Point-evaluation functionals on algebras of symmetric functions on $(L_\infty)^2$

It is known that every continuous symmetric (invariant under the composition of its argument with each Lebesgue measurable bijection of $[0,1]$ that preserve the Lebesgue measure of measurable sets) polynomial on the Cartesian power of the complex Banach space $L_\infty$ of all Lebesgue measurable essentially bounded complex-valued functions on $[0,1]$ can be uniquely represented as an algebraic combination, i.e., a linear combination of products, of the so-called elementary symmetric polynomials. Consequently, every continuous complex-valued linear multiplicative functional (character) of an arbitrary topological algebra of the functions on the Cartesian power of $L_\infty,$ which contains the algebra of continuous symmetric polynomials on the Cartesian power of $L_\infty$ as a dense subalgebra, is uniquely determined by its values on elementary symmetric polynomials. Therefore, the problem of the description of the spectrum (the set of all characters) of such an algebra is equivalent to the problem of the description of sets of the above-mentioned values of characters on elementary symmetric polynomials. In this work, the problem of the description of sets of values of characters, which are point-evaluation functionals, on elementary symmetric polynomials on the Cartesian square of $L_\infty$ is completely solved. We show that sets of values of point-evaluation functionals on elementary symmetric polynomials satisfy some natural condition. Also, we show that for any set $c$ of complex numbers, which satisfies the above-mentioned condition, there exists an element $x$ of the Cartesian square of $L_\infty$ such that values of the point-evaluation functional at $x$ on elementary symmetric polynomials coincide with the respective elements of the set $c.$

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Topology on the spectrum of the algebra of entire symmetric functions of bounded type on the complex $L_\infty$

Carpathian Mathematical Publications ◽

10.15330/cmp.9.1.22-27 ◽

2017 ◽

Vol 9 (1) ◽

pp. 22-27 ◽

Cited By ~ 3

Author(s):

T.V. Vasylyshyn

Keyword(s):

Symmetric Functions ◽

Complex Banach Space ◽

Continuous Homomorphism ◽

The Other ◽

Quotient Topology ◽

Abelian Topological Group ◽

Point Evaluation ◽

Other Hand ◽

Continuous Complex ◽

Complex Valued

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 Hausdorffand that $M_{bs}$ with the operation of coordinate-wise addition of sequences forms an abelian topological group.

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Metric on the spectrum of the algebra of entire symmetric functions of bounded type on the complex $L_\infty$

Carpathian Mathematical Publications ◽

10.15330/cmp.9.2.198-201 ◽

2018 ◽

Vol 9 (2) ◽

pp. 198-201 ◽

Cited By ~ 1

Author(s):

T.V. Vasylyshyn

Keyword(s):

Symmetric Functions ◽

Complex Banach Space ◽

Quotient Topology ◽

Natural Representation ◽

Abelian Topological Group ◽

Point Evaluation ◽

Continuous Complex ◽

Fréchet Algebra ◽

Quotient Set ◽

Complex Valued

It is known that every complex-valued homomorphism of the Fréchet algebra $H_{bs}(L_\infty)$ of all entire symmetric functions of bounded type on the complex Banach space $L_\infty$ is a point-evaluation functional $\delta_x$ (defined by $\delta_x(f) = f(x)$ for $f \in H_{bs}(L_\infty)$) at some point $x \in L_\infty.$ Therefore, 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 $L_\infty/_\sim,$ where an equivalence relation "$\sim$'' on $L_\infty$ is defined by $x\sim y \Leftrightarrow \delta_x = \delta_y.$ Consequently, $M_{bs}$ can be endowed with the quotient topology. On the other hand, $M_{bs}$ has a natural representation as a set of sequences which endowed with the coordinate-wise addition and the quotient topology forms an Abelian topological group. We show that the topology on $M_{bs}$ is metrizable and it is induced by the metric $d(\xi, \eta) = \sup_{n\in\mathbb{N}}\sqrt[n]{|\xi_n-\eta_n|},$ where $\xi = \{\xi_n\}_{n=1}^\infty,\eta = \{\eta_n\}_{n=1}^\infty \in M_{bs}.$

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Algebras of symmetric analytic functions on Cartesian powers of Lebesgue integrable in a power $p\in [1,+\infty)$ functions

