Adrian Duma
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 Posted: Sat Jul 12, 2008 11:03 pm Post subject: Functoriality and the Edgar ordering |
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Hello, NG !
Twenty-five years ago, G.A.Edgar introduced a binary transitive and reflexive relation "<" on the class of Banach
spaces, namely, X < Y iff any a in X** s.t. T**(a) belongs to Y for all T in L(X,Y), must be in X. One say that
X ~ Y iff X < Y, and Y < X, simultaneously.
Now, let Ban_1 be the category of Banach spaces and nonexpansive linear operators.
My (somewhat naive) question is: is it true that any covariant endofunctor F: Ban_1 --> Ban_1 that is (on the objects)
both preserving and reversing the Edgar ordering (i.e., X < Y iff F(X) < F(Y)) admits at least a fixed point,
i.e., there is some Banach space E s.t. F(E) ~ E ?
Best regards,
Ady. |
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