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Let A and B be Banach function algebras on compact Hausdorff spaces X and Y, respectively, and ρ , τ : I ⟶ A , S , T : I ⟶ B be maps on a non-empty set I whose ranges are closed under multiplication and contain exponential functions. In this paper, we first show that if ‖ S (p) ‖ Y = ‖ ρ (p) ‖ X , ‖ T (p) ‖ Y = ‖ τ (p) ‖ X and ‖ S (p) T (q) ‖ Y = ‖ ρ (p) τ (q) ‖ X , for all p , q ∈ I , then there exists a homeomorphism φ from Šilov boundary ∂ A of A onto ∂ B such that for each x ∈ ∂ A and p ∈ I , | S (p) (φ (x)) | = | ρ (p) (x) | and | T (p) (φ (x)) | = | τ (p) (x) | . Then we prove that, if for some ε ≥ 0 , Ran π (T (p) S (q)) is contained in an ε ‖ τ (p) ρ (q) ‖ -neighborhood of Ran π (τ (p) ρ (q)) , for all p , q ∈ I , then, under a certain condition, there exist continuous functions α , β ∈ B such that | α (φ (x)) T (p) (φ (x)) - τ (p) (x) | ⩽ 4 ε | τ (p) (x) | and | β (φ (x)) S (p) (φ (x)) - ρ (p) (x) | ⩽ 4 ε | ρ (p) (x) | , for all p ∈ I and x ∈ ∂ (A) . Our results can be applied for Banach algebras of Lipschitz functions. [ABSTRACT FROM AUTHOR]

In this paper, BSE properties of some Banach function algebras are studied. We show that Lipschitz algebras Lipα (X, d) and Dales-Davie algebras D(X, M) are BSE-algebras for certain underlying plane sets X. Moreover, we investigate BSE properties of certain subalgebras of Lipα(X, d)suchas LipA (X, α), Lipn (X, α) and Lip(X, M, α). BSE properties of Bloch type spaces ℬ α and Zygmund type spaces Zα are also investigated in different cases of α ∈ ℝ. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce new classes of Lipschitz algebras of infinitely differentiable functions which are extensions of the standard Lipschitz algebras of infinitely differentiable functions. Then we determine the maximal ideal space of these extended algebras. Finally, we show that if X and K are uniformly regular subsets in the complex plane, then R(X;K) is natural. [ABSTRACT FROM AUTHOR]

Let X be a perfect, compact subset of the complex plane, and let D(1)(X) denote the (complex) algebra of continuously complex-differentiable functions on X. Then D(1)(X) is a normed algebra of functions but, in some cases, fails to be a Banach function algebra. Bland and the second author ([3]) investigated the completion of the algebra D(1)(X), for certain sets X and collections F of paths in X, by considering F-differentiable functions on X. In this paper, we investigate composition, the chain rule, and the quotient rule for this notion of differentiability. We give an example where the chain rule fails, and give a number of sufficient conditions for the chain rule to hold. Where the chain rule holds, we observe that the Faá di Bruno formula for higher derivatives is valid, and this allows us to give some results on homomorphisms between certain algebras of F-differentiable functions. [ABSTRACT FROM AUTHOR]

We exhibit a singly generated, semisimple commutative operator algebra with a contractive approximate identity such that the spectrum of the generator is a null sequence and zero, but the algebra is not the closed linear span of the idempotents associated with the null sequence and obtained from the analytic functional calculus. Moreover the multiplication on the algebra is neither compact nor weakly compact. Thus we construct a 'large' operator algebra of orthogonal idempotents, which may be viewed as a dense subalgebra of c0. [ABSTRACT FROM AUTHOR]

We exhibit a singly generated, semisimple commutative operator algebra with a contractive approximate identity such that the spectrum of the generator is a null sequence and zero, but the algebra is not the closed linear span of the idempotents associated with the null sequence and obtained from the analytic functional calculus. Moreover the multiplication on the algebra is neither compact nor weakly compact. Thus we construct a 'large' operator algebra of orthogonal idempotents, which may be viewed as a dense subalgebra of c0. [ABSTRACT FROM AUTHOR]

Let A and B be Banach function algebras on compact Hausdorff spaces X and Y and let |.| and |.| 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| = | fg|, 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| = | Tf Tg + α|, 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, . In particular, if T satisfies the stronger condition R( fg + α) = R( Tf Tg + α), where R(.) 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| = | fg|, f, g ∈ A, induces a homeomorphism between quotient spaces of particular subsets of X and Y by some equivalence relations. [ABSTRACT FROM AUTHOR]

Let K be a compact subset of the complex plane and 0<α≤1. In this article we determine the maximal ideal space of Li pA(K, α). [ABSTRACT FROM AUTHOR]