scholarly journals Continuity of derivations on topological algebras of power series

1969 ◽  
Vol 1 (3) ◽  
pp. 419-424 ◽  
Author(s):  
R.J. Loy
Keyword(s):  
Power Series ◽  
Banach Algebra ◽  
Formal Power Series ◽  
Topological Algebra ◽  
Fréchet Space ◽  
Frechet Space ◽  
Complex Field ◽  
Topological Algebras ◽  
Space Topology ◽  
Formal Power

Let A be an algebra of formal power series in one indeterminate over the complex field, D a derivation on A. It is shown that if A has a Fréchet space topology under which it is a topological algebra, then D is necessarily continuous provided the coordinate projections satisfy a certain equicontinuity condition. This condition is always satisfied if A is a Banach algebra and the projections are continuous. A second result is given, with weaker hypothesis on the projections and correspondingly weaker conclusion.

scholarly journals Uniqueness of the Fréchet space topology on certain topological algebras

1971 ◽  
Vol 4 (1) ◽  
pp. 1-7 ◽  
Author(s):  
R. J. Loy
Keyword(s):  
Power Series ◽  
Banach Algebra ◽  
Formal Power Series ◽  
Fréchet Space ◽  
Frechet Space ◽  
Topological Algebras ◽  
Norm Topology ◽  
Space Topology ◽  
Formal Power

It is well known that the complete norm topology on a Banach algebra is not unique in general, though semisimplicity is sufficient (but not necessary) for uniqueness. In this note we consider a class of topological algebras of formal power series which have unique Fréchet space topology. The structure of these algebras in the Banach algebra case will be considered in a later paper.

Local analytic structure in certain commutative topological algebras

1972 ◽  
Vol 6 (2) ◽  
pp. 161-167 ◽  
Author(s):  
R.J. Loy
Keyword(s):  
Power Series ◽  
Topological Algebra ◽  
Fréchet Space ◽  
Analytic Structure ◽  
Frechet Space ◽  
Topological Algebras ◽  
Space Topology ◽  
Formal Power ◽  
Locally Convex Topology ◽  
Locally Convex

Let B be a topological algebra with Fréchet space topology, A an algebra with locally convex topology and an algebra of formal power series over A in n commuting indeterminates which carries a Fréchet space topology. In a previous paper the author showed, for the case n = 1, that a homomorphism of B into whose range contains polynomials is necessarily continuous provided the coordinate projections of into A satisfy a certain equicontinuity condition. This result is here extended to the case of general n, and also to weaker topological assumptions.

scholarly journals Completion of normed algebras of polunomials

1975 ◽  
Vol 20 (4) ◽  
pp. 504-510 ◽  
Author(s):  
H. G. Dales ◽  
J. P. McClure
Keyword(s):  
Power Series ◽  
Banach Algebra ◽  
Formal Power Series ◽  
Complex Field ◽  
Linear Functionals ◽  
Bounded Linear ◽  
One To One ◽  
Formal Power

Let P be the algebra of polynomials in one inderminate x over the complex field C. Suppose ∥ · ∥ is a norm on P such that the coefficient functionals cj: ∑αix1 → αj (j = 0,1,2,…) are all continuous with respect to ∥·∥, and Let K ⊂ C be the set of characters on P which are ∥·∥-continuous. then K is compact, C\K is connected, and 0∈K. K. Let A be the completion of P with respect to ∥·∥. Then A is a singly generated Banach algebra, with space of characters (homeomorphic with) K. The functionals cj have unique extensions to bounded linear functionals on A, and the map a →∑Ci(a)xi (a ∈ A) is a homomorphism from A onto an algebra of formal power series with coefficients in C. We say that A is an algebra of power series if this homomorphism is one-to-one, that is if a ∈ A and a≠O imply cj(a)≠ 0 for some j.

Banach algebras of power series

1974 ◽  
Vol 17 (3) ◽  
pp. 263-273 ◽  
Author(s):  
Richard J. Loy
Keyword(s):  
Power Series ◽  
Banach Algebra ◽  
Formal Power Series ◽  
Weak Topology ◽  
Present Author ◽  
Banach Algebras ◽  
Complex Field ◽  
Formal Power ◽  
Special Case

Let C[[t]] denote the algebra of all formal power series over the complex field C in a commutative indeterminate t with the weak topology determined by the projections pj: Σαiti ↦αj. A subalgebra A of C[[t]] is a Banach algebra of power series if it contains the polynomials and is a Banach algebra under a norm such that the inclusion map A ⊂ C[[t]] is continuous. Such algebras were first introduced in [13] when considering algebras with one generator, and studied, in a special case, in [23]. For a partial bibliography of their subsequent study and application see the references of [9] (note that the usage of the term Banach algebra of power series in [9] differs from that here), and also [2], [3], [11]. Indeed, an examination of their use in [11], under more general topological conditions than here, led the present author to the results of [14], [15], [16], [17].

Formal Power Series Over Commutative N-Algebras

1978 ◽  
Vol 30 (01) ◽  
pp. 66-84 ◽  
Author(s):  
Ernst August Behrens
Keyword(s):  
Power Series ◽  
Banach Algebra ◽  
Formal Power Series ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Identity Element ◽  
Closed Ideal ◽  
Supremum Norm ◽  
Maximal Ideals ◽  
Formal Power

A Banach algebra P over C with identity element is called an N-algebra if any closed ideal in P is the intersection of maximal ideals. An example is given by the algebra of the continuous C-valued functions on a compact Hausdorff space X under the supremum norm; two others are discussed in § 3.

scholarly journals On transformations of formal power series

2003 ◽  
Vol 184 (2) ◽  
pp. 369-383 ◽  
Author(s):  
Manfred Droste ◽  
Guo-Qiang Zhang
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Formal Power

scholarly journals On the Square Root of a Bell Matrix

2021 ◽  
Vol 76 (1) ◽  
Author(s):  
Donatella Merlini
Keyword(s):  
Power Series ◽  
Closed Form ◽  
Formal Power Series ◽  
Computational Method ◽  
Square Root ◽  
Riordan Arrays ◽  
Formal Power

AbstractIn the context of Riordan arrays, the problem of determining the square root of a Bell matrix $$R={\mathcal {R}}(f(t)/t,\ f(t))$$ R = R ( f ( t ) / t , f ( t ) ) defined by a formal power series $$f(t)=\sum _{k \ge 0}f_kt^k$$ f ( t ) = ∑ k ≥ 0 f k t k with $$f(0)=f_0=0$$ f ( 0 ) = f 0 = 0 is presented. It is proved that if $$f^\prime (0)=1$$ f ′ ( 0 ) = 1 and $$f^{\prime \prime }(0)\ne 0$$ f ″ ( 0 ) ≠ 0 then there exists another Bell matrix $$H={\mathcal {R}}(h(t)/t,\ h(t))$$ H = R ( h ( t ) / t , h ( t ) ) such that $$H*H=R;$$ H ∗ H = R ; in particular, function h(t) is univocally determined by a symbolic computational method which in many situations allows to find the function in closed form. Moreover, it is shown that function h(t) is related to the solution of Schröder’s equation. We also compute a Riordan involution related to this kind of matrices.

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