scholarly journals Biderivations and Commuting Linear Maps on Topologically Simple ℒ ∗ -Algebras

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Abdelaziz Ben Yahya
Keyword(s):  
Arbitrary Dimension ◽  
Linear Mapping ◽  
Identity Mapping ◽  
Scalar Multiple ◽  
Linear Maps

Let ℒ be a topologically simple ℒ ∗ -algebra of arbitrary dimension. In this paper, we introduce the notion of semi-inner biderivation in order to prove that every continuous commuting linear mapping on ℒ is a scalar multiple of the identity mapping.

Minimal homeomorphisms on low-dimensional tori

2009 ◽  
Vol 29 (5) ◽  
pp. 1515-1528
Author(s):  
N. M. DOS SANTOS ◽  
R. URZÚA-LUZ
Keyword(s):  
Affine Transformation ◽  
Homology Group ◽  
Arbitrary Dimension ◽  
Linear Part ◽  
Linear Mapping ◽  
Sufficient Condition ◽  
Skew Product ◽  
Lefschetz Fixed Point Theorem ◽  
Low Dimensional ◽  
Minimal Homeomorphisms

AbstractWe study minimal homeomorphisms (all orbits are dense) of the tori Tn, n≤4. The linear part of a homeomorphism φ of Tn is the linear mapping L induced by φ on the first homology group of Tn. It follows from the Lefschetz fixed point theorem that 1 is an eigenvalue of L if φ minimal. We show that if φ is minimal and n≤4, then L is quasi-unipontent, that is, all of the eigenvalues of L are roots of unity and conversely if L∈GL(n,ℤ) is quasi-unipotent and 1 is an eigenvalue of L, then there exists a C∞ minimal skew-product diffeomorphism φ of Tn whose linear part is precisely L. We do not know whether these results are true for n≥5. We give a sufficient condition for a smooth skew-product diffeomorphism of a torus of arbitrary dimension to be smoothly conjugate to an affine transformation.

scholarly journals Linear Maps Preserving Rank-additivity and Rank-sum-minimal on Tensor Products of Matrix Spaces

2019 ◽  
pp. 1-10
Author(s):  
Lele Gao ◽  
Yang Zhang ◽  
Jinli Xu
Keyword(s):  
Matrix Theory ◽  
Tensor Products ◽  
Linear Mapping ◽  
Linear Map ◽  
The Core ◽  
Research Areas ◽  
Linear Maps ◽  
Matrix Spaces

The problems of characterizing maps that preserve certain invariant on given sets are called the preserving problems, which have become one of the core research areas in matrix theory. If for any a linear map, , as established, there is we say that  preserves the rank-additivity. If for any , and a linear map,  established, there is  we say that rank-sum-miminal. In this paper, we characterize the form of linear mapping .

scholarly journals Left Derivable Maps at Non-Trivial Idempotents on Nest Algebras

2019 ◽  
Vol 33 (1) ◽  
pp. 97-105
Author(s):  
Hoger Ghahramani ◽  
Saman Sattari
Keyword(s):  
Identity Element ◽  
Complex Banach Space ◽  
Linear Mapping ◽  
Continuous Linear ◽  
Nest Algebra ◽  
Continuous Linear Mapping ◽  
Nest Algebras ◽  
Left Derivation ◽  
Linear Maps ◽  
Generalized Jordan Left Derivation

AbstractLet Alg 𝒩 be a nest algebra associated with the nest 𝒩 on a (real or complex) Banach space 𝕏. Suppose that there exists a non-trivial idempotent P ∈ Alg 𝒩 with range P (𝕏) ∈ 𝒩, and δ : Alg 𝒩 → Alg 𝒩 is a continuous linear mapping (generalized) left derivable at P, i.e. δ (ab) = aδ (b) + bδ (a) (δ (ab) = aδ(b) + bδ(a) − baδ(I)) for any a, b ∈ Alg 𝒩 with ab = P, where I is the identity element of Alg 𝒩. We show that is a (generalized) Jordan left derivation. Moreover, in a strongly operator topology we characterize continuous linear maps on some nest algebras Alg 𝒩 with the property that δ (P ) = 2Pδ (P ) or δ (P ) = 2P δ (P ) − Pδ (I) for every idempotent P in Alg 𝒩.

