Generalized Derivations on (Semi-)Prime Rings and Noncommutative Banach Algebras
Lie derivations of dual extensions of algebras
Let K be a field and Γ a finite quiver without oriented cycles. Let Λ := K(Γ,ρ) be the quotient algebra of the path algebra KΓ by the ideal generated by ρ, and let 𝒟(Λ) be the dual extension of Λ. We prove that each Lie derivation of 𝒟(Λ) is of the standard form.
Posner's second theorem and annihilator conditions with generalized skew derivations
Let be a prime ring of characteristic different from 2, be its right Martindale quotient ring and be its extended centroid. Suppose that is a non-zero generalized skew derivation of and f(x₁,..., xₙ) is a non-central multilinear polynomial over with n non-commuting variables. If there exists a non-zero element a of such that a[ (f(r₁,..., rₙ)),f(r₁, ..., rₙ)] = 0 for all r₁, ..., rₙ ∈ , then one of the following holds: (a) there exists λ ∈ such that (x) = λx for all x ∈ ; (b) there exist and...
Nonlinear Lie-type derivations of von Neumann algebras and related topics
Motivated by the powerful and elegant works of Miers (1971, 1973, 1978) we mainly study nonlinear Lie-type derivations of von Neumann algebras. Let 𝓐 be a von Neumann algebra without abelian central summands of type I₁. It is shown that every nonlinear Lie n-derivation of 𝓐 has the standard form, that is, can be expressed as a sum of an additive derivation and a central-valued mapping which annihilates each (n-1)th commutator of 𝓐. Several potential research topics related to our work are also...
Centralizing traces and Lie-type isomorphisms on generalized matrix algebras: a new perspective
Let be a commutative ring, be a generalized matrix algebra over with weakly loyal bimodule and be the center of . Suppose that is an -bilinear mapping and that is a trace of . The aim of this article is to describe the form of satisfying the centralizing condition (and commuting condition ) for all . More precisely, we will revisit the question of when the centralizing trace (and commuting trace) has the so-called proper form from a new perspective. Using the aforementioned...
An observation on spaces with a zeroset diagonal
We say that a space has the discrete countable chain condition (DCCC for short) if every discrete family of nonempty open subsets of is countable. A space has a zeroset diagonal if there is a continuous mapping with , where . In this paper, we prove that every first countable DCCC space with a zeroset diagonal has cardinality at most .
A note on star Lindelöf, first countable and normal spaces
A topological space is said to be star Lindelöf if for any open cover of there is a Lindelöf subspace such that . The “extent” of is the supremum of the cardinalities of closed discrete subsets of . We prove that under every star Lindelöf, first countable and normal space must have countable extent. We also obtain an example under , which shows that a star Lindelöf, first countable and normal space may not have countable extent.
On Schur convexity of some symmetric functions.
Inequalities for generalized logarithmic means.
Two sharp inequalities for power mean, geometric mean, and harmonic mean.
Schur-convexity for a class of symmetric functions and its applications.
Some results on spaces with -calibre
We prove that, assuming , if is a space with -calibre and a zeroset diagonal, then is submetrizable. This gives a consistent positive answer to the question of Buzyakova in Observations on spaces with zeroset or regular -diagonals, Comment. Math. Univ. Carolin. 46 (2005), no. 3, 469–473. We also make some observations on spaces with -calibre.
Spaces with property
We prove that if is a first countable space with property and with a -diagonal then the cardinality of is at most . We also show that if is a first countable, DCCC, normal space then the extent of is at most .
Some results on semi-stratifiable spaces
We study relationships between separability with other properties in semi-stratifiable spaces. Especially, we prove the following statements: (1) If is a semi-stratifiable space, then is separable if and only if is ; (2) If is a star countable extent semi-stratifiable space and has a dense metrizable subspace, then is separable; (3) Let be a -monolithic star countable extent semi-stratifiable space. If and , then is hereditarily separable. Finally, we prove that for any -space...
Cardinalities of DCCC normal spaces with a rank 2-diagonal
A topological space has a rank 2-diagonal if there exists a diagonal sequence on of rank , that is, there is a countable family of open covers of such that for each , . We say that a space satisfies the Discrete Countable Chain Condition (DCCC for short) if every discrete family of nonempty open subsets of is countable. We mainly prove that if is a DCCC normal space with a rank 2-diagonal, then the cardinality of is at most . Moreover, we prove that if is a first countable...
The optimal upper and lower power mean bounds for a convex combination of the arithmetic and logarithmic means.
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