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Roberto Conti - One of the best experts on this subject based on the ideXlab platform.
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Yang–Baxter Endomorphisms
Journal of the London Mathematical Society, 2020Co-Authors: Roberto Conti, Gandalf LechnerAbstract:Every unitary solution of the Yang-Baxter equation (R-matrix) in dimension $d$ can be viewed as a unitary element of the Cuntz algebra $\CO_d$ and as such defines an Endomorphism of $\CO_d$. These Yang-Baxter Endomorphisms restrict and extend to Endomorphisms of several other $C^*$- and von Neumann algebras and furthermore define a II$_1$ factor associated with an extremal character of the infinite braid group. This paper is devoted to a detailed study of such Yang-Baxter Endomorphisms. Among the topics discussed are characterizations of Yang-Baxter Endomorphisms and the relative commutants of the various subfactors they induce, an Endomorphism perspective on algebraic operations on R-matrices such as tensor products and cabling powers, and properties of characters of the infinite braid group defined by R-matrices. In particular, it is proven that the partial trace of an R-matrix is an invariant for its character by a commuting square argument. Yang-Baxter Endomorphisms also supply information on R-matrices themselves, for example it is shown that the left and right partial traces of an R-matrix coincide and are normal, and that the spectrum of an R-matrix can not be concentrated in a small disc. Upper and lower bounds on the minimal and Jones indices of Yang-Baxter Endomorphisms are derived, and a full characterization of R-matrices defining ergodic Endomorphisms is given. As examples, so-called simple R-matrices are discussed in any dimension~$d$, and the set of all Yang-Baxter Endomorphisms in $d=2$ is completely analyzed.
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Endomorphisms of o n which preserve the canonical uhf subalgebra
Journal of Functional Analysis, 2010Co-Authors: Roberto Conti, Mikael Rordam, Wojciech SzymanskiAbstract:Abstract Unital Endomorphisms of the Cuntz algebra O n which preserve the canonical UHF-subalgebra F n ⊆ O n are investigated. We give examples of such Endomorphisms λ = λ u for which the associated unitary element u in O n (which satisfies λ ( S j ) = u S j for all j) does not belong to F n . One such example, in the case where n = 2 , arises from a construction of a unital Endomorphism on O 2 which preserves the canonical UHF-subalgebra and where the relative commutant of its image in O 2 contains a copy of O 2 .
Gandalf Lechner - One of the best experts on this subject based on the ideXlab platform.
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Yang–Baxter Endomorphisms
Journal of the London Mathematical Society, 2020Co-Authors: Roberto Conti, Gandalf LechnerAbstract:Every unitary solution of the Yang-Baxter equation (R-matrix) in dimension $d$ can be viewed as a unitary element of the Cuntz algebra $\CO_d$ and as such defines an Endomorphism of $\CO_d$. These Yang-Baxter Endomorphisms restrict and extend to Endomorphisms of several other $C^*$- and von Neumann algebras and furthermore define a II$_1$ factor associated with an extremal character of the infinite braid group. This paper is devoted to a detailed study of such Yang-Baxter Endomorphisms. Among the topics discussed are characterizations of Yang-Baxter Endomorphisms and the relative commutants of the various subfactors they induce, an Endomorphism perspective on algebraic operations on R-matrices such as tensor products and cabling powers, and properties of characters of the infinite braid group defined by R-matrices. In particular, it is proven that the partial trace of an R-matrix is an invariant for its character by a commuting square argument. Yang-Baxter Endomorphisms also supply information on R-matrices themselves, for example it is shown that the left and right partial traces of an R-matrix coincide and are normal, and that the spectrum of an R-matrix can not be concentrated in a small disc. Upper and lower bounds on the minimal and Jones indices of Yang-Baxter Endomorphisms are derived, and a full characterization of R-matrices defining ergodic Endomorphisms is given. As examples, so-called simple R-matrices are discussed in any dimension~$d$, and the set of all Yang-Baxter Endomorphisms in $d=2$ is completely analyzed.
Orgest Zaka - One of the best experts on this subject based on the ideXlab platform.
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skew field of trace preserving Endomorphisms of translation group in affine plane
Proyecciones (antofagasta), 2020Co-Authors: Orgest Zaka, Mohanad A MohammedAbstract:We will show how to constructed an Skew-Field with trace-preserving Endomorphisms of the affine plane. Earlier in my paper, we doing a detailed description of Endomorphisms algebra and trace-preserving Endomorphisms algebra in an affine plane, and we have constructed an associative unitary ring for which trace-preserving Endomorphisms. In this paper we formulate and prove an important Lemma, which enables us to construct a particular trace-preserving Endomorphism, with the help of which we can construct the inverse trace-preserving Endomorphisms of every trace-preserving Endomorphism. At the end of this paper we have proven that the set of tracepreserving Endomorphisms together with the actions of ’addition’ and ’composition’ (which is in the role of ’multiplication’) forms a skewfield.
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the Endomorphisms algebra of translations group and associative unitary ring of trace preserving Endomorphisms in affine plane
Proyecciones (antofagasta), 2020Co-Authors: Orgest Zaka, Mohanad A MohammedAbstract:A description of Endomorphisms of the translation group is introduced in an affine plane, will define the addition and composition of the set of Endomorphisms and specify the neutral elements associated with these two actions and present the Endomorphism algebra thereof will distinguish the Trace-preserving Endomorphism algebra in affine plane, and prove that the set of Trace-preserving Endomorphism associated with the ’addition’ action forms a commutative group. We also try to prove that the set of trace-preserving Endomorphism, together with the two actions, in it, ’addition’ and ’composition’ forms an associative and unitary ring.
