ROOT EXTENSIONS OF DITKIN ALGEBRA

Abstract

Let us start with some standard definitions. Definition 1.1. Let S be a non-empty set, and let ƒ be a bounded, complex-valued function on S. For each non-empty set E contained in S, the uniform norm off on E, denoted by IfE, is defined by fle sup{f(x) | :x € E}. Throughout this paper, a compact space X is a compact Hausdorff topological space. Notation. Let X be a compact space. We shall denote by C(X) the algebra of all continuous functions from X into the field of complex numbers C. Definition 1.2. Let A be a subset of C(X). Then A is called separates the points of X if for each x,ye X with xy, there exists fe A with f(x) = f(y). Definition 1.3. Let X be a compact space. A uniform algebra A on X is a closed subalgebra of C(X) which contains the constant functions, and separates the points of X. Definitions 1.4. Let A be a uniform algebra on a compact space X. Let xe X. We define the following ideals in A by setting