UDC 517.98
Essential amenability of Banach algebras have been defined and investigated. Here, this concept will be introduced for Frechet algebras. Then a number of well-known results of essential amenability of Banach algebras are generalized for Fréchet algebras. Moreover, related results about Segal–Fréchet algebras are provided. As the main result, it is provedthat if
(
𝒜
,
p
ℓ
)
is an amenable Fréchet algebra with a uniformly bounded approximate identity, then every symmetric Segal – Fréchet algebra in
(
𝒜
,
p
ℓ
)
is essentially amenable.
Let (A, p ℓ ) ℓ∈N be a Fréchet algebra. In this paper, we introduce the concept of Segal Fréchet algebra and investigate known results about abstract Segal algebras, for Segal Fréchet algebras. Also we recall the concept of approximate identities for topological algebras and provide some remarkable results for Segal Fréchet algebras. Moreover, we verify ideal theorem for Fréchet algebras and characterize closed ideals of Segal Fréchet algebra (B, qm) m∈N in (A, p ℓ ) ℓ∈N .2000 Mathematics Subject Classification. 46A03, 46A04. Key words and phrases. Abstract Segal algebra, Banach algebra, Fréchet algebra.
In this paper, we recall the concept of Segal Fréchet algebra in a Fréchet algebra (A, p ) and show that in some cases, every continuous linear left multiplier on (A, p ) is a continuous linear left multiplier of any Segal Fréchet algebra (B, q m ) in (A, p ). As the main result, we prove that if A is a commutative Fréchet Q-algebra with an approximate identity, A is semisimple if and only if B is semisimple.
Let T be a Banach algebra homomorphism from a Banach algebra B to a Banach algebra A with T ≤ 1. Recently it has been obtained some results about Arens regularity and also various notions of amenability of A ×T B, in the case where A is commutative. In the present paper, most of these results have been generalized and proved for an arbitrary Banach algebra A.
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