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]**S. J. Bhatt and G. M. Deheri,*Köthe spaces and topological algebras with bases*, Proc. Indian Acad. Sci. Math. Sci.**100**(1990), 259-273. MR**1081710 (91m:46017)****[2]**F. F. Bonsall and J. Duncan,*Complete normed algebras*, Springer-Verlag, Berlin, Heidelberg, and New York, 1973. MR**0423029 (54:11013)****[3]**T. Husain and S. Watson,*Topological algebras with orthogonal Schauder bases*, Pacific J. Math.**91**(1980), 339-347. MR**615682 (82h:46064)****[4]**R. Larsen,*Banach algebras*, Marcel Dekker, New York, 1973. MR**0487369 (58:7010)****[5]**C. E. Rickart,*General theory of Banach algebras*, Van Nostrand, Princeton, NJ, 1960. MR**0115101 (22:5903)**

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Additional Information

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