On the von Neumann algebras associated to Yang–Baxter operators

2020 ◽  
pp. 1-24
Author(s):  
Panchugopal Bikram ◽  
Rahul Kumar ◽  
Rajeeb Mohanta ◽  
Kunal Mukherjee ◽  
Diptesh Saha
Keyword(s):  
Hilbert Space ◽  
Special Form ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Adjoint Operator ◽  
Complex Hilbert Space ◽  
Von Neumann ◽  
Finite Von Neumann Algebra

Bożejko and Speicher associated a finite von Neumann algebra M T to a self-adjoint operator T on a complex Hilbert space of the form $\mathcal {H}\otimes \mathcal {H}$ which satisfies the Yang–Baxter relation and $ \left\| T \right\| < 1$ . We show that if dim $(\mathcal {H})$ ⩾ 2, then M T is a factor when T admits an eigenvector of some special form.

scholarly journals ON EQUALITY CONDITION FOR TRACE JENSEN INEQUALITY IN SEMI-FINITE VON NEUMANN ALGEBRAS

2008 ◽  
Vol 19 (04) ◽  
pp. 481-501 ◽  
Author(s):  
TETSUO HARADA ◽  
HIDEKI KOSAKI
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Adjoint Operator ◽  
Strict Convexity ◽  
Jensen Inequality ◽  
Von Neumann ◽  
Convexity Assumption ◽  
Finite Von Neumann Algebra ◽  
Equality Condition ◽  
Finite Von Neumann Algebras

Let τ be a faithful semi-finite normal trace on a semi-finite von Neumann algebra, and f(t) be a convex function with f(0) = 0. The trace Jensen inequality states τ(f(a* xa)) ≤ τ(a* f(x)a) for a contraction a and a self-adjoint operator x. Under certain strict convexity assumption on f(t), we will study when this inequality reduces to the equality.

Nonlinear ∗-Jordan triple derivations on von Neumann algebras

2018 ◽  
Vol 68 (1) ◽  
pp. 163-170 ◽  
Author(s):  
Fangfang Zhao ◽  
Changjing Li
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
New Product ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Von Neumann

AbstractLetB(H) be the algebra of all bounded linear operators on a complex Hilbert spaceHand 𝓐 ⊆B(H) be a von Neumann algebra with no central summands of typeI1. ForA,B∈ 𝓐, define byA∙B=AB+BA∗a new product ofAandB. In this article, it is proved that a map Φ: 𝓐 →B(H) satisfies Φ(A∙B∙C) = Φ(A) ∙B∙C+A∙ Φ(B) ∙C+A∙B∙Φ(C) for allA,B,C∈ 𝓐 if and only if Φ is an additive *-derivation.

scholarly journals Some aspects of quantum sufficiency for finite and full von Neumann algebras

2021 ◽  
Vol 111 (4) ◽  
Author(s):  
Andrzej Łuczak
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Linear Operators ◽  
Bounded Linear ◽  
Von Neumann ◽  
Finite Von Neumann Algebra ◽  
Pure States ◽  
Normal States ◽  
Full Algebra

AbstractSome features of the notion of sufficiency in quantum statistics are investigated. Three kinds of this notion are considered: plain sufficiency (called simply: sufficiency), strong sufficiency and Umegaki’s sufficiency. It is shown that for a finite von Neumann algebra with a faithful family of normal states the minimal sufficient von Neumann subalgebra is sufficient in Umegaki’s sense. Moreover, a proper version of the factorization theorem of Jenčová and Petz is obtained. The structure of the minimal sufficient subalgebra is described in the case of pure states on the full algebra of all bounded linear operators on a Hilbert space.

scholarly journals Gradient forms and strong solidity of free quantum groups

2020 ◽  
Author(s):  
Martijn Caspers
Keyword(s):  
Quantum Groups ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Dirichlet Forms ◽  
Von Neumann ◽  
Finite Von Neumann Algebra ◽  
Crucial Part

