# Topological algebras with an orthogonal total sequence

Colloquium Mathematicae (1997)

• Volume: 72, Issue: 2, page 215-222
• ISSN: 0010-1354

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## Abstract

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The aim of this paper is an investigation of topological algebras with an orthogonal sequence which is total. Closed prime ideals or closed maximal ideals are kernels of multiplicative functionals and the continuous multiplicative functionals are given by the “coefficient functionals”. Our main result states that an orthogonal total sequence in a unital Fréchet algebra is already a Schauder basis. Further we consider algebras with a total sequence ${\left({x}_{n}\right)}_{n\in ℕ}$ satisfying ${x}_{n}^{2}={x}_{n}$ and ${x}_{n}{x}_{n+1}={x}_{n+1}$ for all n ∈ ℕ.

## How to cite

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Render, Hermann. "Topological algebras with an orthogonal total sequence." Colloquium Mathematicae 72.2 (1997): 215-222. <http://eudml.org/doc/210460>.

@article{Render1997,
abstract = {The aim of this paper is an investigation of topological algebras with an orthogonal sequence which is total. Closed prime ideals or closed maximal ideals are kernels of multiplicative functionals and the continuous multiplicative functionals are given by the “coefficient functionals”. Our main result states that an orthogonal total sequence in a unital Fréchet algebra is already a Schauder basis. Further we consider algebras with a total sequence $(x_n)_\{n∈ℕ\}$ satisfying $x^2_n=x_n$ and $x_n x_\{n+1\} = x_\{n+1\}$ for all n ∈ ℕ.},
author = {Render, Hermann},
journal = {Colloquium Mathematicae},
keywords = {orthogonal basis; Hadamard product; topological algebra; closed prime ideals; coefficient functionals; topological algebras; closed maximal ideals; kernels of multiplicative functionals; orthogonal total sequence; unital Fréchet algebra; Schauder basis},
language = {eng},
number = {2},
pages = {215-222},
title = {Topological algebras with an orthogonal total sequence},
url = {http://eudml.org/doc/210460},
volume = {72},
year = {1997},
}

TY - JOUR
AU - Render, Hermann
TI - Topological algebras with an orthogonal total sequence
JO - Colloquium Mathematicae
PY - 1997
VL - 72
IS - 2
SP - 215
EP - 222
AB - The aim of this paper is an investigation of topological algebras with an orthogonal sequence which is total. Closed prime ideals or closed maximal ideals are kernels of multiplicative functionals and the continuous multiplicative functionals are given by the “coefficient functionals”. Our main result states that an orthogonal total sequence in a unital Fréchet algebra is already a Schauder basis. Further we consider algebras with a total sequence $(x_n)_{n∈ℕ}$ satisfying $x^2_n=x_n$ and $x_n x_{n+1} = x_{n+1}$ for all n ∈ ℕ.
LA - eng
KW - orthogonal basis; Hadamard product; topological algebra; closed prime ideals; coefficient functionals; topological algebras; closed maximal ideals; kernels of multiplicative functionals; orthogonal total sequence; unital Fréchet algebra; Schauder basis
UR - http://eudml.org/doc/210460
ER -

## References

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1. [1] M. Akkar, M. El Azhari and M. Oudadess, Continuité des caractères dans les algèbres de Fréchet à bases, Canad. Math. Bull. 31 (1988), 168-174. Zbl0663.46046
2. [2] R. M. Brooks, A ring of analytic functions, Studia Math. 24 (1964), 191-210. Zbl0199.46201
3. [3] R. M. Brooks, A ring of analytic functions, II, ibid. 39 (1971), 199-208. Zbl0213.40303
4. [4] R. Brück and J. Müller, Invertible elements in a convolution algebra of holomorphic functions, Math. Ann. 294 (1992), 421-438. Zbl0769.30002
5. [5] R. Brück and J. Müller, Closed ideals in a convolution algebra of holomorphic functions, Canad. J. Math. 47 (1995), 915-928. Zbl0836.30002
6. [6] S. El-Helaly and T. Husain, Orthogonal bases are Schauder bases and a characterization of Φ-algebras, Pacific J. Math. 132 (1988), 265-275. Zbl0654.46011
7. [7] S. El-Helaly and T. Husain, Orthogonal bases characterizations of the Banach algebras ${l}_{1}$ and ${c}_{0}$, Math. Japon. 37 (1992), 649-655. Zbl0763.46040
8. [8] H. Goldmann, Uniform Fréchet Algebras, North-Holland, Amsterdam, 1990.
9. [9] T. Husain, Positive functionals on topological algebras with bases, Math. Japon. 28 (1983), 683-687. Zbl0535.46023
10. [10] T. Husain and J. Liang, Multiplicative functionals on Fréchet algebras with bases, Canad. J. Math. 29 (1977), 270-276. Zbl0348.46036
11. [11] T. Husain and S. Watson, Topological algebras with orthogonal bases, Pacific J. Math. 91 (1980), 339-347. Zbl0477.46042
12. [12] T. Husain and S. Watson, Algebras with unconditional orthogonal bases, Proc. Amer. Math. Soc. 79 (1980), 539-545. Zbl0434.46029
13. [13] H. Render and A. Sauer, Algebras of holomorphic functions with Hadamard multiplication, Studia Math. 118 (1996), 77-100. Zbl0855.46032
14. [14] S. W. Warsi and T. Husain, Pil-algebras, Math. Japon. 36 (1991), 983-986.
15. [15] W. Żelazko, Banach Algebras, Elsevier, Amsterdam, 1973.
16. [16] W. Żelazko, Metric generalizations of Banach algebras, Dissertationes Math. 47 (1965). Zbl0131.13005
17. [17] W. Żelazko, Functional continuity of commutative m-convex ${B}_{0}$-algebras with countable maximal ideal spaces, Colloq. Math. 51 (1987), 395-399. Zbl0632.46041

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