Algebras generated by special symmetric polynomials on $\ell_1$

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.

Spectra of some algebras of entire functions of bounded type, generated by a sequence of polynomials

In this work, we investigate the properties of the topological algebra of entire functions of bounded type, generated by a countable set of homogeneous polynomials on a complex Banach space. Let $X$ be a complex Banach space. We consider a subalgebra $H_{b\mathbb{P}}(X)$ of the Fréchet algebra of entire functions of bounded type $H_b(X),$ generated by a countable set of algebraically independent homogeneous polynomials $\mathbb{P}.$ We show that each term of the Taylor series expansion of entire function, which belongs to the algebra $H_{b\mathbb{P}}(X),$ is an algebraic combination of elements of $\mathbb{P}.$ We generalize the theorem for computing the radius function of a linear functional on the case of arbitrary subalgebra of the algebra $H_b(X)$ on the space $X.$ Every continuous linear multiplicative functional, acting from $H_{b\mathbb{P}}(X)$ to $\mathbb{C}$ is uniquely determined by the sequence of its values on the elements of $\mathbb{P}.$ Consequently, there is a bijection between the spectrum (the set of all continuous linear multiplicative functionals) of the algebra $H_{b\mathbb{P}}(X)$ and some set of sequences of complex numbers. We prove the upper estimate for sequences of this set. Also we show that every function that belongs to the algebra $H_{b\mathbb{P}}(X),$ where $X$ is a closed subspace of the space $\ell_{\infty}$ such that $X$ contains the space $c_{00},$ can be uniquely analytically extended to $\ell_{\infty}$ and algebras $H_{b\mathbb{P}}(X)$ and $H_{b\mathbb{P}}(\ell)$ are isometrically isomorphic. We describe the spectrum of the algebra $H_{b\mathbb{P}}(X)$ in this case for some special form of the set $\mathbb{P}.$ Results of the paper can be used for investigations of the algebra of symmetric analytic functions on Banach spaces.

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 $c.$

Topological stable rank of $$\mathcal {E}'(\mathbb {R})$$

The set $$\mathcal {E}'(\mathbb {R})$$ E ′ ( R ) of all compactly supported distributions, with the operations of addition, convolution, multiplication by complex scalars, and with the strong dual topology is a topological algebra. In this article, it is shown that the topological stable rank of $$\mathcal {E}'(\mathbb {R})$$ E ′ ( R ) is 2.

PROFINITENESS IN FINITELY GENERATED VARIETIES IS UNDECIDABLE

Profinite algebras are exactly those that are isomorphic to inverse limits of finite algebras. Such algebras are naturally equipped with Boolean topologies. A variety ${\cal V}$ is standard if every Boolean topological algebra with the algebraic reduct in ${\cal V}$ is profinite.We show that there is no algorithm which takes as input a finite algebra A of a finite type and decide whether the variety $V\left( {\bf{A}} \right)$ generated by A is standard. We also show the undecidability of some related properties. In particular, we solve a problem posed by Clark, Davey, Freese, and Jackson.We accomplish this by combining two results. The first one is Moore’s theorem saying that there is no algorithm which takes as input a finite algebra A of a finite type and decides whether $V\left( {\bf{A}} \right)$ has definable principal subcongruences. The second is our result saying that possessing definable principal subcongruences yields possessing finitely determined syntactic congruences for varieties. The latter property is known to yield standardness.

Multipliers of Uniform Topological Algebras

Let E be a complete uniform topological algebra with Arens-Michael normed factors within an algebra isomorphism ϕ. If each factor Eα is complete, then every multiplier of E is continuous and ϕ is a topological algebra isomorphism where M(E) is endowed with its seminorm topology.