Embedding the Algebra of Formal Power Series in Several Variables into a Banach Algebra

1988 ◽  
Vol s2-37 (2) ◽  
pp. 331-350
Author(s):  
Elizabeth Strouse
Keyword(s):  
Power Series ◽  
Banach Algebra ◽  
Formal Power Series ◽  
Several Variables ◽  
Formal Power

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.

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 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 Some Enumerations on Non-Decreasing Dyck Paths

10.37236/3941 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Éva Czabarka ◽  
Rigoberto Flórez ◽  
Leandro Junes
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Maximum Element ◽  
Dyck Paths ◽  
Several Variables ◽  
Formal Power

We construct a formal power series on several variables that encodes many statistics on non-decreasing Dyck paths. In particular, we use this formal power series to count peaks, pyramid weights, and indexed sums of pyramid weights for all non-decreasing Dyck paths of length $2n.$ We also show that an indexed sum on pyramid weights depends only on the size and maximum element of the indexing set.

Substitution Groups of Formal Power Series

1954 ◽  
Vol 6 ◽  
pp. 325-340 ◽  
Author(s):  
S. A. Jennings
Keyword(s):  
Power Series ◽  
Commutative Ring ◽  
Formal Power Series ◽  
Point Of View ◽  
Group Operation ◽  
Several Variables ◽  
Special Cases ◽  
Image Position ◽  
Formal Power ◽  
Complex Coefficients

In this paper we are concerned with the group of formal power series of the form,the coefficients being elements of a commutative ring R and the group operation being substitution. Little seems to be known of the properties of groups of this type, except in special cases, although groups of formal power series in several variables with complex coefficients have been investigated from a different point of view by Bochner and Martin (1, chap. I) and Gotô (2).

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.

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