scholarly journals On Uniform Topological Algebras

2015 ◽  
Vol 0 (0) ◽  
Author(s):  
M. El Azhari
Keyword(s):  
Banach Algebra ◽  
Topological Algebra ◽  
Uniform Norm ◽  
Commutative Banach Algebra ◽  
Normed Algebra ◽  
Topological Algebras

Abstract The uniform norm on a uniform normed Q-algebra is the only uniform Q-algebra norm on it. The uniform norm on a regular uniform normed Q-algebra with unit is the only uniform norm on it. Let A be a uniform topological algebra whose spectrum M(A) is equicontinuous, then A is a uniform normed algebra. Let A be a regular semisimple commutative Banach algebra, then every algebra norm on A is a Q-algebra norm on A.

scholarly journals Continuity of derivations on topological algebras of power series

1969 ◽  
Vol 1 (3) ◽  
pp. 419-424 ◽  
Author(s):  
R.J. Loy
Keyword(s):  
Power Series ◽  
Banach Algebra ◽  
Formal Power Series ◽  
Topological Algebra ◽  
Fréchet Space ◽  
Frechet Space ◽  
Complex Field ◽  
Topological Algebras ◽  
Space Topology ◽  
Formal Power

Let A be an algebra of formal power series in one indeterminate over the complex field, D a derivation on A. It is shown that if A has a Fréchet space topology under which it is a topological algebra, then D is necessarily continuous provided the coordinate projections satisfy a certain equicontinuity condition. This condition is always satisfied if A is a Banach algebra and the projections are continuous. A second result is given, with weaker hypothesis on the projections and correspondingly weaker conclusion.

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.

Generalized Frobenius Algebras and Hopf Algebras

2014 ◽  
Vol 66 (1) ◽  
pp. 205-240 ◽  
Author(s):  
Miodrag Cristian Iovanov
Keyword(s):  
Hopf Algebras ◽  
Topological Algebra ◽  
Homological Algebra ◽  
General Concept ◽  
Frobenius Algebra ◽  
Theoretic Approach ◽  
Topological Algebras ◽  
Infinite Dimensional ◽  
Finite Case ◽  
Frobenius Algebras

Abstract“Co-Frobenius” coalgebras were introduced as dualizations of Frobenius algebras. We previously showed that they admit left-right symmetric characterizations analogous to those of Frobenius algebras. We consider the more general quasi-co-Frobenius (QcF) coalgebras. The first main result in this paper is that these also admit symmetric characterizations: a coalgebra is QcF if it is weakly isomorphic to its (left, or right) rational dual Rat(C*) in the sense that certain coproduct or product powers of these objects are isomorphic. Fundamental results of Hopf algebras, such as the equivalent characterizations of Hopf algebras with nonzero integrals as left (or right) co-Frobenius, QcF, semiperfect or with nonzero rational dual, as well as the uniqueness of integrals and a short proof of the bijectivity of the antipode for such Hopf algebras all follow as a consequence of these results. This gives a purely representation theoretic approach to many of the basic fundamental results in the theory of Hopf algebras. Furthermore, we introduce a general concept of Frobenius algebra, which makes sense for infinite dimensional and for topological algebras, and specializes to the classical notion in the finite case. This will be a topological algebra A that is isomorphic to its complete topological dual Aν. We show that A is a (quasi)Frobenius algebra if and only if A is the dual C* of a (quasi)co-Frobenius coalgebra C. We give many examples of co-Frobenius coalgebras and Hopf algebras connected to category theory, homological algebra and the newer q-homological algebra, topology or graph theory, showing the importance of the concept.

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.

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.

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.

A New Proof of a Watson's Formula

1988 ◽  
Vol 31 (4) ◽  
pp. 414-418
Author(s):  
Krzysztof Stempak
Keyword(s):  
Banach Algebra ◽  
Function Theory ◽  
Laguerre Polynomials ◽  
Complex Function ◽  
Product Formula ◽  
Commutative Banach Algebra ◽  
Heisenberg Groups ◽  
Complex Function Theory ◽  
Radial Functions

AbstractA new proof of a product formula for Laguerre polynomials, due originally to Watson, is given. Considering the commutative Banach algebra of radial functions on the Heisenberg groups Hn, n ≧ 2, we observe that Watson's formula holds for z = 1,2, 3, …. Then, applying a complex function theory argument, we establish the validity of this formula for other complex values of z, i.e. for Re z > - 1/2.

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].

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