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

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.

Essential amenability of Fréchet algebras

UDC 517.98 Essential amenability of Banach algebras have been defined and investigated. Here, this concept will be introduced for Frechet algebras. Then a number of well-known results of essential amenability of Banach algebras are generalized for Fréchet algebras. Moreover, related results about Segal–Fréchet algebras are provided. As the main result, it is provedthat if ( 𝒜 , p ℓ ) is an amenable Fréchet algebra with a uniformly bounded approximate identity, then every symmetric Segal – Fréchet algebra in ( 𝒜 , p ℓ ) is essentially amenable.

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.

Comparison and continuity of Wick-type star products on certain coadjoint orbits

In this paper, we discuss continuity properties of the Wick-type star product on the 2-sphere, interpreted as a coadjoint orbit. Star products on coadjoint orbits in general have been constructed by different techniques. We compare the constructions of Alekseev–Lachowska and Karabegov, and we prove that they agree in general. In the case of the 2-sphere, we establish the continuity of the star product, thereby allowing for a completion to a Fréchet algebra.

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

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 $\xi = \{\xi_n\}_{n=1}^\infty,\eta = \{\eta_n\}_{n=1}^\infty \in M_{bs}.$

Hypercyclic operators on algebra of symmetric analytic functions on $\ell_p$

In the paper, it is proposed a method of construction of hypercyclic composition operators on $H(\mathbb{C}^n)$ using polynomial automorphisms of $\mathbb{C}^n$ and symmetric analytic functions on $\ell_p.$ In particular, we show that an "symmetric translation" operator is hypercyclic on a Frechet algebra of symmetric entire functions on $\ell_p$ which are bounded on bounded subsets.