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On the Arens product and annihilator algebras


Author: Pak-ken Wong
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
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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 [Enhancements On Off] (What's this?)

  • [1] R. Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2 (1951), 839-848. MR 13, 659. MR 0045941 (13:659f)
  • [2] B. A. Barnes, Modular annihilator algebra, Canad. J. Math. 18 (1966), 566-578. MR 33 #2681. MR 0194471 (33:2681)
  • [3] R. C. Busby, Double centralizers and extensions of $ {C^ \ast }$-algebras, Trans. Amer. Math. Soc. 132 (1968), 79-99. MR 37 #770. MR 0225175 (37:770)
  • [4] P. Civin and B. Yood, The second conjugate space of a Banach algebra as an algebra, Pacific J. Math. 11 (1961), 847-870. MR 26 #622. MR 0143056 (26:622)
  • [5] P. Civin, Ideals in the second conjugate algebra of a group algebra, Math. Scand. 11 (1962), 161-174. MR 27 #5139. MR 0155200 (27:5139)
  • [6] J. Dixmier, Les $ {C^\ast}$-algèbres et leurs représentations, Cahiers Scientifiques, fasc. 29, Gauthier-Villars, Paris, 1964. MR 30 #1404. MR 0171173 (30:1404)
  • [7] L. T. Gardner, On isomorphisms of $ {C^\ast}$-algebras, Amer. J. Math. 87 (1965), 384-396. MR 31 #3883. MR 0179637 (31:3883)
  • [8] G. W. Mackey, Isomorphisms of normed linear spaces, Ann. of Math. (2) 43 (1942), 244-260. MR 4, 12. MR 0006604 (4:12j)
  • [9] C. E. Rickart, General theory of Banach algebras, University Series in Higher Math., Van Nostrand, Princeton, N. J., 1960. MR 22 #5903. MR 0115101 (22:5903)
  • [10] B. J. Tomiuk and P. K. Wong, The Arens product and duality in $ {B^\ast}$-algebras, Proc. Amer. Math. Soc. 25 (1970), 529-535. MR 0259620 (41:4256)
  • [11] P. K. Wong, The Arens product and duality in $ {B^ \ast }$-algebras. II, Proc. Amer. Math. Soc. 27 (1971), 535-538. MR 0275176 (43:933)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0281005-2
Keywords: Dual $ {B^ \ast }$-algebra, Arens product, double centralizer, annihilator algebra, minimal idempotent, group algebra of a compact group
Article copyright: © Copyright 1971 American Mathematical Society

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