Boson mapping of symplectic algebras with Abelian subalgebra mapped as coordinates

1995 ◽  
Vol 36 (3) ◽  
pp. 1123-1135 ◽  
Author(s):  
P. O. Hess ◽  
J. C. López
Keyword(s):  
Abelian Subalgebra ◽  
Symplectic Algebras

Normal Operators on the Banach Space Lp(-∞,∞). Part I

1961 ◽  
Vol 13 ◽  
pp. 505-518 ◽  
Author(s):  
Gregers L. Krabbe
Keyword(s):  
Banach Space ◽  
Boolean Algebra ◽  
Abelian Subalgebra ◽  
The Family ◽  
Normal Operators ◽  
Image Position

Let be the Boolean algebra of all finite unions of subcells of the plane. Denote by εpthe algebra of all linear bounded transformations of Lp(— ∞, ∞) into itself. Suppose for a moment that p = 2, and let Rp be an involutive abelian subalgebra of εp if Rp is also a Banach space and if Tp ∈ Rp, then:(i) The family of all homomorphic mappings of into the algebra Rp contains a member EPT such that(1)

scholarly journals CUNTZ–KRIEGER ALGEBRAS AND ONE-SIDED CONJUGACY OF SHIFTS OF FINITE TYPE AND THEIR GROUPOIDS

2019 ◽  
Vol 109 (3) ◽  
pp. 289-298
Author(s):  
KEVIN AGUYAR BRIX ◽  
TOKE MEIER CARLSEN
Keyword(s):  
Conjugacy Class ◽  
Finite Type ◽  
Negative Answer ◽  
The Other ◽  
Completely Positive Map ◽  
Completely Positive ◽  
Positive Map ◽  
Shift Of Finite Type ◽  
Abelian Subalgebra ◽  
The One

AbstractA one-sided shift of finite type $(\mathsf{X}_{A},\unicode[STIX]{x1D70E}_{A})$ determines on the one hand a Cuntz–Krieger algebra ${\mathcal{O}}_{A}$ with a distinguished abelian subalgebra ${\mathcal{D}}_{A}$ and a certain completely positive map $\unicode[STIX]{x1D70F}_{A}$ on ${\mathcal{O}}_{A}$. On the other hand, $(\mathsf{X}_{A},\unicode[STIX]{x1D70E}_{A})$ determines a groupoid ${\mathcal{G}}_{A}$ together with a certain homomorphism $\unicode[STIX]{x1D716}_{A}$ on ${\mathcal{G}}_{A}$. We show that each of these two sets of data completely characterizes the one-sided conjugacy class of $\mathsf{X}_{A}$. This strengthens a result of Cuntz and Krieger. We also exhibit an example of two irreducible shifts of finite type which are eventually conjugate but not conjugate. This provides a negative answer to a question of Matsumoto of whether eventual conjugacy implies conjugacy.

Quantum Feller semigroup in terms of quantum Bernoulli noises

2020 ◽  
pp. 2150015
Author(s):  
Jinshu Chen
Keyword(s):  
Local Structure ◽  
Identity Operator ◽  
Sufficient Condition ◽  
Markov Semigroups ◽  
Feller Semigroup ◽  
Abelian Subalgebra ◽  
Feller Property ◽  
The Family ◽  
Invariant States ◽  
Creation Operators

Quantum Bernoulli noises (QBN) are the family of annihilation and creation operators acting on Bernoulli functionals, which satisfy a canonical anti-commutation relation in equal-time. In this paper, we aim to investigate quantum Feller semigroups in terms of QBN. We first investigate local structure of the algebra generated by identity operator and QBN. We then use our new results obtained here to construct a class of quantum Markov semigroups from QBN which enjoy Feller property. As an application of our results, we examine a special quantum Feller semigroup associated with QBN, when it reduced to a certain Abelian subalgebra, the semigroup gives rise to the semigroup generated by Ornstein–Uhlenbeck operator. Finally, we find a sufficient condition for the existence of faithful invariant states that are diagonal for the semigroup.

scholarly journals Two special subgroups of the universal sofic group

2018 ◽  
Vol 39 (12) ◽  
pp. 3250-3261
Author(s):  
MATTEO CAVALERI ◽  
RADU B. MUNTEANU ◽  
LIVIU PĂUNESCU
Keyword(s):  
Probability Space ◽  
Group Of Automorphisms ◽  
Abelian Subalgebra ◽  
Standard Probability

We define a subgroup of the universal sofic group, obtained as the normalizer of a separable abelian subalgebra. This subgroup can be obtained as an extension by the group of automorphisms on a standard probability space. We show that each sofic representation can be conjugated inside this subgroup.

