Lie algebras generated by Jordan operators

Peng Cao; Shanli Sun

Displaying similar documents to “Lie algebras generated by Jordan operators”

On the Jordan model $C_{0}$ operators

Hari Bercovici (1977)

Studia Mathematica

Similarity:

Jordan isomorphisms and maps preserving spectra of certain operator products

Jinchuan Hou, Chi-Kwong Li, Ngai-Ching Wong (2008)

Studia Mathematica

Similarity:

Let ₁, ₂ be (not necessarily unital or closed) standard operator algebras on locally convex spaces X₁, X₂, respectively. For k ≥ 2, consider different products $T ₁ * \dots * T_{k}$ on elements in $_{i}$ , which covers the usual product $T ₁ * \dots * T_{k} = T ₁ \dots T_{k}$ and the Jordan triple product T₁ ∗ T₂ = T₂T₁T₂. Let Φ: ₁ → ₂ be a (not necessarily linear) map satisfying $σ (Φ (A ₁) * \dots * Φ (A_{k})) = σ (A ₁ * \dots * A_{k})$ whenever any one of $A_{i}$ ’s has rank at most one. It is shown that if the range of Φ contains all rank one and rank two operators then Φ must be a Jordan isomorphism multiplied...

Product of operators and numerical range preserving maps

Chi-Kwong Li, Nung-Sing Sze (2006)

Studia Mathematica

Similarity:

Let V be the C*-algebra B(H) of bounded linear operators acting on the Hilbert space H, or the Jordan algebra S(H) of self-adjoint operators in B(H). For a fixed sequence (i₁, ..., iₘ) with i₁, ..., iₘ ∈ 1, ..., k, define a product of $A ₁, . . ., A_{k} \in V$ by $A ₁ * \dots * A_{k} = A_{i ₁} \dots A_{i ₘ}$ . This includes the usual product $A ₁ * \dots * A_{k} = A ₁ \dots A_{k}$ and the Jordan triple product A*B = ABA as special cases. Denote the numerical range of A ∈ V by W(A) = (Ax,x): x ∈ H, (x,x) = 1. If there is a unitary operator U and a scalar μ satisfying $μ^{m} = 1$ such that ϕ: V → V has...

On linked Jordan curves in $R^{3}$

M. Mateljević (1975)

Matematički Vesnik

Similarity:

Non-hyperreflexive reflexive spaces of operators

Roman V. Bessonov, Janko Bračič, Michal Zajac (2011)

Studia Mathematica

Similarity:

We study operators whose commutant is reflexive but not hyperreflexive. We construct a C₀ contraction and a Jordan block operator $S_{B}$ associated with a Blaschke product B which have the above mentioned property. A sufficient condition for hyperreflexivity of $S_{B}$ is given. Some other results related to hyperreflexivity of spaces of operators that could be interesting in themselves are proved.

Equidecomposability of Jordan domains under groups of isometries

M. Laczkovich (2003)

Fundamenta Mathematicae

Similarity:

Let $G_{d}$ denote the isometry group of $ℝ^{d}$ . We prove that if G is a paradoxical subgroup of $G_{d}$ then there exist G-equidecomposable Jordan domains with piecewise smooth boundaries and having different volumes. On the other hand, we construct a system $ℱ_{d}$ of Jordan domains with differentiable boundaries and of the same volume such that $ℱ_{d}$ has the cardinality of the continuum, and for every amenable subgroup G of $G_{d}$ , the elements of $ℱ_{d}$ are not G-equidecomposable; moreover, their interiors are not G-equidecomposable...

On James and Jordan-von Neumann constants and the normal structure coefficient of Banach spaces

Mikio Kato, Lech Maligranda, Yasuji Takahashi (2001)

Studia Mathematica

Similarity:

Some relations between the James (or non-square) constant J(X) and the Jordan-von Neumann constant $C_{N J} (X)$ , and the normal structure coefficient N(X) of Banach spaces X are investigated. Relations between J(X) and J(X*) are given as an answer to a problem of Gao and Lau [16]. Connections between $C_{N J} (X)$ and J(X) are also shown. The normal structure coefficient of a Banach space is estimated by the $C_{N J} (X)$ -constant, which implies that a Banach space with $C_{N J} (X)$ -constant less than 5/4 has the fixed point property. ...

A Menon-type identity using Klee's function

Arya Chandran, Neha Elizabeth Thomas, K. Vishnu Namboothiri (2022)

Czechoslovak Mathematical Journal

Similarity:

Menon’s identity is a classical identity involving gcd sums and the Euler totient function $φ$ . A natural generalization of $φ$ is the Klee’s function $Φ_{s}$ . We derive a Menon-type identity using Klee’s function and a generalization of the gcd function. This identity generalizes an identity given by Y. Li and D. Kim (2017).

