Topology on the spectrum of the algebra of entire symmetric functions of bounded type on the complex $L_\infty$
Journal Title: Карпатські математичні публікації - Year 2017, Vol 9, Issue 1
Abstract
It is known that the so-called elementary symmetric polynomials $R_n(x) = \int_{[0,1]}(x(t))^n\,dt$ form an algebraic basis in the algebra of all symmetric continuous polynomials on the complex Banach space $L_\infty,$ which is dense in the Fr\'{e}chet algebra $H_{bs}(L_\infty)$ of all entire symmetric functions of bounded type on $L_\infty.$ Consequently, every continuous homomorphism $\varphi: H_{bs}(L_\infty) \to \mathbb{C}$ is uniquely determined by the sequence $\{ \varphi (R_n) \}_{n=1}^\infty$. By the continuity of the homomorphism $\varphi,$ the sequence $\{\sqrt[n]{|\varphi(R_n)|}\}_{n=1}^\infty$ is bounded. On the other hand, for every sequence $\{\xi_n\}_{n=1}^\infty \subset \mathbb{C},$ such that the sequence $\{\sqrt[n]{|\xi_n|}\}_{n=1}^\infty$ is bounded, there exists $x_\xi \in L_\infty$ such that $R_n(x_\xi) = \xi_n$ for every $n \in \mathbb{N}.$ Therefore, for the point-evaluation functional $\delta_{x_\xi}$ we have $\delta_{x_\xi}(R_n) = \xi_n$ for every $n \in \mathbb{N}.$ Thus, every continuous complex-valued homomorphism of $H_{bs}(L_\infty)$ is a point-evaluation functional at some point of $L_\infty.$ Note that such a point is not unique. We can consider an equivalence relation on $L_\infty,$ defined by $x\sim y Leftrightarrow \delta_x = \delta_y.$ The spectrum (the set of all continuous complex-valued homomorphisms) $M_{bs}$ of the algebra $H_{bs}(L_\infty)$ is one-to-one with the quotient set $L_\infty/_\sim.$ Consequently, $M_{bs}$ can be endowed with the quotient topology. On the other hand, it is naturally to identify $M_{bs}$ with the set of all sequences $\{\xi_n\}_{n=1}^\infty \subset \mathbb{C}$ such that the sequence $\{\sqrt[n]{|\xi_n|}\}_{n=1}^\infty$ is bounded. We show that the quotient topology is Hausdorff and that $M_{bs}$ with the operation of coordinate-wise addition of sequences forms an abelian topological group.
Authors and Affiliations
T. V. Vasylyshyn
Keywords
Related Articles
- EP ID EP324781
- DOI 10.15330/cmp.9.1.22-27
- Views 65
- Downloads 0
Continuously differentiable solutions of one boundary value problem for a systems of linear difference differential equations of neutral type
Conditions of the existence of continuously differentiable bounded for t∈R+ solutions of one boundary value problem for a systems of linear and nonlinear difference differential equations of neutral type have been obtain...
Poincare series for the algebras of joint invariants and covariants of n quadratic forms
We consider one of the fundamental objects of classical invariant theory, namely the Poincare series for an algebra of invariants of Lie group $SL_2$. The first two terms of the Laurent series expansion of Poincar\'e se...
Hilbert polynomials of the algebras of SL2 -invariants
We consider one of the fundamental problems of classical invariant theory, the research of Hilbert polynomials for an algebra of invariants of Lie group SL2. Form of the Hilbert polynomials gives us important information...
Generalized types of the growth of Dirichlet series
Let A∈(−∞,+∞] and Φ be a continuously on [σ0,A) function such that Φ(σ)→+∞ as σ→A−0. We establish a necessary and sufficient condition on a nonnegative sequence λ=(λn), increasing to +∞, under which the equality ¯¯¯¯¯¯¯¯...
Commutative Bezout domains in which any nonzero prime ideal is contained in a finite set of maximal ideals
We investigate commutative Bezout domains in which any nonzero prime ideal is contained in a finite set of maximal ideals. In particular, we have described the class of such rings, which are elementary divisor rings....