AbstractConsider the free orthogonal quantum groups $$O_N^+(F)$$ O N + ( F ) and free unitary quantum groups $$U_N^+(F)$$ U N + ( F ) with $$N \ge 3$$ N ≥ 3 . In the case $$F = \text {id}_N$$ F = id N it was proved both by Isono and Fima-Vergnioux that the associated finite von Neumann algebra $$L_\infty (O_N^+)$$ L ∞ ( O N + ) is strongly solid. Moreover, Isono obtains strong solidity also for $$L_\infty (U_N^+)$$ L ∞ ( U N + ) . In this paper we prove for general $$F \in GL_N(\mathbb {C})$$ F ∈ G L N ( C ) that the von Neumann algebras $$L_\infty (O_N^+(F))$$ L ∞ ( O N + ( F ) ) and $$L_\infty (U_N^+(F))$$ L ∞ ( U N + ( F ) ) are strongly solid. A crucial part in our proof is the study of coarse properties of gradient bimodules associated with Dirichlet forms on these algebras and constructions of derivations due to Cipriani–Sauvageot.

scholarly journals The Spectral Scale of a Self-Adjoint Operator in a Semifinite von Neumann Algebra

2011 ◽  
Vol 2011 ◽  
pp. 1-24
Author(s):  
Christopher M. Pavone
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Adjoint Operator ◽  
Von Neumann ◽  
Selfadjoint Operators ◽  
Normal Operators ◽  
Spectral Scale ◽  
Finite Setting

We extend Akemann, Anderson, and Weaver'sSpectral Scaledefinition to include selfadjoint operators fromsemifinitevon Neumann algebras. New illustrations of spectral scales in both the finite and semifinite von Neumann settings are presented. A counterexample to a conjecture made by Akemann concerning normal operators and the geometry of the their perspective spectral scales (in the finite setting) is offered.

scholarly journals Generators of reflexive algebras

1975 ◽  
Vol 20 (2) ◽  
pp. 159-164
Author(s):  
W. E. Longstaff
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Complex Hilbert Space ◽  
Bounded Operators ◽  
Finitely Generated ◽  
Separable Space ◽  
Von Neumann ◽  
Underlying Space ◽  
Weakly Closed ◽  
Reflexive Algebras

For any collection of closed subspaces of a complex Hilbert space the set of bounded operators that leave invariant all the members of the collection is a weakly-closed algebra. The class of such algebras is precisely the class of reflexive algebras as defined for example in Radjavi and Rosenthal (1969) and contains the class of von Neumann algebras.In this paper we consider the problem of when such algebras are finitely generated as weakly-closed algebras. It is to be hoped that analysis of this problem may shed some light on the famous unsolved problem of whether every von Neumann algebra on a separable Hilbert space is finitely generated. The case where the underlying space is separable and the collection of subspaces is totally ordered is dealt with in Longstaff (1974). In the present paper the result of Longstaff (1974) is generalized to the case of a direct product of countably many totally ordered collections each on a separable space. Also a method of obtaining non-finitely generated reflexive algebras is given.

scholarly journals 2-LOCAL DERIVATIONS ON SEMI-FINITE VON NEUMANN ALGEBRAS

2013 ◽  
Vol 56 (1) ◽  
pp. 9-12 ◽  
Author(s):  
SHAVKAT AYUPOV ◽  
FARKHAD ARZIKULOV
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Von Neumann ◽  
Local Derivation ◽  
Finite Von Neumann Algebra ◽  
Finite Von Neumann Algebras

AbstractIn the present paper we prove that every 2-local derivation on a semi-finite von Neumann algebra is a derivation.