The Embeddability of c₀ in Spaces of Operators

Ioana Ghenciu, Paul Lewis (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

Results of Emmanuele and Drewnowski are used to study the containment of c₀ in the space $K_{w *} (X *, Y)$ , as well as the complementation of the space $K_{w *} (X *, Y)$ of w*-w compact operators in the space $L_{w *} (X *, Y)$ of w*-w operators from X* to Y.

SCAP-subalgebras of Lie algebras

Sara Chehrazi, Ali Reza Salemkar (2016)

Czechoslovak Mathematical Journal

Similarity:

A subalgebra $H$ of a finite dimensional Lie algebra $L$ is said to be a $SCAP$ -subalgebra if there is a chief series $0 = L_{0} \subset L_{1} \subset ... \subset L_{t} = L$ of $L$ such that for every $i = 1, 2, ..., t$ , we have $H + L_{i} = H + L_{i - 1}$ or $H \cap L_{i} = H \cap L_{i - 1}$ . This is analogous to the concept of $SCAP$ -subgroup, which has been studied by a number of authors. In this article, we investigate the connection between the structure of a Lie algebra and its $SCAP$ -subalgebras and give some sufficient conditions for a Lie algebra to be solvable or supersolvable.

Product decompositions of quasirandom groups and a Jordan type theorem

Nikolay Nikolov, László Pyber (2011)

Journal of the European Mathematical Society

Similarity:

We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If $k$ is the minimal degree of a representation of the finite group $G$ , then for every subset $B$ of $G$ with $| B | > | G | / k^{1 / 3}$ we have $B^{3} = G$ . We use this to obtain improved versions of recent deep theorems of Helfgott and of Shalev concerning product decompositions of finite simple groups, with much simpler proofs. On the other hand, we prove a version of Jordan’s theorem which implies that if $k \geq 2$ , then $G$ has a...

Why Jordan algebras are natural in statistics: quadratic regression implies Wishart distributions

G. Letac, J. Wesołowski (2011)

Bulletin de la Société Mathématique de France

Similarity:

If the space $𝒬$ of quadratic forms in $ℝ^{n}$ is splitted in a direct sum $𝒬_{1} \oplus ... \oplus 𝒬_{k}$ and if $X$ and $Y$ are independent random variables of $ℝ^{n}$ , assume that there exist a real number $a$ such that $E (X | X + Y) = a (X + Y)$ and real distinct numbers $b_{1}, . . ., b_{k}$ such that $E (q (X) | X + Y) = b_{i} q (X + Y)$ for any $q$ in $𝒬_{i} .$ We prove that this happens only when $k = 2$ , when $ℝ^{n}$ can be structured in a Euclidean Jordan algebra and when $X$ and $Y$ have Wishart distributions corresponding to this structure.

Spaces of compact operators on $C (2^{} \times [0, α])$ spaces

Elói Medina Galego (2011)

Colloquium Mathematicae

Similarity:

We classify, up to isomorphism, the spaces of compact operators (E,F), where E and F are the Banach spaces of all continuous functions defined on the compact spaces $2^{} \times [0, α]$ , the topological products of Cantor cubes $2^{}$ and intervals of ordinal numbers [0,α].

$𝔤$ -quasi-Frobenius Lie algebras

David N. Pham (2016)

Archivum Mathematicum

Similarity:

A Lie version of Turaev’s $\bar{G}$ -Frobenius algebras from 2-dimensional homotopy quantum field theory is proposed. The foundation for this Lie version is a structure we call a $𝔤$ -quasi-Frobenius Lie algebra for $𝔤$ a finite dimensional Lie algebra. The latter consists of a quasi-Frobenius Lie algebra $(𝔮, β)$ together with a left $𝔤$ -module structure which acts on $𝔮$ via derivations and for which $β$ is $𝔤$ -invariant. Geometrically, $𝔤$ -quasi-Frobenius Lie algebras are the Lie algebra structures associated to...

$\mathcal{L}$ -homomorphisms of Lie algebras

Aleksander A. Lashkhi (1981)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

Similarity:

Si studiano gli omomorfismi reticolari ( $\mathcal{L}$ -omomorfismi) di algebre di Lie sopra anelli commutativi con unità. Le algebre di Lie sopra un campo e le $p$ -algebre di Lie non ammettono $\mathcal{L}$ -omomorfismi propri. Si assegnano condizioni necessarie e sufficienti affinchè un'algebra di Lie periodica o mista possieda un « $\mathcal{L}$ -omomorfismo su una catena di lunghezza $n$ .

