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.

In the spectrum of the algebra of symmetric analytic functions of bounded type on p, 1 p < +∞, and along the same lines as the general non-symmetric case, we define and study a convolution operation and give a formula for the 'radius' function. It is also proved that the algebra of analytic functions of bounded type on 1 is isometrically isomorphic to an algebra of symmetric analytic functions on a polydisc of 1 . We also consider the existence of algebraic projections between algebras of symmetric polynomials and the corresponding subspace of subsymmetric polynomials.

In this paper we introduce and study the notions of isotropic mapping and essential kernel. In addition some theorems on the Borel graph and Baire mapping for polynomial operators are proved. It is shown that a polynomial functional from an infinite dimensional complex linear space into the field of complex numbers vanishes on some infinite dimensional affine subspace.

Abstract. We construct non-separable subspaces in the kernel of every quadratic functional on some classes of complex and real Banach spaces.1. Introduction. Investigation of quadratic functionals is an old story [12], [6], [7]. According to [11], for any polynomial functional p with p(0) = 0 defined on an infinite-dimensional complex linear space X there is an infinite-dimensional subspace X 0 in the kernel ker(p) = p −1 (0) of p. Quantitative finite-dimensional versions of this fact (estimations of dim X 0 depending on dim X and the degree of the polynomial) are contained inThe paper [2] started the consideration of subspaces in kernels of polynomials on non-separable spaces. In particular, the authors of [2] proved that if a real Banach space X admits no positive quadratic continuous functional, then every quadratic continuous functional on X vanishes on some infinite-dimensional subspace. They pose the problem of whether in this statement one can replace "infinite-dimensional" by "non-separable" (see also [1, Question 4.8]). Our note continues the investigations of [2]. In particular, we shall construct a non-separable subspace in the kernel of every quadratic functional on a complex Banach space having weak * non-separable dual and on a real Banach space which has controlled separable projection property and admits no positive quadratic continuous functional. On the other hand, we construct a CH-example of a quadratic functional on the normed space l f 1 (ω 1 ) whose kernel contains no non-separable linear subspace.We use the standard notation; in particular dens X stands for the density of a Banach space X, F ⊥ = {x ∈ X : ∀f ∈ F f (x) = 0} is the annihilator of a subspace F ⊂ X * in X, S(X) is the unit sphere of X, and [M ] denotes the closed linear span of a subset M ⊂ X. We shall identify cardinals with initial ordinals and will denote by α the cardinality of an ordinal α. Elements x α ∈ X form a transfinite basic sequence if there is a constant c > 0 such