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This paper is devoted to the study of the algebra of bounded-type block-symmetric analytic functions on the Banach space l1(Cs). In particular, it presents a description of the spectrum of this algebra in terms of exceptional characters ϕα and characters that can be associated with exponential-type functions of several variables with plane zeros. Due to this representation, it is proven that every element of the spectrum is a convolution of an exceptional character with a point evaluation functional.

We investigate algebraic and topological properties of subsymmetric polynomials on finite- and infinite-dimensional spaces. In particular, we focus on the problem of the existence of an algebraic basis in the algebra of subsymmetric polynomials, as well as possible extensions of subsymmetric polynomials and analytic functions to larger spaces. We consider algebras of subsymmetric analytic functions of bounded type and their spectra, and study linear subspaces in the zero-sets of subsymmetric polynomials, as well as subspaces where a subsymmetric polynomial is symmetric. In addition, we propose some possible applications of subsymmetric polynomials in cryptography and in operator theory.

We construct a countable algebraic basis of the algebra of all symmetric continuous polynomials on the Cartesian product ℓp1×…×ℓpn, where p1,…,pn∈[1,+∞), and ℓp is the complex Banach space of all p-power summable sequences of complex numbers for p∈[1,+∞).

In this paper, we consider a supersymmetric version of block-symmetric polynomials on a Banach space of two-sided absolutely summing series of vectors in Cs for some positive integer s>1. We describe some sequences of generators of the algebra of block-supersymmetric polynomials and algebraic relations between the generators for the finite-dimensional case and construct algebraic bases of block-supersymmetric polynomials in the infinite-dimensional case. Furthermore, we propose some consequences for algebras of block-supersymmetric analytic functions of bounded type and their spectra. Finally, we consider some special derivatives in algebras of block-symmetric and block-supersymmetric analytic functions and find related Appell-type sequences of polynomials.