Semisimple Segal Fréchet algebras

Abstract

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.

Initial objects, terminal objects, zero objects, and equalizers in the category Seg of Segal topological algebras

“…The usual context of Segal algebras during the last 25 years has been the setting of Banach algebras. About 5 years ago appeared some new generalizations to the case of locally multiplicatively convex (shortly, lmc) Fréchet algebras by Abtahi, Rahnama, and Rejali (see [10] and [11]), who called their generalization Segal Fréchet algebra, and to the case of complete lmc algebras by Yousofzadeh (see [16,17]).…”

Frechet algebras in abstract Harmonic analysis

“…G 3.6. Following [9] and also [10], a Fréchet algebra (A, p ℓ ) ℓ∈N is called a Segal Fréchet algebra in the Fréchet algebra (B, q n ) n∈N if the following conditions hold:…”

“…They also studied the concept of ideal amenability for vector-valued Fréchet Lipschitz algebras (ibid.). Furthermore, Rejali et al [9,10] introduced and studied the notions of Segal and semisimple Segal Fréchet algebras. They showed that every continuous linear left multiplier of a Fréchet algebra (B, q n ) n∈N is also a continuous linear left multiplier of any Segal Fréchet algebra (A, p ℓ ) ℓ∈N in (B, q n ) n∈N .…”

“…Moreover, they proved that if B is a commutative Fréchet Q-algebra with an approximate identity, then the spaces of all modular maximal closed ideals of B and any Segal Fréchet algebra A in B are homeomorphic. Particularly, B is semisimple if and only if A is semisimple [10]. In [2], we joint with Abtahi introduced and studied the strict, uniform, and compact-open locally convex topologies on an arbitrary algebra B, by the fundamental system of seminorms of a locally convex subalgebra (A, p α ).…”