this paper is to study the product space $E({ℳ})\odot F({ℳ})$ and the space of multipliers from $E({ℳ})$ to $F({ℳ})$, i.e. $M(E({ℳ}),F({ℳ}))$. These spaces share many properties with their classical counterparts. Let $0<{\alpha }_0,{\alpha }_1<\infty .$ It is shown that if $F$ is ${\alpha }_1$-convex fully symmetric and $E$ is ${\alpha }_0$-convex, then $M(E({ℳ}),F({ℳ}))=M(E,F)({ℳ})$, where $M(E,F)({ℳ})=\{x\in L_0({ℳ}):\mu (x)\in M(E,F)\}$ and $M(E,F)$ is the space of multipliers from $E$ to $F.$ As an application, we give conditions on when $M(E({ℳ}),F({ℳ}))\odot E({ℳ})=F({ℳ}).$ Moreover, we show that the product space can be described with the help of complex interpolation method.

We provide a comprehensive study on uniformly left $\psi$-bounded (respectively, $(\phi ,\psi )$-bounded) orthonormal bases in infinite-dimensional cyclic bimodules associated with c.p. maps between two von Neumann algebras $M$ and $N$, where $\phi$ and $\psi$ are faithful normal states on $M$ and $N$, respectively. Separate investigations on cyclic bimodules associated with Markov maps and arbitrary c.p. maps are also provided since the results differ. This generalizes the results of the authors’ previous work on Kadison’s unitary basis problem.

It is known that every semigroup of normal completely positive maps P = {P_t:t≥ 0} of ℬ(H), satisfying P_t(1) = 1 for every t ≥ 0, has a minimal dilation to an E₀ acting on ℬ(K) for some Hilbert space K⊇H. The minimal dilation of P is unique up to conjugacy. In a previous paper a numerical index was introduced for semigroups of completely positive maps and it was shown that the index of P agrees with the index of its minimal dilation to an E₀-semigroup. However, no examples were discussed, and no computations were made.

In this paper we calculate the index of a unital completely positive semigroup whose generator is a bounded operator

An extension of Cencov’s categorical description of classical inference theory to the domain of quantum systems is presented. It provides a novel categorical foundation to the theory of quantum information that embraces both classical and quantum information theories in a natural way, while also allowing to formalize the notion of quantum environment. A first application of these ideas is provided by extending the notion of statistical manifold to incorporate categories, and investigating a possible, uniparametric Cramer–Rao inequality in this setting.

There are notions of L²-Betti numbers for discrete groups (Cheeger–Gromov, Lück), for type II₁-factors (recent work of Connes-Shlyakhtenko) and for countable standard equivalence relations (Gaboriau). Whereas the first two are algebraically defined using Lück's dimension theory, Gaboriau's definition of the latter is inspired by the work of Cheeger and Gromov. In this work we give a definition of L²-Betti numbers of discrete measured groupoids that is based on Lück's dimension theory, thereby encompassing the cases of groups, equivalence relations and holonomy groupoids with an invariant measure for a complete transversal. We show that with our definition, like with Gaboriau's, the L²-Betti numbers of a countable group G coincide with the L²-Betti numbers of the orbit equivalence relation of a free action of G on a probability space. This yields a new proof of the fact the L²-Betti numbers of groups with orbit equivalent actions coincide.

We show that the L²-Betti numbers of equivalence relations defined by R. Sauer coincide with those defined by D. Gaboriau.

For a von Neumann algebra ࡕ acting on a Hilbert space ℋ with a cyclic and separating vector ξ₀, we investigate the structure of Dirichlet forms on the natural standard form associated with the pair (ࡕ, ξ₀). For a general bounded Lindblad type generator L of a conservative quantum dynamical semigroup on ࡕ, we give sufficient conditions so that the bounded operator H induced by L via the symmetric embedding of ࡕ into ℋ to be self-adjoint. It turns out that the self-adjoint operator H can be written in the form of a Dirichlet operator associated to a Dirichlet form given in Ref. 23. In order to make the connection possible, we also extend the range of applications of the formula in Ref. 23.

Several possible generalizations of the classical notion of markovianity are given for states defined on a von Neumann algebras generated on a triple of subalgebras. Their mutual relation is discussed in the particular case in which they mutually commute, and the generalization of the classical; time reversal theorem is proved. A structure theorem for a class of Markov chains is also proved.

We examine the free entropy and free entropy dimension for projections, and obtain a sufficient condition for the factoriality of the von Neumann algebra generated by projections in terms of their free entropy dimension. This corresponds to Voiculescu's result for self-adjoint elements.

The present paper is a survey of recent results concerning derivations on various algebras of measurable operators affiliated with von Neumann algebras. A complete description of derivation is obtained in the case of type I von Neumann algebras. A special section is devoted to the Abelian case, namely to the existence of nontrivial derivations on algebras of measurable function. Local derivations on the above algebras are also considered.

Recently, Brannan and Vergnioux showed that the orthogonal free quantum group factors ${ℒ}𝔽O_M$ have Jung’s strong $1$-boundedness property, and hence are not isomorphic to free group factors. We prove an analogous result for the other unimodular case, where the parameter matrix is the standard symplectic matrix in $2N$ dimensions $J_{2N}$. We compute free derivatives of the defining relations by introducing self-adjoint generators through a decomposition of the fundamental representation in terms of Pauli matrices, resulting in $1$-boundedness of these generators. Moreover, we prove that under certain conditions, one can add elements to a $1$-bounded set without losing $1$-boundedness. In particular, this allows us to include the character of the fundamental representation, proving strong $1$-boundedness.

In this paper, we prove that the mixed $q$-deformed Araki-Woods von Neumann algebras are factors for dim$({{\mathscr{H}}}_{ℝ})\ge 3$.

The paper is devoted to local derivations on the algebra of τ-measurable operators affiliated with a von Neumann algebra and a faithful normal semi-finite trace τ. We prove that every local derivation on which is continuous in the measure topology, is in fact a derivation. In the particular case of type I von Neumann algebras, they all are inner derivations. It is proved that for type I finite von Neumann algebras without an abelian direct summand, and also for von Neumann algebras with the atomic lattice of projections, the continuity condition on local derivations in the above results is redundant. Finally we give necessary and sufficient conditions on a commutative von Neumann algebra for the algebra to admit local derivations which are not derivations.