Note on separately symmetric polynomials on the Cartesian product of l1

Abstract

In the paper, we describe algebraic bases in algebras of separately symmetric polynomials which are defined on Cartesian products of n copies of ℓ 1. Also, we describe spectra of algebras of entire functions, generated by these polynomials as Cartesian products of spectra of algebras of symmetric analytic functions of bounded type on ℓ 1. Finally, we consider algebras of separately symmetric analytic functions of bounded type on infinite direct sums of copies of ℓ 1. In particular, we show that there is a homomorphism from such algebra onto the algebra of all analytic functions of bounded type on a Banach space X with an unconditional basis.

Block-Supersymmetric Polynomials on Spaces of Absolutely Convergent Series

“…Let us show that H vsup b supports characters that are not point evaluation functionals. Similar results for different algebras were obtained in [8,20,27,29,34].…”

Supersymmetric Polynomials on the Space of Absolutely Convergent Series

“…Spectra of algebras H bs ( p ) were investigated also in [13,14]. Polynomials which are symmetric with respect to some other representations of the group of permutations of natural numbers were considered in [15][16][17].…”

Rings of Multisets and Integer Multinumbers

“…Then, actually, the private key is the pair v, p. For encoding, we have to make the multiplication w = nu ∈ N and reduce each component of the multinumber w modulo q = 9. That is, w = nu = (2,3,4,4,6,6,7,7,9,12,12,18,21,21); (2,3,3,3,3,3,4,4,6,6,7).…”

“…Now, let q be secret and p be public. Then, having w as above, we reduce the repetition of each component of w modulo p. We have I (3,•) (w) = (2,3,4,4,6,6,7,7,9,12,12,18,21,21).…”

“…Such a ring of multisets M 0 was constructed in [6] using symmetric and supersymmetric polynomials on a Banach space (see also [7]). More details about algebras of symmetric polynomials on Banach spaces can be found in [8][9][10][11][12][13][14]. The combinatorial approach to symmetric polynomials can be found in [15].…”