Closed ideals with bounded approximate identities in some Banach algebras

2020 ◽  
Vol 148 (8) ◽  
pp. 3473-3478
Author(s):  
Issa Mohamadi
Keyword(s):  
Banach Algebras ◽  
Approximate Identities

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.

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.

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.

scholarly journals A characterization ofB∗-algebras

1997 ◽  
Vol 20 (3) ◽  
pp. 483-486
Author(s):  
A. K. Gaur
Keyword(s):  
Banach Algebras ◽  
Approximate Identities

A characterization ofB∗-algebras amongst all Banach algebras with bounded approximate identities is obtained.

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.

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.

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