### Algebraic basis of the algebra of block-symmetric polynomials on $\ell_1 \oplus \ell_{\infty}$

#### Abstract

We concider so called block-symmetric polynomials on sequence spaces $\ell_1\oplus \ell_{\infty},\ell_1\oplus c, \ell_1\oplus c_0,$ that is, polynomials which are symmetric with respect to permutationsof elements of the sequences. It is proved that every continuous block-symmetric polynomialson $\ell_1\oplus \ell_{\infty}$can be uniquelly represented as an algebraic combination of some special block-symmetric polynomials whichform an algebraic basis. It is interesting to notethat the algebra of block-symmetric polynomials is infinite-generated while $\ell_{\infty}$ admits no symmetricpolynomials. Algebraic bases ofthe algebras of block-symmetric polynomials on $\ell_1\oplus \ell_{\infty}$ and $\ell_1\oplus c_0$ described.

#### Keywords

symmetric polynomials, block-symmetric polynomials, algebraic basis, algebra

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