scholarly journals Bounded Approximate Identities in Ternary Banach Algebras

2012 ◽  
Vol 2012 ◽  
pp. 1-6 ◽  
Author(s):  
Madjid Eshaghi Gordji ◽  
Ali Jabbari ◽  
Gwang Hui Kim
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Approximate Identities ◽  
Left And Right

LetAbe a ternary Banach algebra. We prove that ifAhas a left-bounded approximating set, thenAhas a left-bounded approximate identity. Moreover, we show that ifAhas bounded left and right approximate identities, thenAhas a bounded approximate identity. Hence, we prove Altman’s Theorem and Dixon’s Theorem for ternary Banach algebras.

scholarly journals Factorization and bounded approximate identities for a class of convolution Banach algebras

1986 ◽  
Vol 28 (2) ◽  
pp. 211-214 ◽  
Author(s):  
S. I. Ouzomgi
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Approximate Identities

An algebra A factors if, for each a ∈ A, there exist b, c ∈ A with a = bc. A bounded approximate identity for a Banach algebra A is a net (eα) ⊂ A such that aeα → a and eαa → a for each a ∈ A and such that sup ‖eα ‖ < ∞. It is well known [2, 11.10] that if A has a bounded approximate identity, then A factors. But a Banach algebra may factor even if it does not have a bounded approximate identity: an example which is non-commutative and separable, and an example which is commutative and nonseparable, are given in [3, §22]. However, we do not know an example of a commutative, separable Banach algebra which factors, but which does not have a bounded approximate identity. See 4 for related work.

Factorization and unbounded approximate identities in Banach algebras

1990 ◽  
Vol 107 (3) ◽  
pp. 557-571 ◽  
Author(s):  
P. G. Dixon
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Factorization Theorem ◽  
Basic Form ◽  
Converse Result ◽  
Approximate Identities

Cohen's Factorization Theorem says, in its basic form, that if A is a Banach algebra with a bounded left approximate identity, then every element x ∈ A may be written as a product x = ay for some a, y ∈ A. Such is the beauty and importance of this result that much interest attaches to the question of whether the hypothesis of a bounded left approximate identity can be weakened, or whether a converse result exists. This paper contributes to the study of that question.

Spectra of approximate identities in Banach algebras

1979 ◽  
Vol 86 (2) ◽  
pp. 271-278 ◽  
Author(s):  
P. G. Dixon
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Unit Interval ◽  
Complex Banach Algebra ◽  
Approximate Identities

1. Introduction. The aim of this paper is to show that, in every complex Banach algebra with a one-sided or two-sided bounded approximate identity, there exists another bounded approximate identity of the same sort whose spectra lie close to the unit interval [0, 1].

Operator Algebras with Contractive Approximate Identities

1994 ◽  
Vol 46 (2) ◽  
pp. 397-414 ◽  
Author(s):  
Yiu-Tung Poon ◽  
Zhong-Jin Ruan
Keyword(s):  
Operator Algebra ◽  
Operator Algebras ◽  
Banach Algebras ◽  
Approximate Identity ◽  
The Other ◽  
Double Centralizer ◽  
Approximate Identities ◽  
Centralizer Algebras ◽  
Matrix Norms ◽  
Centralizer Algebra

AbstractWe study operator algebras with contractive approximate identities and their double centralizer algebras. These operator algebras can be characterized as L∞- Banach algebras with contractive approximate identities. We provide two examples, which show that given a non-unital operator algebra A with a contractive approximate identity, its double centralizer algebra M(A) may admit different operator algebra matrix norms, with which M(A) contains A as an M-ideal. On the other hand, we show that there is a unique operator algebra matrix norm on the unitalization algebra A1 of A such that A1 contains A as an M-ideal.

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].

Approximate identities in extensions of topologically nilpotent Banach algebras

1992 ◽  
Vol 122 (1-2) ◽  
pp. 45-52 ◽  
Author(s):  
P. G. Dixon ◽  
G. A. Willis
Keyword(s):  
Banach Algebras ◽  
Approximate Identity ◽  
Nilpotent Radical ◽  
Global Identity ◽  
Approximate Identities ◽  
Commutative Banach Algebras

SynopsisIn commutative Banach algebras with factorisation, the existence of an identity (bounded approximate identity) modulo a topologically nilpotent radical implies the existence of a global identity (bounded approximate identity), respectively.

scholarly journals Existence of a bounded approximate identity in a tensor product

1972 ◽  
Vol 6 (3) ◽  
pp. 443-445 ◽  
Author(s):  
David A. Robbins
Keyword(s):  
Tensor Product ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Approximate Identities

It has been shown that the existence of a (left) approximate identity in the tensor product A ⊗ B of Banach algebras A and B, where α is an admissible algebra norm on A ⊗ B, implies the existence of approximate identities in A and B. The question has been raised as to whether the boundedness of the approximate identity in A ⊗αB implies the boundedness of the approximate identities in A and B. This paper answers the question affirmatively with a being the greatest cross-norm.

Weak Amenability of a Class of Banach Algebras

2001 ◽  
Vol 44 (4) ◽  
pp. 504-508 ◽  
Author(s):  
Yong Zhang
Keyword(s):  
Banach Algebra ◽  
Left Ideal ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Dual Algebra ◽  
Weak Amenability ◽  
Second Dual

AbstractWe show that, if a Banach algebra is a left ideal in its second dual algebra and has a left bounded approximate identity, then the weak amenability of implies the (2m+ 1)-weak amenability of for all m ≥ 1.

scholarly journals Invariance and normality of projections in the dual of Banach algebras

10.53733/132 ◽  
2021 ◽  
Vol 51 ◽  
pp. 115-118
Author(s):  
Ana Lucía Barrenechea ◽  
Carlos Peña
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Natural Projection ◽  
Existence Problem ◽  
Approximate Identities

We study the classes of invariant and natural projections in the dual of a Banach algebra $A$. These type of projections are relevant by their connections with the existence problem of bounded approximate identities in closed ideals of Banach algebras. It is known that any invariant projection is a natural projection. In this article we consider the issue of when a natural projection is an invariant projection.

Approximate amenability of Fréchet algebras

2008 ◽  
Vol 145 (2) ◽  
pp. 403-418 ◽  
Author(s):  
P. LAWSON ◽  
C. J. READ
Keyword(s):  
Banach Algebras ◽  
Approximate Identity ◽  
Approximate Identities ◽  
Continuous Point ◽  
Approximate Amenability

AbstractThe notion of approximate amenability was introduced by Ghahramani and Loy, in the hope that it would yield Banach algebras without bounded approximate identity which nonetheless had a form of amenability. So far, however, all known approximately amenable Banach algebras have bounded approximate identities (b.a.i.). In this paper we define approximate amenability and contractibility of Fréchet algebras, and we prove the analogue of the result for Banach algebras that these properties are equivalent. We give examples of Fréchet algebras which are approximately contractible, but which do not have a bounded approximate identity. For a good many Fréchet algebras without b.a.i., we find either that the algebra is approximately amenable, or it is “obviously” not approximately amenable because it has continuous point derivations. So the situation for Fréchet algebras is quite close to what was hoped for Banach algebras.

Sign in / Sign up

Export Citation Format

Share Document