For an increasing sequence (ωₙ) of algebra weights on ℝ⁺ we study various properties of the Fréchet algebra A(ω) = ⋂ ₙ L¹(ωₙ) obtained as the intersection of the weighted Banach algebras L¹(ωₙ). We show that every endomorphism of A(ω) is standard, if for all n ∈ ℕ there exists m ∈ ℕ such that ${\omega }_m\left(t\right)/\omega ₙ\left(t\right)\to \infty$ as t → ∞. Moreover, we characterise the continuous derivations on this algebra: Let M(ωₙ) be the corresponding weighted measure algebras and let B(ω) = ⋂ ₙM(ωₙ). If for all n ∈ ℕ there exists m ∈ ℕ such that...

Let A be a commutative Banach algebra with Gelfand space ∆ (A). Denote by Aut (A) the group of all continuous automorphisms of A. Consider a σ(A,∆(A))-continuous group representation α:G → Aut(A) of a locally compact abelian group G by automorphisms of A. For each a ∈ A and φ ∈ ∆(A), the function ${\phi }_a\left(t\right):=\phi \left({\alpha }_{t}a\right)$ t ∈ G is in the space C(G) of all continuous and bounded functions on G. The weak-star spectrum ${\sigma }_w*\left({\phi }_a\right)$ is defined as a closed subset of the dual group Ĝ of G. For φ ∈ ∆(A) we define ${Ʌ}_{\phi }^a$ to be the union of all...

Let A and B be unital, semisimple commutative Banach algebras with the maximal ideal spaces M A and M B, respectively, and let r(a) be the spectral radius of a. We show that if T: A → B is a surjective mapping, not assumed to be linear, satisfying r(T(a) + T(b)) = r(a + b) for all a; b ∈ A, then there exist a homeomorphism φ: M B → M A and a closed and open subset K of M B such that $$\widehat{T\left(a\right)}\left(y\right)=\left\{\begin{array}{c}\widehat{T\left(e\right)}\left(y\right)\widehat{a}\left(\phi \left(y\right)\right)y\in K\\ \widehat{T\left(e\right)}\left(y\right)\overline{\widehat{a}\left(\phi \left(y\right)\right)}y\in M_{ℬ}\setminus K\end{array}\right.$$ for all a ∈ A, where e is unit element of A. If, in addition, $$\widehat{T\left(e\right)}=1$$ and $$\widehat{T\left(ie\right)}=i$$ on M B, then T is an algebra isomorphism.

A systematic investigation of algebras of holomorphic functions endowed with the Hadamard product is given. For example we show that the set of all non-invertible elements is dense and that each multiplicative functional is continuous, answering some questions in the literature.

Let A be a complex commutative Banach algebra with unit 1 and δ > 0. A linear map ϕ: A → ℂ is said to be δ-almost multiplicative if |ϕ(ab) - ϕ(a)ϕ(b)| ≤ δ||a|| ||b|| for all a,b ∈ A. Let 0 < ϵ < 1. The ϵ-condition spectrum of an element a in A is defined by ${\sigma }_{ϵ}\left(a\right):=\lambda \in ℂ:\left|\right|\lambda -{a\left|\right|\left|\right|(\lambda -a)}^{-1}\left|\right|\ge 1/ϵ$ with the convention that $\left|\right|\lambda -{a\left|\right|\left|\right|(\lambda -a)}^{-1}\left|\right|=\infty$ when λ - a is not invertible. We prove the following results connecting these two notions: (1) If ϕ(1) = 1 and ϕ is δ-almost multiplicative, then $\varphi \left(a\right)\in {\sigma }_{\delta }\left(a\right)$ for all a in A. (2) If ϕ is linear and $\varphi \left(a\right)\in {\sigma }_{ϵ}\left(a\right)$ for all a in A,...

We construct an example of a Fréchet m-convex algebra which is a principal ideal domain, and has the unit disk as the maximal ideal space.

We study asymptotics of a class of extremal problems rₙ(A,ε) related to norm controlled inversion in Banach algebras. In a general setting we prove estimates that can be considered as quantitative refinements of a theorem of Jan-Erik Björk [1]. In the last section we specialize further and consider a class of analytic Beurling algebras. In particular, a question raised by Jan-Erik Björk in [1] is answered in the negative.