scholarly journals An extension of orthogonality relations based on norm derivatives

2018 ◽  
Vol 70 (2) ◽  
pp. 379-393 ◽  
Author(s):  
Ali Zamani ◽  
Mohammad Sal Moslehian
Keyword(s):  
Linear Mapping ◽  
Inner Product ◽  
Product Spaces ◽  
Normed Spaces ◽  
Inner Product Spaces ◽  
Scalar Multiple ◽  
Essential Properties ◽  
Orthogonality Relations ◽  
Norm Derivatives

Abstract We introduce the relation ρλ-orthogonality in the setting of normed spaces as an extension of some orthogonality relations based on norm derivatives and present some of its essential properties. Among other things, we give a characterization of inner product spaces via the functional ρλ. Moreover, we consider a class of linear mappings preserving this new kind of orthogonality. In particular, we show that a linear mapping preserving ρλ-orthogonality has to be a similarity, that is, a scalar multiple of an isometry.

THE HEAT FLOW OF THE CCR ALGEBRA

2002 ◽  
Vol 34 (1) ◽  
pp. 73-83 ◽  
Author(s):  
WILLIAM ARVESON
Keyword(s):  
Degrees Of Freedom ◽  
Integral Operators ◽  
Linear Mapping ◽  
Interaction Theory ◽  
Dense Domain ◽  
Smooth Kernels ◽  
Ccr Algebra ◽  
Linear Maps ◽  
Invariant States ◽  
Positive Linear Maps

Let Pf(x) =−if′(x) and Qf(x) = xf(x) be the canonical operators acting on an appropriate common dense domain in L2(ℝ). The derivations DP(A) = i(PA−AP) and DQ(A) = i(QA−AQ) act on the *-algebra [Ascr ] of all integral operators having smooth kernels of compact support, for example, and one may consider the noncommutative ‘Laplacian’, L = D2P+D2Q, as a linear mapping of [Ascr ] into itself.L generates a semigroup of normal completely positive linear maps on [Bscr ](L2(ℝ)), and this paper establishes some basic properties of this semigroup and its minimal dilation to an E0-semigroup. In particular, the author shows that its minimal dilation is pure and has no normal invariant states, and he discusses the significance of those facts for the interaction theory introduced in a previous paper.There are similar results for the canonical commutation relations with n degrees of freedom, where 1 [les ] n < 1.

Simplex Free Adaptive Tree Fast Sweeping and Evolution Methods for Solving Level Set Equations in Arbitrary Dimension

2005 ◽  
Author(s):  
T. C. Cecil ◽  
S. J. Osher ◽  
J. Qian
Keyword(s):  
Level Set ◽  
Arbitrary Dimension ◽  
Fast Sweeping

scholarly journals Multiplicative Chow–Künneth decompositions and varieties of cohomological K3 type

2021 ◽  
Author(s):  
Lie Fu ◽  
Robert Laterveer ◽  
Charles Vial
Keyword(s):  
Projective Variety ◽  
Arbitrary Dimension ◽  
Smooth Projective Variety ◽  
The Other ◽  
Fano Variety ◽  
Fano Varieties ◽  
Intersection Product ◽  
Canonical Class ◽  
Main Input

AbstractGiven a smooth projective variety, a Chow–Künneth decomposition is called multiplicative if it is compatible with the intersection product. Following works of Beauville and Voisin, Shen and Vial conjectured that hyper-Kähler varieties admit a multiplicative Chow–Künneth decomposition. In this paper, based on the mysterious link between Fano varieties with cohomology of K3 type and hyper-Kähler varieties, we ask whether Fano varieties with cohomology of K3 type also admit a multiplicative Chow–Künneth decomposition, and provide evidence by establishing their existence for cubic fourfolds and Küchle fourfolds of type c7. The main input in the cubic hypersurface case is the Franchetta property for the square of the Fano variety of lines; this was established in our earlier work in the fourfold case and is generalized here to arbitrary dimension. On the other end of the spectrum, we also give evidence that varieties with ample canonical class and with cohomology of K3 type might admit a multiplicative Chow–Künneth decomposition, by establishing this for two families of Todorov surfaces.

Linear maps preserving structured singular values of matrices

2021 ◽  
Author(s):  
Constantin Costara
Keyword(s):  
Singular Values ◽  
Linear Maps
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