Commutators in Factors of Type III

1966 ◽  
Vol 18 ◽  
pp. 1152-1160 ◽  
Author(s):  
Arlen Brown ◽  
Carl Pearcy
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Complex Hilbert Space ◽  
Identity Operator ◽  
Von Neumann ◽  
Type Iii ◽  
Underlying Space ◽  
Special Cases ◽  
Weakly Closed ◽  
Algebra Of Operators

Let denote a separable, complex Hilbert space, and let R be a von Neumann algebra acting on . (A von Neumann algebra is a weakly closed, self-adjoint algebra of operators that contains the identity operator on its underlying space.) An element A of R is a commutator in R if there exist operators B and C in R such that A = BC — CB. The problem of specifying exactly which operators are commutators in R has been solved in certain special cases; e.g. if R is an algebra of type In (n < ∞) (2), and if R is a factor of type I∞ (1). It is the purpose of this note to treat the same problem in case R is a factor of type III. Our main result is the following theorem.

scholarly journals Quantum theory in quaternionic Hilbert space: How Poincaré symmetry reduces the theory to the standard complex one

2019 ◽  
Vol 31 (04) ◽  
pp. 1950013 ◽  
Author(s):  
Valter Moretti ◽  
Marco Oppio
Keyword(s):  
Hilbert Space ◽  
Quantum System ◽  
Hilbert Spaces ◽  
Von Neumann Algebras ◽  
Complex Structure ◽  
Complex Hilbert Space ◽  
Von Neumann ◽  
The Real ◽  
Quaternionic Hilbert Space ◽  
Adjoint Operators

As earlier conjectured by several authors and much later established by Solèr, from the lattice-theory point of view, Quantum Mechanics may be formulated in real, complex or quaternionic Hilbert spaces only. On the other hand, no quantum systems seem to exist that are naturally described in a real or quaternionic Hilbert space. In a previous paper [23], we showed that any quantum system which is elementary from the viewpoint of the Poincaré symmetry group and it is initially described in a real Hilbert space, it can also be described within the standard complex Hilbert space framework. This complex description is unique and more precise than the real one as, for instance, in the complex description, all self-adjoint operators represent observables defined by the symmetry group. The complex picture fulfils the thesis of Solér’s theorem and permits the standard formulation of the quantum Noether’s theorem. The present work is devoted to investigate the remaining case, namely, the possibility of a description of a relativistic elementary quantum system in a quaternionic Hilbert space. Everything is done exploiting recent results of the quaternionic spectral theory that were independently developed. In the initial part of this work, we extend some results of group representation theory and von Neumann algebra theory from the real and complex cases to the quaternionic Hilbert space case. We prove the double commutant theorem also for quaternionic von Neumann algebras (whose proof requires a different procedure with respect to the real and complex cases) and we extend to the quaternionic case a result established in the previous paper concerning the classification of irreducible von Neumann algebras into three categories. In the second 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 quaternionic 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 quaternionic one, all self-adjoint operators represent observables in agreement with Solèr’s thesis, the standard quantum version of Noether theorem may be formulated and the notion of composite system may be given in terms of tensor product of elementary systems. In the third part of the paper, 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. The overall conclusion is that relativistic elementary systems are naturally and better described in complex Hilbert spaces even if starting from a real or quaternionic Hilbert space formulation and this complex description is uniquely fixed by physics.

scholarly journals On the entropy in II1 von Neumann algebras

1981 ◽  
Vol 1 (4) ◽  
pp. 419-429 ◽  
Author(s):  
O. Besson
Keyword(s):  
Lebesgue Space ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Point Spectrum ◽  
Crossed Product ◽  
Pure Point ◽  
Pure Point Spectrum ◽  
Von Neumann ◽  
Finite Von Neumann Algebra ◽  
Spectrum Transformation

AbstractLet α be an automorphism of a finite von Neumann algebra and let H(α) be its Connes-Størmer's entropy. We show that H(α) = 0 if α is the induced automorphism on the crossed product of a Lebesgue space by a pure point spectrum transformation. We also show that H is not continuous in α and we compute H(α) for some α.

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