Algebras of weakly symmetric functions on spaces of Lebesgue measurable functions

In this work, we investigate algebras of block-symmetric and weakly symmetric polynomials and analytic functions on complex Banach spaces of Lebesgue measurable functions, for which the pth power of the absolute value is Lebesgue integrable, where p is an element of [1, +oo), and Lebesgue measurable essentially bounded functions on [0, 1]. We construct generating systems of algebras of all weakly symmetric continuous complex-valued polynomials on these spaces. Also we establish conditions under which sets of weakly symmetric analytic functions are algebras.