Let $H^{\infty }\left(\right)$ denote the usual commutative Banach algebra of bounded analytic functions on the open unit disc of the finite complex plane, under Hadamard product of power series. We construct a Boehmian space which includes the Banach algebra A where A is the commutative Banach algebra with unit containing $H^{\infty }\left(\right)$. The Gelfand transform theory is extended to this setup along with the usual classical properties. The image is also a Boehmian space which includes the Banach algebra C(Δ) of continuous functions on...

Cet article est consacré à l’étude des espaces $A_{\omega }=ℱL^2({\mathbf{R}}^n;\omega \left(x\right)dx)$ qui sont des algèbres de Banach. On démontre que les multiplicateurs ponctuels de $A_{\omega }$ sont les fonctions qui appartiennent localement et uniformément à $A_{\omega }$ si et seulement si $A_{\omega }$ contient des fonctions à support compact.

We study subalgebras of $C_b\left(X\right)$ equipped with topologies that generalize both the uniform and the strict topology. In particular, we study the Stone-Weierstrass property and describe the ideal structure of these algebras.

Let (X, d X) and (Y,d Y) be pointed compact metric spaces with distinguished base points e X and e Y. The Banach algebra of all $𝕂$-valued Lipschitz functions on X - where $𝕂$ is either‒or ℝ - that map the base point e X to 0 is denoted by Lip0(X). The peripheral range of a function f ∈ Lip0(X) is the set Ranµ(f) = f(x): |f(x)| = ‖f‖∞ of range values of maximum modulus. We prove that if T 1, T 2: Lip0(X) → Lip0(Y) and S 1, S 2: Lip0(X) → Lip0(X) are surjective mappings such that $Ra{n}_{\pi }\left(T_1\left(f\right)T_2\left(g\right)\right)\cap Ra{n}_{\pi }\left(S_1\left(f\right)S_2\left(g\right)\right)\ne \varnothing$ for all f, g ∈ Lip0(X),...

In 1971, Grauert and Remmert proved that a commutative, complex, Noetherian Banach algebra is necessarily finite-dimensional. More precisely, they proved that a commutative, complex Banach algebra has finite dimension over ℂ whenever all the closed ideals in the algebra are (algebraically) finitely generated. In 1974, Sinclair and Tullo obtained a non-commutative version of this result. In 1978, Ferreira and Tomassini improved the result of Grauert and Remmert by showing that the statement...

Let G be a Lie group and A(G) the Fourier algebra of G. We describe sufficient conditions for complex-valued functions to operate on elements u ∈ A(G) of certain differentiability classes in terms of the dimension of the group G. Furthermore, generalizing a result of Kirsch and Müller [Ark. Mat. 18 (1980), 145-155] we prove that closed subsets E of a smooth m-dimensional submanifold of a Lie group G having a certain cone property are sets of smooth spectral synthesis. For such sets we give an estimate...