Vasylyshyn T. Analysis on spectra of algebras of analytic and smooth functions on a Banach space

The thesis is devoted to the study of spectra (sets of nontrivial continuous scalar-valued homomorphisms) of topological algebras of analytic functions on complex Banach spaces and topological algebras of smooth functions on real Banach spaces. It is constructed countable algebraic bases of algebras of continuous symmetric (invariant under the composition of its argument with any measurable automorphism of the domain of the argument) scalar-valued polynomials on the complex Banach space of all complex-valued Lebesgue measurable essentially bounded functions on a segment, on the complex Banach space of all complex-valued Lebesgue integrable essentially bounded functions on the semi-axis, on the Cartesian powers of the real and complex Banach spaces of all Lebesgue measurable essentially bounded functions on a segment. Also it is constructed finite algebraic bases of algebras of complex-valued symmetric continuous polynomials on Cartesian powers of complex Banach spaces of complex-valued Lebesgue integrable in a power p functions on the segment and on the semi-axis. It is described spectra of Fréchet algebras of symmetric entire analytic functions of bounded type on these spaces with the topology of uniform convergence on bounded sets. Also it is represented these Fréchet algebras as Fréchet algebras of entire analytic functions on their spectra.