Closed ideals with bounded approximate identities in some Banach algebras

2020 ◽  
Vol 148 (8) ◽  
pp. 3473-3478
Author(s):  
Issa Mohamadi
1994 ◽  
Vol 46 (2) ◽  
pp. 397-414 ◽  
Author(s):  
Yiu-Tung Poon ◽  
Zhong-Jin Ruan

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.


Author(s):  
P. G. Dixon ◽  
G. A. Willis

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.


1972 ◽  
Vol 6 (3) ◽  
pp. 443-445 ◽  
Author(s):  
David A. Robbins

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.


10.53733/132 ◽  
2021 ◽  
Vol 51 ◽  
pp. 115-118
Author(s):  
Ana Lucía Barrenechea ◽  
Carlos Peña

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.


1997 ◽  
Vol 20 (3) ◽  
pp. 483-486
Author(s):  
A. K. Gaur

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


1986 ◽  
Vol 28 (2) ◽  
pp. 211-214 ◽  
Author(s):  
S. I. Ouzomgi

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.


2008 ◽  
Vol 145 (2) ◽  
pp. 403-418 ◽  
Author(s):  
P. LAWSON ◽  
C. J. READ

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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