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.

In this paper we introduce and study the notions of isotropic mapping and essential kernel. In addition some theorems on the Borel graph and Baire mapping for polynomial operators are proved. It is shown that a polynomial functional from an infinite dimensional complex linear space into the field of complex numbers vanishes on some infinite dimensional affine subspace.

We formulate a general theory of positions for subspaces of a Banach space: we define equivalent and isomorphic positions, study the automorphy index a(Y, X) that measures how many non-equivalent positions Y admits in X, and obtain estimates of a(Y, X) for X a classical Banach space such as p , L p , L 1 , C(ω ω ) or C[0, 1]. Then, we study different aspects of the automorphic space problem posed by Lindenstrauss and Rosenthal; namely, does there exist a separable automorphic space different from c 0 or 2 ? Recall that a Banach space X is said to be automorphic if every subspace Y admits only one position in X; i.e., a(Y, X) = 1 for every subspace Y of X. We study the notion of extensible space and uniformly finitely extensible space (UFO), which are relevant since every automorphic space is extensible and every extensible space is UFO. We obtain a dichotomy theorem: Every UFO must be either an L ∞ -space or a weak type 2 near-Hilbert space with the Maurey projection property. We show that a Banach space all of whose subspaces are UFO (called hereditarily UFO spaces) must be asymptotically Hilbertian; while a Banach space for which both X and X * are UFO must be weak Hilbert. We then refine the dichotomy theorem for Banach spaces with some additional structure. In particular, we show that an UFO with unconditional basis must be either c 0 or a superreflexive weak type 2 space; that a hereditarily UFO Köthe function space must be Hilbert; and that a rearrangement invariant space UFO must be either L ∞ or a superreflexive type 2 Banach lattice.

Abstract. We construct non-separable subspaces in the kernel of every quadratic functional on some classes of complex and real Banach spaces.1. Introduction. Investigation of quadratic functionals is an old story [12], [6], [7]. According to [11], for any polynomial functional p with p(0) = 0 defined on an infinite-dimensional complex linear space X there is an infinite-dimensional subspace X 0 in the kernel ker(p) = p −1 (0) of p. Quantitative finite-dimensional versions of this fact (estimations of dim X 0 depending on dim X and the degree of the polynomial) are contained inThe paper [2] started the consideration of subspaces in kernels of polynomials on non-separable spaces. In particular, the authors of [2] proved that if a real Banach space X admits no positive quadratic continuous functional, then every quadratic continuous functional on X vanishes on some infinite-dimensional subspace. They pose the problem of whether in this statement one can replace "infinite-dimensional" by "non-separable" (see also [1, Question 4.8]). Our note continues the investigations of [2]. In particular, we shall construct a non-separable subspace in the kernel of every quadratic functional on a complex Banach space having weak * non-separable dual and on a real Banach space which has controlled separable projection property and admits no positive quadratic continuous functional. On the other hand, we construct a CH-example of a quadratic functional on the normed space l f 1 (ω 1 ) whose kernel contains no non-separable linear subspace.We use the standard notation; in particular dens X stands for the density of a Banach space X, F ⊥ = {x ∈ X : ∀f ∈ F f (x) = 0} is the annihilator of a subspace F ⊂ X * in X, S(X) is the unit sphere of X, and [M ] denotes the closed linear span of a subset M ⊂ X. We shall identify cardinals with initial ordinals and will denote by α the cardinality of an ordinal α. Elements x α ∈ X form a transfinite basic sequence if there is a constant c > 0 such