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.

A Theorem on Henselian Rings

1968 ◽  
Vol 11 (2) ◽  
pp. 275-277 ◽  
Author(s):  
N. Sankaran
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Maximal Ideal ◽  
Identity Element ◽  
Maximal Ideals ◽  
Formal Power

It is known that if K is a field, then the ring of formal power series in one or more variables, with coefficients in K, is Henselian at its maximal ideal. In this note we show that if R is a ring (commutative and with identity element) which is Henselian at the maximal ideals M1, M2, … then R[[x]] - the ring of formal power series in x with coefficients from R - is also Henselian at the maximal ideals M1 ⋅ R[[x]] + x⋅ R[[x]], etc.

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.

Functionals on Real C(S)

1978 ◽  
Vol 30 (03) ◽  
pp. 490-498 ◽  
Author(s):  
Nicholas Farnum ◽  
Robert Whitley
Keyword(s):  
Banach Algebra ◽  
Compact Hausdorff Space ◽  
Linear Functional ◽  
Hausdorff Space ◽  
Function Algebra ◽  
Continuous Functions ◽  
Codimension One ◽  
Maximal Ideals ◽  
Invertible Elements ◽  
Complex Valued

The maximal ideals in a commutative Banach algebra with identity have been elegantly characterized [5; 6] as those subspaces of codimension one which do not contain invertible elements. Also, see [1]. For a function algebra A, a closed separating subalgebra with constants of the algebra of complex-valued continuous functions on the spectrum of A, a compact Hausdorff space, this characterization can be restated: Let F be a linear functional on A with the property: (*) For each ƒ in A there is a point s, which may depend on f, for which F(f) = f(s).

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.

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].

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
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