Carpathian Mathematical Publications ◽

10.15330/cmp.13.2.340-351 ◽

2021 ◽

Vol 13 (2) ◽

pp. 340-351

Author(s):

T.V. Vasylyshyn

Keyword(s):

Banach Space ◽

Analytic Functions ◽

Entire Functions ◽

Complex Banach Space ◽

Cartesian Power ◽

Entire Complex ◽

Continuous Complex ◽

Fréchet Algebra ◽

Algebraic Basis ◽

Complex Valued

The work is devoted to the study of Fréchet algebras of symmetric (invariant under the composition of every of components of its argument with any measure preserving bijection of the domain of components of the argument) analytic functions on Cartesian powers of complex Banach spaces of Lebesgue integrable in a power $p\in [1,+\infty)$ complex-valued functions on the segment $[0,1]$ and on the semi-axis. We show that the Fréchet algebra of all symmetric analytic entire complex-valued functions of bounded type on the $n$th Cartesian power of the complex Banach space $L_p[0,1]$ of all Lebesgue integrable in a power $p\in [1,+\infty)$ complex-valued functions on the segment $[0,1]$ is isomorphic to the Fréchet algebra of all analytic entire functions on $\mathbb C^m,$ where $m$ is the cardinality of the algebraic basis of the algebra of all symmetric continuous complex-valued polynomials on this Cartesian power. The analogical result for the Fréchet algebra of all symmetric analytic entire complex-valued functions of bounded type on the $n$th Cartesian power of the complex Banach space $L_p[0,+\infty)$ of all Lebesgue integrable in a power $p\in [1,+\infty)$ complex-valued functions on the semi-axis $[0,+\infty)$ is proved.

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Generalized Numerical Index of Function Algebras

Journal of Function Spaces ◽

10.1155/2019/9080867 ◽

2019 ◽

Vol 2019 ◽

pp. 1-6

Author(s):

Han Ju Lee

Keyword(s):

Banach Space ◽

Function Algebra ◽

Complex Banach Space ◽

Continuous Functions ◽

Function Algebras ◽

Bounded Complex ◽

Closed Subalgebra ◽

Bounded Continuous Functions ◽

Complex Valued ◽

Numerical Index

Let X be a complex Banach space and Cb(Ω:X) be the Banach space of all bounded continuous functions from a Hausdorff space Ω to X, equipped with sup norm. A closed subspace A of Cb(Ω:X) is said to be an X-valued function algebra if it satisfies the following three conditions: (i) A≔{x⁎∘f:f∈A, x⁎∈X⁎} is a closed subalgebra of Cb(Ω), the Banach space of all bounded complex-valued continuous functions; (ii) ϕ⊗x∈A for all ϕ∈A and x∈X; and (iii) ϕf∈A for every ϕ∈A and for every f∈A. It is shown that k-homogeneous polynomial and analytic numerical index of certain X-valued function algebras are the same as those of X.

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Towards the Calculation of Jack Polynomials

10.26686/wgtn.16945804.v1 ◽

2021 ◽

Author(s):

◽

Leigh Alan Roberts

Keyword(s):

Montel Algebras on the Plane

Canadian Journal of Mathematics ◽

10.4153/cjm-1970-014-5 ◽

1970 ◽

Vol 22 (1) ◽

pp. 116-122 ◽

Cited By ~ 5

Author(s):

W. E. Meyers

Keyword(s):

Topological Algebra ◽

Maximum Modulus ◽

Continuous Homomorphism ◽

Continuous Functions ◽

Maximum Modulus Principle ◽

Compact Open Topology ◽

Bounded Functions ◽

Continuous Complex ◽

Complex Valued

The results of Rudin in [7] show that under certain conditions, the maximum modulus principle characterizes the algebra A (G) of functions analytic on an open subset G of the plane C (see below). In [2], Birtel obtained a characterization of A(C) in terms of the Liouville theorem; he proved that every singly generated F-algebra of continuous functions on C which contains no non-constant bounded functions is isomorphic to A(C) in the compact-open topology. In this paper we show that the Montel property of the topological algebra A (G) also characterizes it. In particular, any Montel algebra A of continuous complex-valued functions on G which contains the polynomials and has continuous homomorphism space M (A) homeomorphic to G is precisely A(G).

