Frechet Algebra Research Articles

Automatic Continuity of Almost Derivations on Frechet $Q$-Algebras

In 1971 R. L. Carpenter proved that every derivation on a semisimple commutative Frechet algebra with identity is continuous. The concept of almost derivations on Frechet algebras is introduced in this article. Also, R. L. Carpenter result motivates us to ask an open question: Is every almost derivation on semisimple commutative Frechet algebras continuous?. Moreover, a partial answer to this open question is derived in the sense that every almost derivation T on semisimple commutative Frechet Q-algebras A, with an additional condition on A, is continuous. Furthermore, an example is provided to illustrate our main result.

The BSE Property for Vector-Valued Frechet Lipschitz Algebras

Let $( X,d )$ be a metric space with at least two elements and $( A , p_l )$ be a commutative semisimple Frechet algebra over the scalar field of complex numbers. The correlation between the BSE-property of the Frechet algebra $( A , p_l )$ and $\Lip_{d}( X , A )$ is assessed. It is found and approved that if $\Lip_{d}( X , A )$ is a BSE-Frechet algebra, then so is $A$. The opposite correlation will hold if $( A , p_l )$ is unital.

Deformation quantization on the cotangent bundle of a Lie group

We develop a complete theory of non-formal deformation quantization on the cotangent bundle of a weakly exponential Lie group. An appropriate integral formula for the star-product is introduced together with a suitable space of functions on which the star-product is well defined. This space of functions becomes a Fréchet algebra as well as a pre-C*-algebra. Basic properties of the star-product are proved, and the extension of the star-product to a Hilbert algebra and an algebra of distributions is given. A C*-algebra of observables and a space of states are constructed. Moreover, an operator representation in position space is presented. Finally, examples of weakly exponential Lie groups are given.

Algebras of Symmetric *-Polynomials in the Space ℂ2

We study symmetric *-polynomials in the space ℂ2. In particular, we construct a system of generators of the algebra of symmetric *-polynomials on ℂ2 and show that elements of this system are algebraically dependent. We also study the algebras obtained as completions of the algebra of symmetric * -polynomials in the space ℂ2. The spectrum and the *-spectrum of the Frechet algebra of absolutely convergent series of symmetric *-polynomials are described.

A $p$-adically entire function with integral values on $\mathbb {Q}_p$ and entire liftings of the $p$-divisible group $\mathbb {Q}_p/\mathbb {Z}_p$

We give a self-contained proof of the fact that, for any prime number $p$, there exists a power series $$\Psi= \Psi_p(T) \in T + T^2\Z[[T]] $$ which trivializes the addition law of the formal group of Witt covectors is $p$-adically entire and assumes values in $\Z_p$ all over $\Q_p$. We actually generalize, following a suggestion of M. Candilera, the previous facts to any fixed unramified extension $\Q_q$ of $\Q_p$ of degree $f$, where $q = p^f$. We show that $\Psi = \Psi_q$ provides a quasi-finite covering of the Berkovich affine line $\A^1_{\Q_p}$ by itself. We prove in section 3 new strong estimates for the growth of $\Psi$, in view of the application to $p$-adic Fourier expansions on $\Q_p$. We locate the zeros of $\Psi$ and to obtain its product expansion. We reconcile the present discussion (for $q =p$) with a previous formal group proof which takes place in the Frechet algebra $\Q_p\{x\}$ of the analytic additive group $\G_{a,\Q_p}$ over $\Q_p$. We show that, for any $\lambda \in \Q_p^\times$, the closure $\sE_\lambda^\circ$ of $\Z_p[\Psi(p^ix/\lambda)\,|\,i=0,1,\dots]$ in $\Q_p\{x\}$ is a Hopf algebra object in the category of Frechet $\Z_p$-algebras. The special fiber of $\sE_\lambda^\circ$ is the affine algebra of the $p$-divisible group $\Q_p/p \lambda \Z_p$ over $\F_p$, while $\sE_\lambda^\circ [1/p]$ is dense in $\Q_p\{x\}$. From $\Z_p[\Psi(\lambda x)\,|\,\lambda \in \Q_p^\times]$ we construct a $p$-adic analog $\AP_{\Q_p}(\Sigma_\rho)$ of the algebra of Dirichlet series holomorphic in a strip $(-\rho, \rho) \times i \R \subset \C$. We start developing this analogy. It turns out that the Banach algebra of almost periodic functions on $\Q_p$ identifies with the topological ring of germs of holomorphic almost periodic functions on strips around $\Q_p$.