$\mathcal{L}$ -homomorphisms of Lie algebras

Aleksander A. Lashkhi (1981)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Similarity:

Interpolation by elementary operators

Bojan Magajna (1993)

Studia Mathematica

Similarity:

Given two n-tuples $a = (a_{1}, . . ., a_{n})$ and $b = (b_{1}, . . ., b_{n})$ of bounded linear operators on a Hilbert space the question of when there exists an elementary operator E such that $E a_{j} = b_{j}$ for all j =1,...,n, is studied. The analogous question for left multiplications (instead of elementary operators) is answered in any C*-algebra A, as a consequence of the characterization of closed left A-submodules in $A^{n}$ .

2-summing multiplication operators

Dumitru Popa (2013)

Studia Mathematica

Similarity:

Let 1 ≤ p < ∞, $= {(X ₙ)}_{n \in ℕ}$ be a sequence of Banach spaces and $l_{p} ()$ the coresponding vector valued sequence space. Let $= {(X ₙ)}_{n \in ℕ}$ , $= {(Y ₙ)}_{n \in ℕ}$ be two sequences of Banach spaces, $= {(V ₙ)}_{n \in ℕ}$ , Vₙ: Xₙ → Yₙ, a sequence of bounded linear operators and 1 ≤ p,q < ∞. We define the multiplication operator $M_{} : l_{p} () \to l_{q} ()$ by $M_{} ({(x ₙ)}_{n \in ℕ}) : = {(V ₙ (x ₙ))}_{n \in ℕ}$ . We give necessary and sufficient conditions for $M_{}$ to be 2-summing when (p,q) is one of the couples (1,2), (2,1), (2,2), (1,1), (p,1), (p,2), (2,p), (1,p), (p,q); in the last case 1 < p < 2, 1 < q < ∞. ...

On local CR-transformations of Levi-degenerate group orbits in compact Hermitian symmetric spaces

Wilhelm Kaup, Dmitri Zaitsev (2006)

Journal of the European Mathematical Society

Similarity:

We present a large class of homogeneous 2-nondegenerate CR-manifolds $M$ , both of hypersurface type and of arbitrarily high CR-codimension, with the following property: Every CR-equivalence between domains $U$ , $V$ in $M$ extends to a global real-analytic CR-automorphism of $M$ . We show that this class contains $G$ -orbits in Hermitian symmetric spaces $Z$ of compact type, where $G$ is a real form of the complex Lie group $Aut {(Z)}^{0}$ and $G$ has an open orbit that is a bounded symmetric domain of tube type. ...

Compact operators whose adjoints factor through subspaces of $l_{p}$

Deba P. Sinha, Anil K. Karn (2002)

Studia Mathematica

Similarity:

For p ≥ 1, a subset K of a Banach space X is said to be relatively p-compact if $K \subset \sum_{n = 1}^{\infty} α ₙ x ₙ : α ₙ \in B a l l (l_{p^{'}})$ , where p’ = p/(p-1) and $x ₙ \in l_{p}^{s} (X)$ . An operator T ∈ B(X,Y) is said to be p-compact if T(Ball(X)) is relatively p-compact in Y. Similarly, weak p-compactness may be defined by considering $x ₙ \in l_{p}^{w} (X)$ . It is proved that T is (weakly) p-compact if and only if T* factors through a subspace of $l_{p}$ in a particular manner. The normed operator ideals $(K_{p}, κ_{p})$ of p-compact operators and $(W_{p}, ω_{p})$ of weakly p-compact operators, arising from these factorizations,...

Generalized derivations on Lie ideals in prime rings

Basudeb Dhara, Sukhendu Kar, Sachhidananda Mondal (2015)

Czechoslovak Mathematical Journal

Similarity:

Let $R$ be a prime ring with its Utumi ring of quotients $U$ and extended centroid $C$ . Suppose that $F$ is a generalized derivation of $R$ and $L$ is a noncentral Lie ideal of $R$ such that $F (u) {[F (u), u]}^{n} = 0$ for all $u \in L$ , where $n \geq 1$ is a fixed integer. Then one of the following holds: (1) ...

Generalized Cesàro operators on certain function spaces

Sunanda Naik (2010)

Annales Polonici Mathematici

Similarity:

Motivated by some recent results by Li and Stević, in this paper we prove that a two-parameter family of Cesàro averaging operators $^{b, c}$ is bounded on the Dirichlet spaces $_{p, a}$ . We also give a short and direct proof of boundedness of $^{b, c}$ on the Hardy space $H^{p}$ for 1 < p < ∞.