# Topological algebras with an orthogonal total sequence

Colloquium Mathematicae (1997)

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

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topRender, 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

top- [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] R. M. Brooks, A ring of analytic functions, Studia Math. 24 (1964), 191-210. Zbl0199.46201
- [3] R. M. Brooks, A ring of analytic functions, II, ibid. 39 (1971), 199-208. Zbl0213.40303
- [4] R. Brück and J. Müller, Invertible elements in a convolution algebra of holomorphic functions, Math. Ann. 294 (1992), 421-438. Zbl0769.30002
- [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] 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] 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] H. Goldmann, Uniform Fréchet Algebras, North-Holland, Amsterdam, 1990.
- [9] T. Husain, Positive functionals on topological algebras with bases, Math. Japon. 28 (1983), 683-687. Zbl0535.46023
- [10] T. Husain and J. Liang, Multiplicative functionals on Fréchet algebras with bases, Canad. J. Math. 29 (1977), 270-276. Zbl0348.46036
- [11] T. Husain and S. Watson, Topological algebras with orthogonal bases, Pacific J. Math. 91 (1980), 339-347. Zbl0477.46042
- [12] T. Husain and S. Watson, Algebras with unconditional orthogonal bases, Proc. Amer. Math. Soc. 79 (1980), 539-545. Zbl0434.46029
- [13] H. Render and A. Sauer, Algebras of holomorphic functions with Hadamard multiplication, Studia Math. 118 (1996), 77-100. Zbl0855.46032
- [14] S. W. Warsi and T. Husain, Pil-algebras, Math. Japon. 36 (1991), 983-986.
- [15] W. Żelazko, Banach Algebras, Elsevier, Amsterdam, 1973.
- [16] W. Żelazko, Metric generalizations of Banach algebras, Dissertationes Math. 47 (1965). Zbl0131.13005
- [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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