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.

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 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.

scholarly journals Bounded approximate identities and tensor products

1972 ◽  
Vol 7 (3) ◽  
pp. 443-445
Author(s):  
J.R. Holub
Keyword(s):  
Tensor Product ◽  
Banach Algebras ◽  
Tensor Products ◽  
Approximate Identity ◽  
General Question ◽  
Approximate Identities

The question has been raised [R.J. Loy, Bull. Austral. Math. Soc. 5 (1970), 253–260] as to whether the existence of a bounded (left) approximate identity in the tensor product A ⊗αB of Banach algebras A and B (for a a crossnorm on A ⊗ B ) implies the existence of a bounded (left) approximate identity in A and B. This is known [David A. Robbins, Bull. Austral. Math. Soc. 6 (1972), 443–445] to be the case for α equal to the greatest crossnorm. This paper answers the general question affirmatively.

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].

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.

Approximate Identities in Banach Algebras of Compact Operators

1993 ◽  
Vol 36 (1) ◽  
pp. 45-53 ◽  
Author(s):  
Niels Grønbæk ◽  
George A. Willis
Keyword(s):  
Banach Space ◽  
Finite Rank ◽  
General Framework ◽  
Banach Algebras ◽  
Compact Operators ◽  
Approximate Identity ◽  
Approximation Properties ◽  
Approximate Identities ◽  
Set Up ◽  
Uniformly Closed

AbstractLet X be a Banach space and let A be a uniformly closed algebra of compact operators on X, containing the finite rank operators. We set up a general framework to discuss the equivalence between Banach space approximation properties and the existence of right approximate identities in A. The appropriate properties require approximation in the dual X* by operators which are adjoints of operators on X. We show that the existence of a bounded right approximate identity implies that of a bounded left approximate identity. We give examples to show that these properties are not equivalent, however. Finally, we discuss the well known result that, if X* has a basis, then X has a shrinking basis. We make some attempts to generalize this to various bounded approximation properties.

scholarly journals Johnson Pseudo-Contractibility and Pseudo-Amenability of θ-Lau Product

2020 ◽  
Vol 44 (4) ◽  
pp. 593-601
Author(s):  
M. ASKARI-SAYAH ◽  
A. POURABBAS ◽  
A. SAHAMI
Keyword(s):  
Banach Algebras ◽  
Complete Characterization ◽  
Approximate Identity ◽  
Approximate Amenability

Given Banach algebras A and B and θ ∈ ∆(B). We shall study the Johnson pseudo-contractibility and pseudo-amenability of the θ-Lau product A×θ B. We show that if A ×θ B is Johnson pseudo-contractible, then both A and B are Johnson pseudo-contractible and A has a bounded approximate identity. In some particular cases, a complete characterization of Johnson pseudo-contractibility of A ×θ B is given. Also, we show that pseudo-amenability of A ×θ B implies the approximate amenability of A and pseudo-amenability of B.

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