On the Arens product and commutative Banach algebras
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- by Pak Ken Wong
- Proc. Amer. Math. Soc. 37 (1973), 111-113
- DOI: https://doi.org/10.1090/S0002-9939-1973-0306912-5
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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 ${X_A}$ and $\pi$ the canonical embedding of A into its second conjugate space ${A^{ \ast \ast }}$ (with the Arens product). We show that if A is a semisimple annihilator algebra, then $\pi (A)$ is a two-sided ideal of ${A^{ \ast \ast }}$. We also obtain that if A is a dense two-sided ideal of ${C_0}({X_A})$, then $\pi (A)$ is a two-sided ideal of ${A^{ \ast \ast }}$ if and only if A is a modular annihilator algebra.References
- Richard Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2 (1951), 839–848. MR 45941, DOI 10.1090/S0002-9939-1951-0045941-1
- Bruce A. Barnes, Modular annihilator algebras, Canadian J. Math. 18 (1966), 566–578. MR 194471, DOI 10.4153/CJM-1966-055-6
- 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 310640
- Paul Civin and Bertram Yood, The second conjugate space of a Banach algebra as an algebra, Pacific J. Math. 11 (1961), 847–870. MR 143056
- 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 115101
- B. J. Tomiuk and Pak-ken Wong, The Arens product and duality in $B^{\ast }$-algebras, Proc. Amer. Math. Soc. 25 (1970), 529–535. MR 259620, DOI 10.1090/S0002-9939-1970-0259620-0
- Pak-ken Wong, On the Arens product and annihilator algebras, Proc. Amer. Math. Soc. 30 (1971), 79–83. MR 281005, DOI 10.1090/S0002-9939-1971-0281005-2
- Pak Ken Wong, Modular annihilator $A^{\ast }$-algebras, Pacific J. Math. 37 (1971), 825–834. MR 305083
Bibliographic Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 37 (1973), 111-113
- MSC: Primary 46H99; Secondary 46J05
- DOI: https://doi.org/10.1090/S0002-9939-1973-0306912-5
- MathSciNet review: 0306912