scholarly journals Maximal ideal spaces of Banach algebras of derivable elements

1970 ◽  
Vol 11 (3) ◽  
pp. 310-312 ◽  
Author(s):  
R. J. Loy
Keyword(s):  
Banach Algebra ◽  
Maximal Ideal ◽  
Present Note ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Supremum Norm ◽  
Simple Consequence ◽  
Maximal Ideals ◽  
Image Position

Let A be a commutative Banach algebra, D a closed derivation defined on a subalgebra Δ of A, and with range in A. The elements of Δ may be called derivable in the obvious sense. For each integer k ≦.l, denote by Δk the domain of Dk (so that Dgr;1 = Δ); it is a simple consequence of Leibniz's formula that each Δk is an algebra. The classical example of this situation is A = C(O, 1) under the supremum norm with D ordinary differentiation, and here Δk = Ck(0, 1) is a Banach algebra under the norm ∥.∥k: Furthermore, the maximal ideals of Ak are precisely those subsets of Δk of the form M ∩ Δk where M is a maximal ideal of A, and = M, the bar denoting closure in A. In the present note we show how this extends to the general case.

scholarly journals Tensor Products of Banach Algebras

1959 ◽  
Vol 11 ◽  
pp. 297-310 ◽  
Author(s):  
Bernard R. Gelbaum
Keyword(s):  
Banach Algebra ◽  
Maximal Ideal ◽  
Banach Algebras ◽  
Cartesian Product ◽  
Commutative Banach Algebra ◽  
Maximal Ideal Space ◽  
Locally Compact Abelian Group ◽  
Ideal Space ◽  
Integrable Functions ◽  
Group A

This paper is concerned with a generalization of some recent theorems of Hausner (1) and Johnson (4; 5). Their result can be summarized as follows: Let G be a locally compact abelian group, A a commutative Banach algebra, B1 = Bl(G,A) the (commutative Banach) algebra of A-valued, Bochner integrable junctions on G, 3m1the maximal ideal space of A, m2the maximal ideal space of L1(G) [the [commutative Banach] algebra of complex-valued, Haar integrable functions on G, m3the maximal ideal space of B1. Then m3and the Cartesian product m1 X m2are homeomorphic when the spaces mi, i = 1, 2, 3, are given their weak* topologies. Furthermore, the association between m3and m1 X m2is such as to permit a description of any epimorphism E3: B1 → B1/m3 in terms of related epimorphisms E1: A → A/M1 and E2:L1(G) → Ll(G)/M2, where M1 is in mi i = 1, 2, 3.

ONE FUNCTIONAL OPERATOR INVERSION FORMULA

2001 ◽  
Vol 6 (1) ◽  
pp. 138-146 ◽  
Author(s):  
P. Plaschinsky
Keyword(s):  
Banach Algebra ◽  
Explicit Form ◽  
Maximal Ideal ◽  
Inversion Formula ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Functional Operator ◽  
Inversion Operator ◽  
Commutative Banach Algebras ◽  
Algebra Theory

Some results about inversion formula of functional operator with generalized dilation are given. By means of commutative Banach algebra theory the explicit form of inversion operator is expressed. Some commutative Banach algebras with countable generator systems are constructed, their maximal ideal spaces are investigated.

scholarly journals Estimations des solutions de l'équation de Bezout dans les algèbres de Beurling analytiques

2005 ◽  
Vol 96 (2) ◽  
pp. 307 ◽  
Author(s):  
O. El-Fallah ◽  
M. Zarrabi
Keyword(s):  
Banach Algebra ◽  
Existence Of Solutions ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Gelfand Transform ◽  
Maximal Ideals ◽  
Bezout Equation ◽  
Beurling Algebras

Let $A$ be a unitary commutative Banach algebra with unit $e$. For $f\in A$ we denote by $\hat f$ the Gelfand transform of $f$ defined on $\hat A$, the set of maximal ideals of $A$. Let $(f_1,\dots,f_n)\in A^n$ be such that $\sum_{i=1}^n\|f_i\|^2 \leq 1$. We study here the existence of solutions $(g_1,\dots,g_n)\in A^n$ to the Bezout equation $f_1g_1+\cdots+f_ng_n=e$, whose norm is controlled by a function of $n$ and $\delta=\inf_{\chi\in\hat A}(|\hat f_1(\chi)|^2+\cdots+|\hat f_n(\chi)|^2)^{1/2}$. We treat this problem for the analytic Beurling algebras and their quotient by closed ideals. The general Banach algebras with compact Gelfand transform are also considered.

scholarly journals Commutators in Banach algebras

1979 ◽  
Vol 22 (3) ◽  
pp. 207-211 ◽  
Author(s):  
Vlastimil Pták
Keyword(s):  
Banach Algebra ◽  
Present Author ◽  
Present Note ◽  
Banach Algebras ◽  
Related Condition ◽  
Image Position

In a recent paper (6) the present author has shown that, for an element a of a Banach algebra A, the conditionfor all x∈A and some constant α is equivalent to [x, a]∈Rad a for all x∈A; it turns out that α may be replaced by |α|σ It is the purpose of the present note to investigate a related condition

Commutative Banach Algebras with Idempotent Maximal Ideals

1969 ◽  
Vol 9 (3-4) ◽  
pp. 275-286 ◽  
Author(s):  
R. J. Loy
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Complex Field ◽  
Linear Combinations ◽  
Maximal Ideals ◽  
Commutative Banach Algebras ◽  
Finite Linear