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Skew-Field of Trace-Preserving Endomorphisms, of Translation Group in Affine Plane
arXiv: General Mathematics, 2020Co-Authors: Orgest ZakaAbstract:In this paper we will show how to constructed an Skew-Field with trace-preserving Endomorphisms of the affine plane. Earlier in my paper, we doing a detailed description of Endomorphisms algebra and trace-preserving Endomorphisms algebra in an affine plane, and we have constructed an associative unitary ring for which trace-preserving Endomorphisms. In this paper we formulate and prove an important Lemma, which enables us to construct a particular trace-preserving Endomorphism, with the help of which we can construct the inverse trace-preserving Endomorphisms of every trace-preserving Endomorphism. At the end of this paper we have proven that the set of trace-preserving Endomorphisms together with the actions of 'addition' and 'composition' (which is in the role of 'multiplication') forms a skew-field.
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the Endomorphisms algebra of translations group and associative unitary ring of trace preserving Endomorphisms in affine plane
arXiv: General Mathematics, 2020Co-Authors: Orgest ZakaAbstract:This paper introduces a description of Endomorphisms of the translation group in an affine plane, will define the addition and composition of the set of Endomorphisms and specify the neutral elements associated with these two actions and present the Endomorphism algebra thereof will distinguish the Trace-preserving Endomorphism algebra in affine plane, and prove that the set of Trace-preserving Endomorphism associated with the 'addition' action forms a commutative group. We also try to prove that the set of trace-preserving Endomorphism, together with the two actions, in it, 'addition' and 'composition' forms an associative and unitary ring.
Khosro Tajbakhsh - One of the best experts on this subject based on the ideXlab platform.
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On the Density of Pre-Orbits under Linear Toral Endomorphisms
Vestnik St. Petersburg University Mathematics, 2020Co-Authors: Saeed Azimi, Khosro TajbakhshAbstract:It is known that if each pre-orbit of a non-injective Endomorphism is dense, the Endomorphism is transitive (i.e., a dense orbit exists). However, it is still unknown whether the pre-orbits of an Anosov map are dense, and the conditions necessary for all pre-orbits to be dense are also unknown. Using the properties of integral lattices, we construct our proof by considering the pre-orbits of linear Endomorphisms. We introduce a class of hyperbolic linear Endomorphisms characterized by the property of absolute hyperbolicity to prove that if A : T ^ m → T ^ m is an absolutely hyperbolic Endomorphism, the pre-orbit of any point is dense in T ^ m .
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Classification of Special Anosov Endomorphisms of Nil-manifolds
Acta Mathematica Sinica English Series, 2019Co-Authors: Seyed Mohsen Moosavi, Khosro TajbakhshAbstract:In this paper we give a classification of special Endomorphisms of nil-manifolds: Let f : N /Γ → N /Γ be a covering map of a nil-manifold and denote by A : N /Γ → N /Γ the nil-Endomorphism which is homotopic to f . If f is a special TA -map, then A is a hyperbolic nil-Endomorphism and f is topologically conjugate to A .
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Topology of pre-images under Anosov Endomorphisms
arXiv: Dynamical Systems, 2017Co-Authors: Mohammad Saeed Azimi, Khosro TajbakhshAbstract:For an Endomorphism it is known that if all the points in the manifold have dense sets of pre-images then the dynamical system is transitive. The inverse has been shown for a residual set of points but the the exact inverse has not yet been investigated before. Here we are going to show that under some conditions it is true for Anosov Endomorphisms on closed manifolds, by using the fact that Anosov Endomorphisms are covering maps.
Mohanad A Mohammed - One of the best experts on this subject based on the ideXlab platform.
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skew field of trace preserving Endomorphisms of translation group in affine plane
Proyecciones (antofagasta), 2020Co-Authors: Orgest Zaka, Mohanad A MohammedAbstract:We will show how to constructed an Skew-Field with trace-preserving Endomorphisms of the affine plane. Earlier in my paper, we doing a detailed description of Endomorphisms algebra and trace-preserving Endomorphisms algebra in an affine plane, and we have constructed an associative unitary ring for which trace-preserving Endomorphisms. In this paper we formulate and prove an important Lemma, which enables us to construct a particular trace-preserving Endomorphism, with the help of which we can construct the inverse trace-preserving Endomorphisms of every trace-preserving Endomorphism. At the end of this paper we have proven that the set of tracepreserving Endomorphisms together with the actions of ’addition’ and ’composition’ (which is in the role of ’multiplication’) forms a skewfield.
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the Endomorphisms algebra of translations group and associative unitary ring of trace preserving Endomorphisms in affine plane
Proyecciones (antofagasta), 2020Co-Authors: Orgest Zaka, Mohanad A MohammedAbstract:A description of Endomorphisms of the translation group is introduced in an affine plane, will define the addition and composition of the set of Endomorphisms and specify the neutral elements associated with these two actions and present the Endomorphism algebra thereof will distinguish the Trace-preserving Endomorphism algebra in affine plane, and prove that the set of Trace-preserving Endomorphism associated with the ’addition’ action forms a commutative group. We also try to prove that the set of trace-preserving Endomorphism, together with the two actions, in it, ’addition’ and ’composition’ forms an associative and unitary ring.
