scholarly journals The Relationship between Two Involutive SemigroupsSandSTIs Defined by a Left MultiplierT

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
S. M. Mohammadi ◽  
J. Laali
Keyword(s):  
Banach Algebra ◽  
Mild Conditions ◽  
Underlying Space ◽  
Second Dual ◽  
Left Multiplier ◽  
The Relationship

LetSbe a semigroup with a left multiplierTonS. There exists a new semigroupST, which depends onSandT, which has the same underlying space asS. We study the question of involutions onSTand a Banach algebraAT. We find a condition ofTunder whichSTand the second dualAT****admit an involution. We will show thatATisC*-algebra if and only ifT:AT→Ais an isometry, under mild conditions. Also,AisC*-algebra if and only if so isAT, under other minor conditions.

scholarly journals Weak amenability for the second dual of Banach modules

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Fatemeh Anousheh ◽  
Davood Ebrahimi Bagha ◽  
Abasalt Bodaghi
Keyword(s):  
Banach Algebra ◽  
Mild Conditions ◽  
Weak Amenability ◽  
Banach Modules ◽  
Second Dual

AbstractLet A be a Banach algebra, E be a Banach A-bimodule and Δ E → A be a bounded Banach A-bimodule homomorphism. It is shown that under some mild conditions, the weakΔ''-amenability of E'' (as an A''-bimodule) necessitates weak Δ-amenability of E (as an A-bimodule). Some examples of weak-amenable Banach modules are provided as well.

scholarly journals Compact linear operators from an algebraic standpoint

1967 ◽  
Vol 8 (1) ◽  
pp. 41-49 ◽  
Author(s):  
F. F. Bonsall
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Linear Operators ◽  
Identity Operator ◽  
Algebra Structure ◽  
Natural Setting ◽  
Norm Topology ◽  
Bounded Linear ◽  
Underlying Space ◽  
Closed Subalgebra

Let B(X) denote the Banach algebra of all bounded linear operators on a Banach space X. Let t be an element of B(X), and let edenote the identity operator on X. Since the earliest days of the theory of Banach algebras, ithas been understood that the natural setting within which to study spectral properties of t is the Banach algebra B(X), or perhaps a closed subalgebra of B(X) containing t and e. The effective application of this method to a given class of operators depends upon first translating the data into terms involving only the Banach algebra structure of B(X) without reference to the underlying space X. In particular, the appropriate topology is the norm topology in B(X) given by the usual operator norm. Theorem 1 carries out this translation for the class of compact operators t. It is proved that if t is compact, then multiplication by t is a compact linear operator on the closed subalgebra of B(X) consisting of operators that commute with t.

Amenability of Banach algebras

1989 ◽  
Vol 105 (2) ◽  
pp. 351-355 ◽  
Author(s):  
Frédéric Gourdeau
Keyword(s):  
Banach Algebra ◽  
Metric Space ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Commutative Banach Algebra ◽  
Cohomology Theory ◽  
Compact Metric Space ◽  
Amenable Banach Algebra ◽  
Second Dual

We consider the problem of amenability for a commutative Banach algebra. The question of amenability for a Banach algebra was first studied by B. E. Johnson in 1972, in [5]. The most recent contributions, to our knowledge, are papers by Bade, Curtis and Dales [1], and by Curtis and Loy [3]. In the first, amenability for Lipschitz algebras on a compact metric space K is studied. Using the fact, which they prove, that LipαK is isometrically isomorphic to the second dual of lipαK, for 0 < α < 1, they show that lipαK is not amenable when K is infinite and 0 < α < 1. In the second paper, the authors prove, without using any serious cohomology theory, some results proved earlier by Khelemskii and Scheinberg [8] using cohomology. They also discuss the amenability of Lipschitz algebras, using the result that a weakly complemented closed two-sided ideal in an amenable Banach algebra has a bounded approximate identity. Their result is stronger than that of [1].

On the Relationship between Interpolation of Banach Algebras and Interpolation of Bilinear Operators

2010 ◽  
Vol 53 (1) ◽  
pp. 51-57 ◽  
Author(s):  
Fernando Cobos ◽  
Luz M. Fernández-Cabrera
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Interpolation Theorem ◽  
Algebra Structure ◽  
Bilinear Interpolation ◽  
Bilinear Operators ◽  
Real Method ◽  
The Relationship

AbstractWe show that if the general real method (· , ·)Γ preserves the Banach-algebra structure, then a bilinear interpolation theorem holds for (· , ·)Γ.

scholarly journals On the closure of the prime radical of a Banach algebra

1993 ◽  
Vol 36 (3) ◽  
pp. 421-425 ◽  
Author(s):  
D. W. B. Somerset ◽  
G. A. Willis
Keyword(s):  
Banach Algebra ◽  
Prime Radical ◽  
Prime Ideals ◽  
The Relationship

The relationship between the prime ideals and the primal ideals of a Banach algebra is investigated. It is shown that the closure of the prime radical of a Banach algebra may be properly contained in the intersection of the closed primal ideals of the algebra.

