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Partially defined σ-derivations on semisimple Banach algebras

Tsiu-Kwen Lee, Cheng-Kai Liu (2009)

Studia Mathematica

Let A be a semisimple Banach algebra with a linear automorphism σ and let δ: I → A be a σ-derivation, where I is an ideal of A. Then Φ(δ)(I ∩ σ(I)) = 0, where Φ(δ) is the separating space of δ. As a consequence, if I is an essential ideal then the σ-derivation δ is closable. In a prime C*-algebra, we show that every σ-derivation defined on a nonzero ideal is continuous. Finally, any linear map on a prime semisimple Banach algebra with nontrivial idempotents is continuous if it satisfies the σ-derivation...

Problems concerning n -weak amenability of a Banach algebra

Alireza Medghalchi, Taher Yazdanpanah (2005)

Czechoslovak Mathematical Journal

In this paper we extend the notion of n -weak amenability of a Banach algebra 𝒜 when n ∈ ℕ . Technical calculations show that when 𝒜 is Arens regular or an ideal in 𝒜 * * , then 𝒜 * is an 𝒜 ( 2 n ) -module and this idea leads to a number of interesting results on Banach algebras. We then extend the concept of n -weak amenability to n ∈ ℤ .

Range inclusion results for derivations on noncommutative Banach algebras

Volker Runde (1993)

Studia Mathematica

Let A be a Banach algebra, and let D : A → A be a (possibly unbounded) derivation. We are interested in two problems concerning the range of D: 1. When does D map into the (Jacobson) radical of A? 2. If [a,Da] = 0 for some a ∈ A, is Da necessarily quasinilpotent? We prove that derivations satisfying certain polynomial identities map into the radical. As an application, we show that if [a,[a,[a,Da]]] lies in the prime radical of A for all a ∈ A, then D maps into the radical. This generalizes a result...

Small deformations of topological algebras

Mati Abel, Krzysztof Jarosz (2003)

Studia Mathematica

We investigate stability of various classes of topological algebras and individual algebras under small deformations of multiplication.

Some simple proofs in holomorphic spectral theory

Graham R. Allan (2007)

Banach Center Publications

This paper gives some very elementary proofs of results of Aupetit, Ransford and others on the variation of the spectral radius of a holomorphic family of elements in a Banach algebra. There is also some brief discussion of a notorious unsolved problem in automatic continuity theory.

The continuity of Lie homomorphisms

Bernard Aupetit, Martin Mathieu (2000)

Studia Mathematica

We prove that the separating space of a Lie homomorphism from a Banach algebra onto a Banach algebra is contained in the centre modulo the radical.

Topological algebras of random elements

Bertram M. Schreiber, M. Victoria Velasco (2016)

Studia Mathematica

Let L₀(Ω;A) be the Fréchet space of Bochner-measurable random variables with values in a unital complex Banach algebra A. We study L₀(Ω;A) as a topological algebra, investigating the notion of spectrum in L₀(Ω;A), the Jacobson radical, ideals, hulls and kernels. Several results on automatic continuity of homomorphisms are developed, including versions of well-known theorems of C. Rickart and B. E. Johnson.

Uniqueness of complete norms for quotients of Banach function algebras

W. Bade, H. Dales (1993)

Studia Mathematica

We prove that every quotient algebra of a unital Banach function algebra A has a unique complete norm if A is a Ditkin algebra. The theorem applies, for example, to the algebra A (Γ) of Fourier transforms of the group algebra L 1 ( G ) of a locally compact abelian group (with identity adjoined if Γ is not compact). In such algebras non-semisimple quotients A ( Γ ) / J ( E ) ¯ arise from closed subsets E of Γ which are sets of non-synthesis. Examples are given to show that the condition of Ditkin cannot be relaxed. We construct...

Uniqueness of the topology on L¹(G)

J. Extremera, J. F. Mena, A. R. Villena (2002)

Studia Mathematica

Let G be a locally compact abelian group and let X be a translation invariant linear subspace of L¹(G). If G is noncompact, then there is at most one Banach space topology on X that makes translations on X continuous. In fact, the Banach space topology on X is determined just by a single nontrivial translation in the case where the dual group Ĝ is connected. For G compact we show that the problem of determining a Banach space topology on X by considering translation operators on X is closely related...

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