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.

Almost multipliers of Fréchet algebras

In this paper, we introduce the notation of almost multiplier of Frechet algebras. We show that, under special hypotheses, every almost multiplier $$T:A\longrightarrow A$$ is either multiplier, or it is continuous. Some useful results about almost multiplier of Frechet algebras are given as well.

Algebras of Entire Symmetric Functions on Spaces of Lebesgue-Measurable Essentially Bounded Functions

It is shown that the Frechet algebra of all entire symmetric functions of bounded type on the complex Banach space of all Lebesgue measurable essentially bounded complex-valued functions defined on the segment [0,1] is isomorphic to the Frechet algebra of all entire symmetric functions of bounded type on the complex Banach space of all Lebesgue integrable essentially bounded complex-valued functions defined on the semiaxis.

Analytic structure on the spectrum of the algebra of symmetric analytic functions on $$L_\infty $$

We consider the subalgebra \(H_{bs}(L_\infty )\) of analytic functions of bounded type on \(L_\infty [0,1]\) which are symmetric, i.e. invariant, with respect to measurable bijections of [0, 1] that preserve the measure. Our main result is that \(H_{bs}(L_\infty )\) is isomorphic to the algebra of all analytic functions on the strong dual of the space of entire functions on the complex plane \({\mathbb {C}}.\) From this result we deduce that \(H_{bs}(L_\infty )\) is a test algebra for Michael problem about the continuity of complex valued homomorphisms on Frechet algebras.

F-cone metric spaces over Fréchet algebra

The paper deals with the achievements of introducing the notion of F-cone metric spaces over Frechet algebra as a generalization of F-cone metric spaces over a Banach algebra, N p -cone metric spac...

Zero product preserving functionals on $$C(\varOmega )$$-valued spaces of functions

Let X be a compact Hausdorff space and $$\varOmega $$ be a locally compact $$\sigma $$ -compact space. In this paper we study (real-linear) continuous zero product preserving functionals $$\varphi : A \longrightarrow {\mathbb {C}}$$ on certain subalgebras A of the Frechet algebra $$C(X,C(\varOmega ))$$ . The case that $$\varphi $$ is continuous with respect to a specified complete metric on A will also be discussed. In particular, for a compact Hausdorff space K we characterize $$\Vert \cdot \Vert $$ -continuous linear zero product preserving functionals on the Banach algebra $$C^1([0,1],C(K))$$ equipped with the norm $$\Vert f\Vert =\Vert f\Vert _{[0,1]}+\Vert f'\Vert _{[0,1]}$$ , where $$\Vert \cdot \Vert _{[0,1]}$$ denotes the supremum norm. An application of the results is given for continuous ring homomorphisms on such subalgebras.

Uniqueness of F-Algebra Topology for Commutative Semisimple Algebras

Let A be a complex commutative semisimple complete LMC algebra with respect to a topology $$\tau _{1}$$ and a complete metrizable topological algebra with respect to a topology $$\tau _{2}$$ . It is proved that every $$\tau _{1}$$ -bounded set is a $$\tau _{2}$$ -bounded set. This generalizes a result of R. L. Carpenter on uniqueness of Frechet algebra topology for complex commutative semisimple algebras.

Some problems concerning algebras of holomorphic functions

Let $X$ be a complex Banach space with open unit ball $B_X.$ We describe some recent work and a number of open problems related to the maximal ideal spaces of the Frechet algebra of holomorphic functions of bounded type on $X$ and the Banach algebra of bounded holomorphic functions on $B_X$.

Some properties of shift operators on algebras generated by $*$-polynomials

A $*$-polynomial is a function on a complex Banach space $X,$ which is a sum of so-called $(p,q)$-polynomials. In turn, for non-negative integers $p$ and $q,$ a $(p,q)$-polynomial is a function on $X,$ which is the restriction to the diagonal of some mapping, defined on the Cartesian power $X^{p+q},$ which is linear with respect to every of its first $p$ arguments and antilinear with respect to every of its other $q$ arguments. The set of all continuous $*$-polynomials on $X$ form an algebra, which contains the algebra of all continuous polynomials on $X$ as a proper subalgebra. So, completions of this algebra with respect to some natural norms are wider classes of functions than algebras of holomorphic functions. On the other hand, due to the similarity of structures of $*$-polynomials and polynomials, for the investigation of such completions one can use the technique, developed for the investigation of holomorphic functions on Banach spaces.
 We investigate the Frechet algebra of functions on a complex Banach space, which is the completion of the algebra of all continuous $*$-polynomials with respect to the countable system of norms, equivalent to norms of the uniform convergence on closed balls of the space. We establish some properties of shift operators (which act as the addition of some fixed element of the underlying space to the argument of a function) on this algebra. In particular, we show that shift operators are well-defined continuous linear operators. Also we prove some estimates for norms of values of shift operators. Using these results, we investigate one special class of functions from the algebra, which is important in the description of the spectrum (the set of all maximal ideals) of the algebra.