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Algebras generated by special symmetric polynomials on $\ell_1$

Carpathian Mathematical Publications ◽

10.15330/cmp.11.2.335-344 ◽

2019 ◽

Vol 11 (2) ◽

pp. 335-344

Author(s):

F. Jawad ◽

H. Karpenko ◽

A.V. Zagorodnyuk

Keyword(s):

Topological Algebra ◽

Algebra Structure ◽

Symmetric Polynomials ◽

Point Evaluation ◽

Complex Spaces ◽

Direct Sum ◽

Quotient Set

Let $X$ be a weighted direct sum of infinity many copies of complex spaces $\ell_1\bigoplus \ell_1.$ We consider an algebra consisting of polynomials on $X$ which are supersymmetric on each term $\ell_1\bigoplus \ell_1.$ Point evaluation functionals on such algebra gives us a relation of equivalence `$\sim$' on $X.$ We investigate the quotient set $X/\sim$ and show that under some conditions, it has a real topological algebra structure.

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Symmetric $*$-polynomials on $\mathbb C^n$

Carpathian Mathematical Publications ◽

10.15330/cmp.10.2.395-401 ◽

2018 ◽

Vol 10 (2) ◽

pp. 395-401

Author(s):

T.V. Vasylyshyn

Keyword(s):

Vector Space ◽

Symmetric Polynomial ◽

Vector Spaces ◽

Symmetric Polynomials ◽

Complex Vector Space ◽

Complex Vector ◽

Cartesian Power ◽

Elementary Symmetric Polynomials

$*$-Polynomials are natural generalizations of usual polynomials between complex vector spaces. A $*$-polynomial is a function between complex vector spaces $X$ and $Y,$ which is a sum of so-called $(p,q)$-polynomials. In turn, for nonnegative integers $p$ and $q,$ a $(p,q)$-polynomial is a function between $X$ and $Y,$ which is the restriction to the diagonal of some mapping, acting from the Cartesian power $X^{p+q}$ to $Y,$ which is linear with respect to every of its first $p$ arguments, antilinear with respect to every of its last $q$ arguments and invariant with respect to permutations of its first $p$ arguments and last $q$ arguments separately. In this work we construct formulas for recovering of $(p,q)$-polynomial components of $*$-polynomials, acting between complex vector spaces $X$ and $Y,$ by the values of $*$-polynomials. We use these formulas for investigations of $*$-polynomials, acting from the $n$-dimensional complex vector space $\mathbb{C}^n$ to $\mathbb{C},$ which are symmetric, that is, invariant with respect to permutations of coordinates of its argument. We show that every symmetric $*$-polynomial, acting from $\mathbb{C}^n$ to $\mathbb{C},$ can be represented as an algebraic combination of some "elementary" symmetric $*$-polynomials. Results of the paper can be used for investigations of algebras, generated by symmetric $*$-polynomials, acting from $\mathbb{C}^n$ to $\mathbb{C}.$

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SOME PROPERTIES OF ELEMENTARY SYMMETRIC POLYNOMIALS ON THE CARTESIAN SQUARE OF THE COMPLEX BANACH SPACE L∞ [ 0,1 ]

PRECARPATHIAN BULLETIN OF THE SHEVCHENKO SCIENTIFIC SOCIETY Number ◽

10.31471/2304-7399-2018-2(46)-9-16 ◽

2018 ◽

pp. 9-16

Author(s):

T. V. Vasylyshyn

Keyword(s):

Banach Space ◽

Complex Banach Space ◽

Symmetric Polynomials ◽

Bounded Functions ◽

Elementary Symmetric Polynomials ◽

Algebraic Basis

We construct the element of the Cartesian square of the complex Banach space L ∞ [ 0,1 ] of all Lebesgue measurable essentially bounded functions on [ 0,1 ] by the predefined values of elementary symmetric polynomials on this element. Results of this work can be applied to the investigation of an algebraic basis of the algebra of continuous symmetric polynomials on the Cartesian square of the complex Banach space L ∞ [ 0,1 ] .

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Towards the Calculation of Jack Polynomials

10.26686/wgtn.16945804 ◽

2021 ◽

Author(s):

◽

Leigh Alan Roberts

Keyword(s):