scholarly journals Solution to a problem of FitzGerald

2021 ◽  
Author(s):  
Jens Hemelaer ◽  
Morgan Rogers
Keyword(s):  
Affirmative Answer ◽  
General Algebra

AbstractFitzGerald identified four conditions (RI), (UR), (RI*) and (UR*) that are necessarily satisfied by an algebra if its monoid of endomorphisms has commuting idempotents. We show that these conditions are not sufficient, by giving an example of an algebra satisfying the four properties, such that its monoid of endomorphisms does not have commuting idempotents. This settles a problem presented by FitzGerald at the Conference and Workshop on General Algebra and Its Applications in 2013 and more recently at the workshop NCS 2018. After giving the counterexample, we show that the properties (UR), (RI*) and (UR*) depend only on the monoid of endomorphisms of the algebra, and that the counterexample we gave is in some sense the easiest possible. Finally, we list some categories in which FitzGerald’s question has an affirmative answer.

A SPHERICAL VERSION OF THE KOWALSKI–SŁODKOWSKI THEOREM AND ITS APPLICATIONS

2020 ◽  
pp. 1-26
Author(s):  
SHIHO OI
Keyword(s):  
Banach Algebra ◽  
Function Space ◽  
Affirmative Answer ◽  
List Type ◽  
Private Communication ◽  
Type Number ◽  
Uniform Algebras ◽  
Surjective Isometry ◽  
Complex Linear ◽  
Local Map

Abstract Li et al. [‘Weak 2-local isometries on uniform algebras and Lipschitz algebras’, Publ. Mat.63 (2019), 241–264] generalized the Kowalski–Słodkowski theorem by establishing the following spherical variant: let A be a unital complex Banach algebra and let $\Delta : A \to \mathbb {C}$ be a mapping satisfying the following properties: (a) $\Delta $ is 1-homogeneous (that is, $\Delta (\lambda x)=\lambda \Delta (x)$ for all $x \in A$ , $\lambda \in \mathbb C$ ); (b) $\Delta (x)-\Delta (y) \in \mathbb {T}\sigma (x-y), \quad x,y \in A$ . Then $\Delta $ is linear and there exists $\lambda _{0} \in \mathbb {T}$ such that $\lambda _{0}\Delta $ is multiplicative. In this note we prove that if (a) is relaxed to $\Delta (0)=0$ , then $\Delta $ is complex-linear or conjugate-linear and $\overline {\Delta (\mathbf {1})}\Delta $ is multiplicative. We extend the Kowalski–Słodkowski theorem as a conclusion. As a corollary, we prove that every 2-local map in the set of all surjective isometries (without assuming linearity) on a certain function space is in fact a surjective isometry. This gives an affirmative answer to a problem on 2-local isometries posed by Molnár [‘On 2-local *-automorphisms and 2-local isometries of B(H)', J. Math. Anal. Appl.479(1) (2019), 569–580] and also in a private communication between Molnár and O. Hatori, 2018.

A proof of a conjecture by Jabara on groups in which all involutions are odd transpositions

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Egle Bettio
Keyword(s):  
Affirmative Answer ◽  
Kourovka Notebook

Abstract In this paper, we prove that if 𝐺 is a group generated by elements of order two with the property that the product of any two such elements has order 1, 2, 3 or 5 with all possibilities occurring, then G ≃ A 5 G\simeq A_{5} or G ≃ PSU ⁢ ( 3 , 4 ) G\simeq\mathrm{PSU}(3,4) . This provides an affirmative answer to Problem 19.36 in the Kourovka notebook.

Justice as a Personal Virtue and Justice as an Institutional Virtue: Mencius’s Confucian Virtue Politics

2020 ◽  
Vol 2019 (4) ◽  
pp. 277-294
Author(s):  
Yong Huang
Keyword(s):  
Political Philosophy ◽  
Virtue Ethics ◽  
Social Institutions ◽  
Affirmative Answer ◽  
The State ◽  
Point Of View ◽  
Character Traits ◽  
Social Policies ◽  
Moral Principles ◽  
The Government