On James' Quasi-Reflexive Banach Space as a Banach Algebra

1980 ◽  
Vol 32 (5) ◽  
pp. 1080-1101 ◽  
Author(s):  
Alfred D. Andrew ◽  
William L. Green
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Reflexive Banach Space ◽  
Equivalent Norm ◽  
New Techniques ◽  
Codimension One ◽  
Gelfand Transform ◽  
Schauder Bases ◽  
Second Dual ◽  
Arens Multiplication

In [4] and [5], R. C. James introduced a non-reflexive Banach space J which is isometric to its second dual. Developing new techniques in the theory of Schauder bases, James identified J**, showed that the canonical image of J in J** is of codimension one, and proved that J** is isometric to J.In Section 2 of this paper we show that J, equipped with an equivalent norm, is a semi-simple (commutative) Banach algebra under point wise multiplication, and we determine its closed ideals. We use the Arens multiplication and the Gelfand transform to identify J**, which is in fact just the algebra obtained from J by adjoining an identity.

scholarly journals The second dual of a Banach algebra

1979 ◽  
Vol 84 (3-4) ◽  
pp. 309-325 ◽  
Author(s):  
J. Duncan ◽  
S. A. R. Hosseiniun
Keyword(s):  
Banach Algebra ◽  
Open Problems ◽  
Current State ◽  
Second Dual ◽  
The Subject

SynopsisWe give a survey of the current state of knowledge on the Arens second dual of a Banach algebra, including some simplified proofs of known results, some new results, some open problems and a full bibliography of the subject.

scholarly journals Amenability and topological centres of the second duals of Banach algebras

2002 ◽  
Vol 65 (2) ◽  
pp. 191-197 ◽  
Author(s):  
F. Ghahramani ◽  
J. Laali
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Dual Algebra ◽  
Arens Product ◽  
Weak Amenability ◽  
Arens Regularity ◽  
Second Dual

Let  be a Banach algebra and let ** be the second dual algebra of  endowed with the first or the second Arens product. We investigate relations between amenability of ** and Arens regularity of  and the rôle topological centres in amenability of **. We also find conditions under which weak amenability of ** implies weak amenability of .

scholarly journals AN IMPROVED METHOD FOR THE PREPARATION OF PHLOROGLUCINOL FROM 1,3,5-TRINITROBENZENE

2021 ◽  
pp. 148-154
Author(s):  
И.А. Щурова ◽  
Н.А. Алексеева ◽  
С.В. Сысолятин ◽  
В.В. Малыхин
Keyword(s):  
Sulfuric Acid ◽  
Molar Ratio ◽  
Improved Method ◽  
Fresh Catalyst ◽  
Pd Catalysts ◽  
Mild Conditions ◽  
Process Cost ◽  
Medicinal Drugs ◽  
The Relationship ◽  
Water Ratio

Статья посвящена способу получения флороглюцина, представляющего интерес в качестве основы для конструирования лекарственных средств, полимеров различного назначения и малочувствительного взрывчатого вещества 1,3,5-триамино-2,4,6-тринитробензола. Современным и наиболее экологичным методом получения флороглюцина является каталитическое гидрирование 1,3,5-тринитробензола на палладиевом катализаторе до 1,3,5-триаминобензола и его последующий гидролиз. Использование палладиевых катализаторов позволяет проводить восстановление в мягких условиях, но их высокая стоимость обуславливает потребность в поиске путей снижения расхода палладия и, соответственно, себестоимости процесса. В данном исследовании показано, что использование 1 %-го Pd/сибунит (50 % к массе субстрата) в сочетании с водно-ацетоновым раствором в качестве среды при проведении гидрирования способствует более длительному сохранению активности катализатора. Установлено, что оптимальное соотношение ацетона и воды находится в диапазоне от 4:1 до 7:1. В этом случае может быть проведено до 20 циклов гидрирования без добавления свежего катализатора, за счет чего удается снизить расход палладия в три раза по сравнению с другими известными методиками. Кроме того, подход позволяет исключить из схемы синтеза токсичный растворитель метанол. Триаминобензол, полученный в ходе гидрирования, без выделения подвергается гидролизу в присутствии серной кислоты с образованием флороглюцина. Изучена зависимость выхода флороглюцина от мольного соотношения серной кислоты и тринитробензола. Установлено, что оптимальное соотношение серная кислота : тринитробензол составляет 2,0-2,4 моль/моль. Суммарный выход флороглюцина составляет 76 % в пересчете на тринитробензол. The study is concerned with a synthetic method for phloroglucinol that is of great concern as a scaffold for designing medicinal drugs, different-purpose polymers and the insensitive explosive 1,3,5-triamino-2,4,6-trinitrobenzene. The current and most eco-benign method for the synthesis of phloroglucinol is through catalytic hydrogenation of 1,3,5-trinitrobenzene over the Pd catalyst to 1,3,5-triaminobenzene followed by its hydrolysis. The use of Pd catalysts allows the reduction under mild conditions, but their high cost necessitates the need to find ways how to spare the Pd usage and, consequently, the process cost. Here we demonstrated that the use of 1% Pd/Sibunite (50% to substrate weight) combined with a water-acetone solution as the medium in hydrogenation allows the catalyst to keep active longer. The optimum acetone-to-water ratio was found to be between 4:1 and 7:1. In this case, as many as 20 hydrogenation runs can be done without a fresh catalyst added whereby the Pd usage can be lowered threefold when compared to the other common methods in use. Besides, this approach allows the toxic solvent methanol to be expelled from the synthetic protocol. Triaminobenzene resulting from the hydrogenation without isolation undergoes hydrolysis in the presence of sulfuric acid to furnish phloroglucinol. The relationship between the phloroglucinol yield and the molar ratio of sulfuric acid and trinitrobenzene was also explored. The optimum sulfuric acid-to-trinitrobenzene ratio was found to be 2.0-2.4 mol/mol. The overall yield of phloroglucinol was 76% on a trinitrobenzene basis.

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