Abstract. We study the algebra of uniformly continuous holomorphic symmetric functions on the ball of p , investigating in particular the spectrum of such algebras. To do so, we examine the algebra of symmetric polynomials on p − spaces as well as finitely generated symmetric algebras of holomorphic functions. Such symmetric polynomials determine the points in p up to a permutation.In recent years, algebras of holomorphic functions on the unit ball of standard complex Banach spaces have been considered by a number of authors and the spectrum of such algebras was studied in [1],[2], [7]. For example, properties of A u (B X ), the algebra of uniformly continuous holomorphic functions on the ball of a complex Banach space X have been studied by Gamelin, et al. Unfortunately, this analogue of the classical disc algebra A(D) has a very complicated, not well understood, spectrum. If X * has the approximation property, the spectrum of A u (B X ) coincides with the closed unit ball of the bidual if, and only if, X * generates a dense subalgebra in A u (B X ) [5]. In a very real sense, however, the problem is that A u (B p ) is usually too large, admitting far too many functions. For instance, ∞ ⊂ A u (B 2 ) isometrically via the mapping a = (a j ) ; P a , whereThis paper addresses this problem, by severely restricting the functions which we admit. Specifically, we limit our attention here to uniformly continuous symmetric holomorphic functions on B p . By a symmetric function on p we mean a function which is invariant under any reordering of the sequence in p . Symmetric polynomials in finite dimensional spaces can be studied in [9] or [12]; in the infinite dimensional Hilbert space they already appear in [11]. Throughout this note P s ( p ) is the space of symmetric polynomials on a complex space p , 1 ≤ p < ∞. Such polynomials determine, as we prove, the points in p up to a permutation. We will use the notation A us (B p ) for the uniform algebra of symmetric holomorphic functions which are uniformly continuous on the open unit ball B p of p and we also study some particular finitely generated subalgebras. The purpose of this paper is to describe such algebras and their spectra, which we identify with certain subsets of ∞ and C m , respectively, and as a result of this we show that A us (B p ) is algebraically and topologically isomorphic to a uniform Banach algebra generated by coordinate projections in ∞ . This is done in Section 3, following algebraic preliminaries and a brief examination of the finite dimensional situation in Sections 1 and 2.
