ALMOST AUTOMORPHIC FUNCTIONS WITH VALUES IN p-FRÉCHET SPACES (original) (raw)
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Israel Journal of Mathematics, 2009
A Banach space X will be called extensible if every operator E → X from a subspace E ⊂ X can be extended to an operator X → X. Denote by dens X. The smallest cardinal of a subset of X whose linear span is dense in X, the space X will be called automorphic when for every subspace E ⊂ X every into isomorphism T : E → X for which dens X/E = dens X/T E can be extended to an automorphism X → X. Lindenstrauss and Rosenthal proved that c 0 is automorphic and conjectured that c 0 and 2 are the only separable automorphic spaces. Moreover, they ask about the extensible or automorphic character of c 0 (Γ), for Γ uncountable. That c 0 (Γ) is extensible was proved by Johnson and Zippin, and we prove here that it is automorphic and that, moreover, every automorphic space is extensible while the converse fails. We then study the local structure of extensible spaces, showing in particular that an infinite dimensional extensible space cannot contain uniformly complemented copies of n p , 1 ≤ p < ∞, p = 2. We derive that infinite dimensional spaces such as Lp(µ), p = 2, C(K) spaces not isomorphic to c 0 for K metric compact, subspaces of c 0 which are not isomorphic to c 0 , the Gurarij space, Tsirelson spaces or the Argyros-Deliyanni HI space cannot be automorphic.
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International Journal of Mathematics and Mathematical Sciences, 2000
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Nonlinear Analysis: Theory, Methods & Applications, 2009
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