On the Arens product and commutative Banach algebras

Author:
Pak Ken Wong

Journal:
Proc. Amer. Math. Soc. **37** (1973), 111-113

MSC:
Primary 46H99; Secondary 46J05

MathSciNet review:
0306912

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Abstract: The purpose of this note is to generalize two recent results by the author for commutative Banach algebras. Let *A* be a commutative Banach algebra with carrier space and the canonical embedding of *A* into its second conjugate space (with the Arens product). We show that if *A* is a semisimple annihilator algebra, then is a two-sided ideal of . We also obtain that if *A* is a dense two-sided ideal of , then is a two-sided ideal of if and only if *A* is a modular annihilator algebra.

**[1]**Richard Arens,*The adjoint of a bilinear operation*, Proc. Amer. Math. Soc.**2**(1951), 839–848. MR**0045941**, 10.1090/S0002-9939-1951-0045941-1**[2]**Bruce A. Barnes,*Modular annihilator algebras*, Canad. J. Math.**18**(1966), 566–578. MR**0194471****[3]**Bruce A. Barnes,*Banach algebras which are ideals in a Banach algebra*, Pacific J. Math.**38**(1971), 1–7; correction, ibid. 39 (1971), 828. MR**0310640****[4]**Paul Civin and Bertram Yood,*The second conjugate space of a Banach algebra as an algebra*, Pacific J. Math.**11**(1961), 847–870. MR**0143056****[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]**B. J. Tomiuk and Pak-ken Wong,*The Arens product and duality in 𝐵*-algebras*, Proc. Amer. Math. Soc.**25**(1970), 529–535. MR**0259620**, 10.1090/S0002-9939-1970-0259620-0**[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**[8]**Pak Ken Wong,*Modular annihilator 𝐴*-algebras*, Pacific J. Math.**37**(1971), 825–834. MR**0305083**

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DOI:
https://doi.org/10.1090/S0002-9939-1973-0306912-5

Keywords:
Arens product,
modular annihilator,
annihilator algebra,
carrier space

Article copyright:
© Copyright 1973
American Mathematical Society