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

MathSciNet review:
958052

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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]**Hari Bercovici,*Note on property (𝐴₁)*, Linear Algebra Appl.**91**(1987), 213–216. MR**888491**, 10.1016/0024-3795(87)90072-3**[2]**Hari Bercovici, Ciprian Foias, and Carl Pearcy,*Dual algebras with applications to invariant subspaces and dilation theory*, CBMS Regional Conference Series in Mathematics, vol. 56, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1985. MR**787041****[3]**B. Chevreau and J. Esterle,*Pettis’ lemma and topological properties of dual algebras*, Michigan Math. J.**34**(1987), no. 1, 143–146. MR**873028**, 10.1307/mmj/1029003491**[4]**Paul J. Cohen,*A counterexample to the closed graph theorem for bilinear maps*, J. Functional Analysis**16**(1974), 235–240. MR**0343043****[5]**H. G. Dales,*Automatic continuity theory*(in preparation).**[6]**P. G. Dixon,*Nonseparable Banach algebras whose squares are pathological*, J. Functional Analysis**26**(1977), no. 2, 190–200. MR**0454639****[7]**P. G. Dixon,*Automatic continuity of positive functionals on topological involution algebras*, Bull. Austral. Math. Soc.**23**(1981), no. 2, 265–281. MR**617069**, 10.1017/S0004972700007127**[8]**Charles Horowitz,*An elementary counterexample to the open mapping principle for bilinear maps*, Proc. Amer. Math. Soc.**53**(1975), no. 2, 293–294. MR**0419813**, 10.1090/S0002-9939-1975-0419813-0**[9]**Richard J. Loy,*Multilinear mappings and Banach algebras*, J. London Math. Soc. (2)**14**(1976), no. 3, 423–429. MR**0454641**

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1988-0958052-0

Keywords:
Open mapping theorem,
bilinear map,
dual algebra

Article copyright:
© Copyright 1988
American Mathematical Society