Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Generalized open mapping theorems for bilinear maps, with an application to operator algebras


Author: P. G. Dixon
Journal: Proc. Amer. Math. Soc. 104 (1988), 106-110
MSC: Primary 46A30; Secondary 47A65, 47D25
DOI: https://doi.org/10.1090/S0002-9939-1988-0958052-0
MathSciNet review: 958052
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Cohen [4] gave an example of a surjective bilinear mapping between Banach spaces which was not open, and Horowitz [8] gave a much simpler example. We build on Horowitz' example to produce a similar result for bilinear mappings such that every element of the target space is a linear combination of $ n$ elements of the range. An immediate application is that Bercovici's construction [1] of an operator algebra with property $ ({\mathbb{A}_1})$ but not $ ({\mathbb{A}_1}(r))$ can be extended to achieve property $ ({\mathbb{A}_{1/n}})$ without $ ({\mathbb{A}_{1/n}}(r))$.


References [Enhancements On Off] (What's this?)

  • [1] H. Bercovici, Note on property $ ({\mathbb{A}_1})$, Linear Algebra Appl. 91 (1987), 213-216. MR 888491 (88e:47086)
  • [2] H. Bercovici, C. Foiaş and C. Pearcy, Dual algebras with applications to invariant subspaces and dilation theory, CBMS Regional Conf. Ser. in Math., no. 56, Amer. Math. Soc., Providence, R. I., 1985. MR 787041 (87g:47091)
  • [3] B. Chevreau and J. Esterle, Pettis' lemma and topological properties of dual algebras, Michigan Math. J. 34 (1987), 143-146. MR 873028 (88g:47088)
  • [4] P. J. Cohen, A counterexample to the closed graph theorem for bilinear maps, J. Funct. Anal. 16 (1974), 235-239. MR 0343043 (49:7787)
  • [5] H. G. Dales, Automatic continuity theory (in preparation).
  • [6] P. G. Dixon, Non-separable Banach algebras whose squares are pathological, J. Funct. Anal. 26 (1977), 190-200. MR 0454639 (56:12888)
  • [7] -, Automatic continuity of positive functional on topological involution algebras, Bull. Austral. Math. Soc. 23 (1981), 265-281. MR 617069 (82f:46061)
  • [8] C. Horowitz, An elementary counterexample to the open mapping principle for bilinear maps, Proc. Amer. Math. Soc. 53 (1975), 293-294. MR 0419813 (54:7831)
  • [9] R. J. Loy, Multilinear mappings and Banach algebras, J. London Math. Soc. (2) 14 (1976), 423-429. MR 0454641 (56:12890)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46A30, 47A65, 47D25

Retrieve articles in all journals with MSC: 46A30, 47A65, 47D25


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0958052-0
Keywords: Open mapping theorem, bilinear map, dual algebra
Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society