Banach algebras in which every element is a topological zero divisor

Authors:
S. J. Bhatt and H. V. Dedania

Journal:
Proc. Amer. Math. Soc. **123** (1995), 735-737

MSC:
Primary 46H05; Secondary 46K05

DOI:
https://doi.org/10.1090/S0002-9939-1995-1224613-0

MathSciNet review:
1224613

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Abstract: Every element of a complex Banach algebra is a topological divisor of zero, if at least one of the following holds: (i) *A* is infinite dimensional and admits an orthogonal basis, (ii) *A* is a nonunital uniform Banach algebra in which the Silov boundary coincides with the Gelfand space ; and (iii) *A* is a nonunital hermitian Banach -algebra with continuous involution. Several algebras of analysis have this property. Examples are discussed to show that (a) neither hermiticity nor can be omitted, and that (b) in case (ii), is not a necessary condition.

**[1]**Subhash J. Bhatt and G. M. Deheri,*Köthe spaces and topological algebra with bases*, Proc. Indian Acad. Sci. Math. Sci.**100**(1990), no. 3, 259–273. MR**1081710****[2]**Frank F. Bonsall and John Duncan,*Complete normed algebras*, Springer-Verlag, New York-Heidelberg, 1973. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 80. MR**0423029****[3]**Taqdir Husain and Saleem Watson,*Topological algebras with orthogonal Schauder bases*, Pacific J. Math.**91**(1980), no. 2, 339–347. MR**615682****[4]**Ronald Larsen,*Banach algebras*, Marcel Dekker, Inc., New York, 1973. An introduction; Pure and Applied Mathematics, No. 24. MR**0487369****[5]**Charles E. Rickart,*General theory of Banach algebras*, The University Series in Higher Mathematics, D. van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR**0115101**

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DOI:
https://doi.org/10.1090/S0002-9939-1995-1224613-0

Keywords:
Topological divisor of zero,
hermitian Banach -algebra,
orthogonal basis,
uniform Banach algebra

Article copyright:
© Copyright 1995
American Mathematical Society