On automorphic Banach spaces

Abstract

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/TE can be extended to an automorphism X → X. Lindenstrauss and Rosenthal proved that c0 is automorphic and conjectured that c0 and l2 are the only separable automorphic spaces. Moreover, they ask about the extensible or automorphic character of c0(Γ), for Γ uncountable. That c0(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 lnp, 1 ≤ p < ∞, p ≠ 2. We derive that infinite dimensional spaces such as Lp(μ), p ≠ 2, C(K) spaces not isomorphic to c0 for K metric compact, subspaces of c0 which are not isomorphic to c0, the Gurarij space, Tsirelson spaces or the Argyros-Deliyanni HI space cannot be automorphic.