On the Arens product and annihilator algebras
HTML articles powered by AMS MathViewer
- by Pak-ken Wong PDF
- Proc. Amer. Math. Soc. 30 (1971), 79-83 Request permission
Abstract:
The purpose of this paper is to generalize two results in a recent paper by B. J. Tomiuk and the author. Let A be a ${B^ \ast }$-algebra and $M(A)$ the algebra of double centralizers of A. We show that A is a dual algebra if and only if $M(A)$ coincides with ${A^{ \ast \ast }}$. We also obtain that if A is an annihilator ${A^\ast }$-algebra, then ${\pi _A}(A)$ is a two-sided ideal of ${A^{ \ast \ast }}$.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
- Robert C. Busby, Double centralizers and extensions of $C^{\ast }$-algebras, Trans. Amer. Math. Soc. 132 (1968), 79–99. MR 225175, DOI 10.1090/S0002-9947-1968-0225175-5
- 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
- Paul Civin, Ideals in the second conjugate algebra of a group algebra, Math. Scand. 11 (1962), 161–174. MR 155200, DOI 10.7146/math.scand.a-10663
- Jacques Dixmier, Les $C^{\ast }$-algèbres et leurs représentations, Cahiers Scientifiques, Fasc. XXIX, Gauthier-Villars & Cie, Éditeur-Imprimeur, Paris, 1964 (French). MR 0171173
- L. Terrell Gardner, On isomorphisms of $C^{\ast }$-algebras, Amer. J. Math. 87 (1965), 384–396. MR 179637, DOI 10.2307/2373010
- George W. Mackey, Isomorphisms of normed linear spaces, Ann. of Math. (2) 43 (1942), 244–260. MR 6604, DOI 10.2307/1968868
- 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
- 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, The Arens product and duality in $B^{\ast }$-algebras. II, Proc. Amer. Math. Soc. 27 (1971), 535–538. MR 275176, DOI 10.1090/S0002-9939-1971-0275176-1
Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 30 (1971), 79-83
- MSC: Primary 46.50
- DOI: https://doi.org/10.1090/S0002-9939-1971-0281005-2
- MathSciNet review: 0281005