A Banach Algebra of Functions With Bounded nth Differences

In this paper we study the symbolic calculus for a Banach function algebra with certain conditions. First we define a class of Banach function algebras which contains the class of all function algebras and the class of all ultraseparating Banach function algebras. Our purpose is to prove the theorem asserting that if A is a non-trivial Banach function algebra in the class, then only analytic functions can operate on A. It is a generalization of theorems of de Leeuw and Katznelson [6], Bernard [2] and Bernard and Dufresnoy [3].

Multiplicatively range-preserving maps between Banach function algebras

Let A and B be Banach function algebras on compact Hausdorff spaces X and Y and let ‖.‖X and ‖.‖Y denote the supremum norms on X and Y, respectively. We first establish a result concerning a surjective map T between particular subsets of the uniform closures of A and B, preserving multiplicatively the norm, i.e. ‖Tf Tg‖Y = ‖fg‖X, for certain elements f and g in the domain. Then we show that if α ∈ ℂ {0} and T: A → B is a surjective, not necessarily linear, map satisfying ‖fg + α‖X = ‖Tf Tg + α‖Y, f,g ∈ A, then T is injective and there exist a homeomorphism φ: c(B) → c(A) between the Choquet boundaries of B and A, an invertible element η ∈ B with η(Y) ⊆ {1, −1} and a clopen subset K of c(B) such that for each f ∈ A, $$ Tf\left( y \right) = \left\{ \begin{gathered} \eta \left( y \right)f\left( {\phi \left( y \right)} \right) y \in K, \hfill \\ - \frac{\alpha } {{\left| \alpha \right|}}\eta \left( y \right)\overline {f\left( {\phi \left( y \right)} \right)} y \in c\left( B \right)\backslash K \hfill \\ \end{gathered} \right. $$. In particular, if T satisfies the stronger condition R π(fg + α) = R π(Tf Tg + α), where R π(.) denotes the peripheral range of algebra elements, then Tf(y) = T1(y)f(φ(y)), y ∈ c(B), for some homeomorphism φ: c(B) → c(A). At the end of the paper, we consider the case where X and Y are locally compact Hausdorff spaces and show that if A and B are Banach function algebras on X and Y, respectively, then every surjective map T: A → B satisfying ‖Tf Tg‖Y = ‖fg‖, f, g ∈ A, induces a homeomorphism between quotient spaces of particular subsets of X and Y by some equivalence relations.

Let K and X be compact plane sets such that K X. Let P (K) be the uniform closure of polynomials on K. Let R(K) be the closure of rational functions K with poles o K. Dene P (X;K) and R(X;K) to be the uniform algebras of functions in C(X) whose restriction to K belongs to P (K) and R(K), respectively. Let CZ(X;K) be the Banach algebra of functions f in C(X) such that fjK = 0. In this paper, we show that every nonzero complex homomorphism ’ on CZ(X;K) is an evaluation homomorphism ez for some z in XnK. By considering this fact, we characterize the maximal ideal space of the uniform algebra P (X;K). Moreover, we show that the uniform algebra R(X;K) is natural.

Automatic Continuity of Lipschitz Algebras

We recently introduced a weighted Banach algebra of functions that are holomorphic on the unit disc D, continuous up to the boundary, and of the class C(n) at all points where the function G does not vanish. Here, G refers to a function of the disc algebra without zeros on D. Then we proved that all closed ideals in with at most countable hull are standard. In this paper, on the assumption that G is an outer function in C(n) having infinite roots in and countable zero set h0(G), we show that all the closed ideals I with hull containing h0(G) are standard.

We show that a real Banach algebra $A$ such that $\Vert a^{2}\Vert =\Vert a\Vert^{2}$ for $a\in A$ is a subalgebra of the algebra $C_{\mathbb{H}}( X) $ of continuous quaternion-valued functions on a compact set $X$.

In this paper we derive the explicit, closed-form, recursion-free formulae for the arbitrary-order Fr?chet derivatives of the exponential and logarithmic functions in unital Banach algebras (complex or real). These computations are obtained via the Bochner integrals for the Banach algebra valued functions, with respect to the standard Lebesgue measure. As an application, we utilize our results in the approximation schemes of the solutions to stochastic functional differential equations.

Throughout this paper we deal only with complex and semi-simple algebras. Let B be such an algebra. We denote the socle of B as SB. B is a modular annihilator algebra if B/SB is a radical algebra, i.e. if every element of B is quasi-regular modulo the socle of B; see (1) or (12). Now assume that B is a modular annihilator algebra and a Banach algebra. Then any semi-simple closed subalgebra of B is a modular annihilator algebra by ((4), Cor. to Theorem 4·2,). It is not true, however, that any semi-simple subalgebra A of B is a modular annihilator algebra, even when A is a Banach algebra in some norm. We give a simple example to illustrate this. Let A be the algebra of all complex functions f, continuous on the closed unit disk D in the complex plane, analytic in the interior of D, and such that f(0) = 0. A is a Banach algebra in the usual sup norm over D. Now consider the norm on A defined byLet B be the completion of A in this norm. A has an involution * defined by and also ‖ff*‖ = ‖f‖2 for all f ∈ A. Therefore B is a B*-algebra. It is not difficult to verify that the only non-zero multiplicative linear functionals on A which are continuous with respect to the norm ‖·‖, are the point evaluations at 1/n, n = 1, 2 …. It follows that every non-zero multiplicative linear functional on B is an extension of one of these point evaluations to B. Thus B can be identified with the algebra of all complex sequences which converge to zero.

In this paper, we study the existence of the solutions of a class of functional integral equations which contain a lot of classical nonlinear integral equations as special cases. We consider the solvability of the equations in the Banach algebra of continuous functions on a closed and bounded interval. The main tools here are the measure of noncompactness and the suitable fixed point theorem for the product of two operators in the Banach algebra.

We give a description of the continuity ideals and the kernels of homomorphisms from the algebras of continuous functions on locally compact spaces into Banach algebras. We also construct families of prime ideals satisfying a certain intriguing property in the algebras of continuous functions.

A commutative Banach algebra is called symmetric if, regarded as a function algebra on its maximal ideal space, it is closed under complex conjugation. Varopoulos, [5], proved the asymmetry of the tensor algebra , where T is the unit circle. It is the object of this paper to prove the asymmetry of the Schur multipliers of the space , where m is the Lebesgue measure. In the second part of the paper we study the Hankel multipliers of the space and give an application to it.1. The asymmetry of. Let C(T) denote the space of continuous functions on T and A(T) be the space of those functions in C(T) that have absolutely convergent Fourier series. Consider the mapping F: C(T) → C(T × T) defined by F(f)(x, y) = f (x + y).

This article is devoted to study of vector functions in Banach algebras and Banach spaces over normed fields. A structure of their Banach algebras is investigated. Banach algebras of vector functions with values in ∗-algebras, finely regular algebras, B∗-algebras, and operator algebras are scrutinized. An approximation of vector functions is investigated. The realizations of these algebras by operator algebras are studied.

In a previous note the author gave an example of a strong Ditkin algebra which does not have bounded relative units in the sense of Dales. In this note we investigate a certain family of Banach function algebras on the one point compactification of ℕ, and see that within this family are many easier examples of strong Ditkin algebras without bounded relative units in the sense of Dales.

In this note we answer a question of W. G. Bade by showing that if a normal, unital Banach function algebra A is strongly regular at one of its characters φ has a bounded approximate identity, then A has bounded relative units at φ. In particular, every strong Ditkin algebra has bounded relative units at all points of its character space. There need not, however, be a global bound available for the relative units.