AbstractIt has been widely observed that virtue ethics, regarded as an ethics of the ancient, in contrast to deontology and consequentialism, seen as an ethics of the modern (Larmore 1996: 19–23), is experiencing an impressive revival and is becoming a strong rival to utilitarianism and deontology in the English-speaking world in the last a few decades. Despite this, it has been perceived as having an obvious weakness in comparison with its two major rivals. While both utilitarianism and deontology can at the same time serve as an ethical theory, providing guidance for individual persons and a political philosophy, offering ways to structure social institutions, virtue ethics, as it is concerned with character traits of individual persons, seems to be ill-equipped to be politically useful. In recent years, some attempts have been made to develop the so-called virtue politics, but most of them, including my own (see Huang 2014: Chapter 5), are limited to arguing for the perfectionist view that the state has the obligation to do things to help its members develop their virtues, and so the focus is still on the character traits of individual persons. However important those attempts are, such a notion of virtue politics is clearly too narrow, unless one thinks that the only job the state is supposed to do is to cultivate its people’s virtues. Yet obviously the government has many other jobs to do such as making laws and social policies, many if not most of which are not for the purpose of making people virtuous. The question is then in what sense such laws and social policies are moral in general and just in particular. Utilitarianism and deontology have their ready answers in the light of utility or moral principles respectively. Can virtue ethics provide its own answer? This paper attempts to argue for an affirmative answer to this question from the Confucian point of view, as represented by Mencius. It does so with a focus on the virtue of justice, as it is a central concept in both virtue ethics and political philosophy.

Simultaneous Triangularization of Algebras of Polynomially Compact Operators

1991 ◽  
Vol 34 (2) ◽  
pp. 260-264 ◽  
Author(s):  
M. Radjabalipour
Keyword(s):  
Hilbert Space ◽  
Jacobson Radical ◽  
Compact Operators ◽  
General Algebra ◽  
Closed Algebra ◽  
Quasinilpotent Operators ◽  
Simultaneous Triangularization

AbstractIf A is a norm closed algebra of compact operators on a Hilbert space and if its Jacobson radical J(A) consists of all quasinilpotent operators in A then A/ J(A) is commutative. The result is not valid for a general algebra of polynomially compact operators.

scholarly journals A NOTE ON WEIGHTED BADLY APPROXIMABLE LINEAR FORMS

2016 ◽  
Vol 59 (2) ◽  
pp. 349-357 ◽  
Author(s):  
STEPHEN HARRAP ◽  
NIKOLAY MOSHCHEVITIN
Keyword(s):  
Diophantine Approximation ◽  
Affirmative Answer ◽  
Famous Conjecture ◽  
Linear Forms ◽  
Badly Approximable

AbstractWe prove a result in the area of twisted Diophantine approximation related to the theory of Schmidt games. In particular, under certain restrictions we give an affirmative answer to the analogue in this setting of a famous conjecture of Schmidt from Diophantine approximation.

Norm inequalities for positive semi-definite matrices and a question of Bourin II

2021 ◽  
pp. 2150043
Author(s):  
Mostafa Hayajneh ◽  
Saja Hayajneh ◽  
Fuad Kittaneh
Keyword(s):  
Affirmative Answer ◽  
Unitarily Invariant Norm ◽  
Norm Inequalities ◽  
Invariant Norm ◽  
Special Case

Let [Formula: see text] and [Formula: see text] be [Formula: see text] positive semi-definite matrices. It is shown that [Formula: see text] for every unitarily invariant norm. This gives an affirmative answer to a question of Bourin in a special case. It is also shown that [Formula: see text] for [Formula: see text] and for every unitarily invariant norm.

Elements of Order Coxeter Number +1 in Chevalley Groups

1982 ◽  
Vol 34 (4) ◽  
pp. 945-951 ◽  
Author(s):  
Bomshik Chang
Keyword(s):  
Weyl Group ◽  
Chevalley Group ◽  
Trivial Solution ◽  
Affirmative Answer ◽  
Chevalley Groups ◽  
General Solutions ◽  
Coxeter Number ◽  
Image Position

Following the notation and the definitions in [1], let L(K) be the Chevalley group of type L over a field K, W the Weyl group of L and h the Coxeter number, i.e., the order of Coxeter elements of W. In a letter to the author, John McKay asked the following question: If h + 1 is a prime, is there an element of order h + 1 in L(C)? In this note we give an affirmative answer to this question by constructing an element of order h + 1 (prime or otherwise) in the subgroup Lz = 〈xτ(1)|r ∈ Φ〉 of L(K), for any K.Our problem has an immediate solution when L = An. In this case h = n + 1 and the (n + l) × (n + l) matrixhas order 2(h + 1) in SLn+1(K). This seemingly trivial solution turns out to be a prototype of general solutions in the following sense.

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