NONCOMMUTATIVE KdV HIERARCHY

2007 ◽  
Vol 19 (07) ◽  
pp. 677-724 ◽  
Author(s):  
FRANÇOIS TREVES
Keyword(s):  
Evolution Equations ◽  
Vries Equation ◽  
Kdv Equation ◽  
Completely Integrable Systems ◽  
Maximal Abelian Subalgebra ◽  
Abelian Subalgebra ◽  
Kdv Hierarchy ◽  
Noncommutative Version ◽  
Constants Of Motion ◽  
Hamiltonian Evolution

The noncommutative version of the Korteweg–de Vries equation studied here is shown to admit infinitely many constants of motion and to give rise to a hierarchy of higher-order Hamiltonian evolution equations, each one the noncommutative version of the commutative KdV equation of the same order. The noncommutative KdV polynomials span, topologically, a maximal Abelian subalgebra of the Lie algebra of noncommutative Bäcklund transformations. Two classes of examples of "completely integrable" systems of evolution equations to which the theory applies are described in the last two sections.

scholarly journals AUTOMORPHISMS OF THE UHF ALGEBRA THAT DO NOT EXTEND TO THE CUNTZ ALGEBRA

2010 ◽  
Vol 89 (3) ◽  
pp. 309-315 ◽  
Author(s):  
ROBERTO CONTI
Keyword(s):  
Sufficient Conditions ◽  
Extension Property ◽  
Cuntz Algebra ◽  
Maximal Abelian Subalgebra ◽  
Abelian Subalgebra ◽  
Matrix Algebras ◽  
The Family ◽  
The Matrix ◽  
Inner Automorphisms ◽  
Necessary And Sufficient

AbstractThe automorphisms of the canonical core UHF subalgebra ℱn of the Cuntz algebra 𝒪n do not necessarily extend to automorphisms of 𝒪n. Simple examples are discussed within the family of infinite tensor products of (inner) automorphisms of the matrix algebras Mn. In that case, necessary and sufficient conditions for the extension property are presented. Also addressed is the problem of extending to 𝒪n the automorphisms of the diagonal 𝒟n, which is a regular maximal abelian subalgebra with Cantor spectrum. In particular, it is shown that there exist product-type automorphisms of 𝒟n that do not extend to (possibly proper) endomorphisms of 𝒪n.

QUANTUM ENTROPY AND INFORMATION IN DISCRETE ENTANGLED STATES

2001 ◽  
Vol 04 (02) ◽  
pp. 137-160 ◽  
Author(s):  
VIACHESLAV P. BELAVKIN ◽  
MASANORI OHYA
Keyword(s):  
Conditional Entropy ◽  
Von Neumann Entropy ◽  
Entangled States ◽  
Commutative Case ◽  
Maximal Abelian Subalgebra ◽  
Von Neumann ◽  
Abelian Subalgebra ◽  
Different Types ◽  
Classical Quantum ◽  
Entropic Measure

Quantum entanglements, describing truly quantum couplings, are studied and classified for discrete compound states. We show that classical-quantum correspondences such as quantum encodings can be treated as d-entanglements leading to a special class of separable compound states. The mutual information for the d-compound and for q-compound (entangled) states leads to two different types of entropies for a given quantum state. The first one is the von Neumann entropy, which is achieved as the supremum of the information over all d-entanglements, and the second one is the dimensional entropy, which is achieved at the standard entanglement, the true quantum entanglement, coinciding with a d-entanglement only in the commutative case. The q-conditional entropy and q-capacity of a quantum noiseless channel, defined as the supremum over all entanglements, is given as the logarithm of the dimensionality of the input von Neumann algebra. It can double the classical capacity, achieved as the supremum over all semiquantum couplings (d-entanglements, or encodings), which is bounded by the logarithm of the dimensionality of a maximal Abelian subalgebra. The entropic measure for essential entanglement is introduced.

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