On automorphic Banach spaces (Q1001436)

A Banach space \(X\) is called automorphic if, for every subspace \(E\) of \(X\), every into isomorphism \(T:E\to X\) extends to an automorphism of \(X\) (provided, of course, that \(X/E\) and \(X/T(E)\) have the same density character). Until now, only Hilbert spaces and \(c_0\) were known to be automorphic. This tour de force shows that \(c_0(\Gamma)\) is also automorphic when \(\Gamma\) is uncountable. A number of Banach spaces are proved not to be automorphic, including Gurariĭ's space, and at least one hereditarily indecomposable space.