The dual and bidual of certain -algebras

Author:
Freda E. Alexander

Journal:
Proc. Amer. Math. Soc. **38** (1973), 571-576

MSC:
Primary 46L15

MathSciNet review:
0318913

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Abstract: It is well known that every -algebra is Arens' regular and that its bidual is a -algebra. Wong has asked whether a dual -algebra of the first kind is Arens' regular. It is shown that this is true in the topologically simple case; in the course of the proof it is shown that in this case the bidual is, modulo its radical, an -algebra of the first kind.

**[1]**F. E. Alexander,*On annihilator and dual algebras*(to appear).**[2]**D. J. H. Garling,*On ideals of operators in Hilbert space*, Proc. London Math. Soc. (3)**17**(1967), 115–138. MR**0208398****[3]**Jesús Gil de Lamadrid,*Topological modules. Banach algebras, tensor products, algebras of kernels*, Trans. Amer. Math. Soc.**126**(1967), 361–419. MR**0205104**, 10.1090/S0002-9947-1967-0205104-X**[4]**Tôzirô Ogasawara and Kyôichi Yoshinaga,*Weakly completely continuous Banach *-algebras*, J. Sci. Hiroshima Univ. Ser. A.**18**(1954), 15–36. MR**0070068****[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****[6]**Robert Schatten,*A Theory of Cross-Spaces*, Annals of Mathematics Studies, no. 26, Princeton University Press, Princeton, N. J., 1950. MR**0036935****[7]**Pak-ken Wong,*On the Arens product and annihilator algebras*, Proc. Amer. Math. Soc.**30**(1971), 79–83. MR**0281005**, 10.1090/S0002-9939-1971-0281005-2

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DOI:
https://doi.org/10.1090/S0002-9939-1973-0318913-1

Keywords:
-algebra,
annihilator algebra,
Arens regular

Article copyright:
© Copyright 1973
American Mathematical Society