FA ] 1 0 D ec 2 01 5 The chain rule for F-differentiation

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. Throughout this paper, we use the term compact plane set to mean a nonempty, compact subset of the complex plane, C. We denote the set of all positive integers by N and the set of all non-negative integers by N0. Let X be a compact Hausdorff space. We denote the algebra of all continuous, complex-valued functions on X by C(X) and we give C(X) the uniform norm |·|X , defined by |f |X = sup x∈X |f(x)| (f ∈ C(X)). This makes C(X) into a commutative, unital Banach algebra. A subset S of C(X) separates the points of X if, for each x, y ∈ X with x 6= y, there exists f ∈ S such that f(x) 6= f(y). A normed function algebra on X a normed algebra (A, ‖·‖) such that A is a subalgebra of C(X), A contains all constant functions and separates the points of X , and, for each f ∈ A, ‖f‖ ≥ |f |X . A Banach function algebra on X is a normed function algebra on X which is complete. We say that such a Banach function algebra A is natural (on X) if every character on A is given by evaluation at some point of X . We refer the reader to [5] (especially Chapter 4) for further information on Banach algebras and Banach function algebras. Let D(X) denote the normed algebra of all continuously (complex) differentiable, complex-valued functions on X , as discussed in [6] and [7]. Furthermore, 2010 Mathematics Subject Classification. Primary 46J10, 46J15; Secondary 46E25.