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

by P.G.Dixon

*Proc. Amer. Math. Soc.,* **104** (1988), 106-110.

1991 Mathematics Subject Classification: primary 46A30; secondary 47D35

## Abstract

Cohen [**2**] gave an example of a surjective bilinear mapping between Banach spaces
which was not open, and Horowitz [**3**] gave a much simpler example. We build on Horowitz's
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 (**A**1) but not
(**A**1(*r*)) can be extended to achieve property (**A**1/*n*) without
(**A**1/*n*(*r*)).

### References

- H. Bercovici, `Note on property (
**A**1)', *Linear Algebra Appl.*, **91**
(1987), 213-216.
- P. J. Cohen, `A counterexample to the closed graph theorem for bilinear maps',
*J. Functional Analysis,* **16** (1974), 235-239.
- C. Horowitz, `An elementary counterexample to the open mapping principle for bilinear maps',
*Proc. Amer. Math. Soc.,* **53** (1975), 293-294.

## Availability

Reprints of the paper available on request: e-mail P.Dixon with extension
@sheffield.ac.uk.

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