Segal Fréchet Algebras

In this paper, we study the concept of a Segal Frechet algebra and investigate and generalize many known results about abstract Segal algebras, for Segal Frechet algebras. Moreover, we characterize closed ideals of Segal Frechet algebras, and show that the ideal theorem is also valid for Frechet algebras.

Almost n-Multiplicative Maps between Frechet Algebras

For the Fr'{e}chet algebras $(A, (p_k))$ and $(B, (q_k))$ and $n in mathbb{N}$, $ngeq 2$, a linear map $T:A rightarrow B$ is called textit{almost $n$-multiplicative}, with respect to $(p_k)$ and $(q_k)$, if there exists $varepsilongeq 0$ such that$$q_k(Ta_1a_2cdots a_n-Ta_1Ta_2cdots Ta_n)leq varepsilon p_k(a_1) p_k(a_2)cdots p_k(a_n),$$for each $kin mathbb{N}$ and $a_1, a_2, ldots, a_nin A$. The linear map $T$ is called textit{weakly almost $n$-multiplicative}, if there exists $varepsilongeq 0$ such that for every $kin mathbb{N}$ there exists $n(k)in mathbb{N}$ with$$q_k(Ta_1a_2cdots a_n-Ta_1Ta_2cdots Ta_n)leq varepsilon p_{n(k)}(a_1) p_{n(k)}(a_2)cdots p_{n(k)}(a_n),$$for each $k in mathbb{N}$ and $a_1, a_2, ldots, a_nin A$.The linear map $T$ is called $n$-multiplicative if$$Ta_{1}a_{2} cdots a_{n} = Ta_{1} Ta_{2} cdots Ta_{n},$$for every $a_{1}, a_{2},ldots, a_{n} in A$.In this paper, we investigate automatic continuity of (weakly) almost $n$-multiplicative maps between certain classes of Fr'{e}chet algebras, including Banach algebras. We show that if $(A, (p_k))$ is a Fr'{e}chet algebra and $T: A rightarrow mathbb{C}$ is a weakly almost $n$-multiplicative linear functional, then either $T$ is $n$-multiplicative, or it is continuous. Moreover, if $(A, (p_k))$ and $(B, (q_k))$ are Fr'{e}chet algebras and $T:A rightarrow B$ is a continuous linear map, then under certain conditions $T$ is weakly almost $n$-multiplicative for each $ngeq 2$. In particular, every continuous linear functional on $A$ is weakly almost $n$-multiplicative for each $ngeq 2$.

An Extension of Segal Algebras

In this paper we study a class of dense ideals in a locally convex algebra. The results concern the relationship between this concept and abstract Segal algebras, which introduced by Reiter in 1071. We define generalized Segal algebras and prove that a finite dimensional semisimple Banach algebra cannot be a proper generalized Segal algebra in a Frechet algebra. //

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.

Compact homomorphisms between uniform Fréchet algebras

We study compact homomorphisms between uniform Frechet algebras, by analyzing the behavior of its spectral adjoint on the underlying spectrum. We prove that every compact homomorphism between uniform Frechet algebras actually ranges into a uniform Banach algebra, and that its spectral adjointmaps τ-bounded subsets into relatively τ-compact subsets, when τ is the strong or the compact-open topology.

Smooth Frechet subalgebras of C ∗ -algebras defined by first order differential seminorms

The differential structure in a C ∗-algebra defined by a dense Frechet subalgebra whose topology is defined by a sequence of differential seminorms of order 1 is investigated. This includes differential Arens–Michael decomposition, spectral invariance, closure under functional calculi as well as intrinsic spectral description. A large number of examples of such Frechet algebras are exhibited; and the smooth structure defined by an unbounded self-adjoint Hilbert space operator is discussed.