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 elements of the range. An immediate application is that Bercovici's construction [**1**] of an operator algebra with property but not can be extended to achieve property without .

**[1]**H. Bercovici,*Note on property*, 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)**

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