Letbe a commutative Banach algebra over the complex fieldC,Man ideal of. Denote byM2the set of all finite linear combinations of products of elements fromM.Mwill be termed idempotent ifM2=M. The purpose of this paper is to investigate the structure of commutative Banach algebras in which all maximal ideals are idempotent.

scholarly journals Higher point derivations on commutative Banach algebras, III

1981 ◽  
Vol 24 (1) ◽  
pp. 31-40 ◽  
Author(s):  
H. G. Dales ◽  
J. P. McClure
Keyword(s):  
Banach Algebra ◽  
Infinite Order ◽  
Infinite Sequence ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Complex Field ◽  
Linear Functionals ◽  
Point Derivation ◽  
Commutative Banach Algebras ◽  
Image Position

Let A be a commutative Banach algebra with identity 1 over the complex field C, and let d0 be a character on A. We recall that a (higher) point derivation of order q on A at d0 is a sequence d1, …, dq of linear functionals on A such that the identitieshold for each choice of f and g in A and k in {1, …, q}. A point derivation of infinite order is an infinite sequence {dk} of linear functionals such that (1.1) holds for all k. A point derivation is continuous if each dk is continuous, totally discontinuous if dk is discontinuous for each k≧1, and degenerate if d1 = 0.

scholarly journals Maximal ideals and the structure of contractible and amenable Banach algebras

2000 ◽  
Vol 62 (2) ◽  
pp. 221-226 ◽  
Author(s):  
Yong Zhang
Keyword(s):  
Banach Algebra ◽  
Maximal Ideal ◽  
Banach Algebras ◽  
Maximal Ideals

Properties of minimal idempotents in contractible and reflexive amenable Banach algebras are exploited to prove that such a kind of Banach algebra is finite demensional if each maximal ideal is contained in a maximal left or a maximal right ideal that is complemented as a Banach subspace. This result covers several known results on this subject.

A characterization of Jordan and 5-Jordan homomorphisms between Banach algebras

2018 ◽  
Vol 11 (02) ◽  
pp. 1850021 ◽  
Author(s):  
A. Zivari-Kazempour
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Jordan Homomorphism ◽  
Jordan Homomorphisms ◽  
Unital Banach Algebra

We prove that each surjective Jordan homomorphism from a Banach algebra [Formula: see text] onto a semiprime commutative Banach algebra [Formula: see text] is a homomorphism, and each 5-Jordan homomorphism from a unital Banach algebra [Formula: see text] into a semisimple commutative Banach algebra [Formula: see text] is a 5-homomorphism.

Amalgamated duplication of the Banach algebra 𝔄 along a 𝔄-bimodule 𝒜

2018 ◽  
Vol 17 (09) ◽  
pp. 1850169 ◽  
Author(s):  
Hossein Javanshiri ◽  
Mehdi Nemati
Keyword(s):  
Banach Algebra ◽  
Cohomology Group ◽  
Banach Algebras ◽  
Multiplier Algebra ◽  
Topological Center ◽  
Maximal Ideals ◽  
Open Questions ◽  
Arens Regularity ◽  
Amalgamated Duplication

Let [Formula: see text] and [Formula: see text] be Banach algebras such that [Formula: see text] is a Banach [Formula: see text]-bimodule with compatible actions. We define the product [Formula: see text], which is a strongly splitting Banach algebra extension of [Formula: see text] by [Formula: see text]. After characterization of the multiplier algebra, topological center, (maximal) ideals and spectrum of [Formula: see text], we restrict our investigation to the study of semisimplicity, regularity, Arens regularity of [Formula: see text] in relation to that of the algebras [Formula: see text], [Formula: see text] and the action of [Formula: see text] on [Formula: see text]. We also compute the first cohomology group [Formula: see text] for all [Formula: see text] as well as the first-order cyclic cohomology group [Formula: see text], where [Formula: see text] is the [Formula: see text]th dual space of [Formula: see text] when [Formula: see text] and [Formula: see text] itself when [Formula: see text]. These results are not only of interest in their own right, but also they pave the way for obtaining some new results for Lau products and module extensions of Banach algebras as well as triangular Banach algebra. Finally, special attention is devoted to the cyclic and [Formula: see text]-weak amenability of [Formula: see text]. In this context, several open questions arise.

scholarly journals Sectional representation of Banach modules and their multipliers

2003 ◽  
Vol 2003 (13) ◽  
pp. 817-825
Author(s):  
Terje Hõim ◽  
D. A. Robbins
Keyword(s):  
Banach Algebra ◽  
Maximal Ideal ◽  
Commutative Banach Algebra ◽  
Maximal Ideal Space ◽  
Ideal Space ◽  
Banach Module ◽  
The Past ◽  
Banach Modules ◽  
Quotient Modules

LetXbe a Banach module over the commutative Banach algebraAwith maximal ideal spaceΔ. We show that there is a norm-decreasing representation ofXas a space of bounded sections in a Banach bundleπ:ℰ→Δ, whose fibers are quotient modules ofX. There is also a representation ofM(X), the space of multipliersT:A→X, as a space of sections in the same bundle, but this representation may not be continuous. These sectional representations subsume results of various authors over the past three decades.

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