We prove that the relation E Kσ is Borel reducible to isomorphism and complemented biembeddability between subspaces of c 0 or l p with 1 ≤ p < 2. We also show that the relation E Kσ ⊗ = + is Borel reducible to isomorphism, complemented biembeddability, and Lipschitz isomorphism between subspaces of L p for 1 ≤ p < 2.

Abstract. A group G is representable in a Banach space X if G is isomorphic to the group of isometries on X in some equivalent norm. We prove that a countable group G is representable in a separable real Banach space X in several general cases, including when G {−1, 1} × H, H finite and dim X ≥ |H|, or when G contains a normal subgroup with two elements and X is of the form c 0 (Y ) or p (Y ), 1 ≤ p < +∞. This is a consequence of a result inspired by methods of S. Bellenot (1986) and stating that under rather general conditions on a separable real Banach space X and a countable bounded group G of isomorphisms on X containing −Id, there exists an equivalent norm on X for which G is equal to the group of isometries on X.We also extend methods of K. Jarosz (1988) to prove that any complex Banach space of dimension at least 2 may be renormed with an equivalent complex norm to admit only trivial real isometries, and that any complexification of a Banach space may be renormed with an equivalent complex norm to admit only trivial and conjugation real isometries. It follows that every real Banach space of dimension at least 4 and with a complex structure may be renormed to admit exactly two complex structures up to isometry, and that every real Cartesian square may be renormed to admit a unique complex structure up to isometry.

Abstract. In relation to some Banach spaces recently constructed by W. T. Gowers and B. Maurey, we introduce the notion of Schroeder-Bernstein index SBi(X) for every Banach space X. This index is related to complemented subspaces of X which contain some complemented copy of X. Then we establish the existence of a Banach space E which is not isomorphic to E n for every n ∈ N, n ≥ 2, but has a complemented subspace isomorphic to E

Let C 0 (K, X) denote the Banach space of all X-valued continuous functions defined on the locally compact Hausdorff space K which vanish at infinity, provided with the supremum norm. We prove that if X is a real Banach space and T is an isomorphismwhere J(X) is the James constant of X, then K 1 is homeomorphic to K 2 . In the complex case, we provide a similar result for reflexive spaces X. In other words, we obtain a vector-valued extension of the classical Amir-Cambern theorem (X = R or X = C) which at the same time unifies and strengthens several generalizations of the classical Banach-Stone theorem due to Cambern (1976) and (1985), BehrendsCambern (1988) andJarosz (1989). In the case where X = l p , 2 ≤ p < ∞, our results are optimal.

Abstract. We prove an extension of the classical isomorphic classification of Banach spaces of continuous functions on ordinals. As a consequence, we give complete isomorphic classifications of some Banach spaces ( , ), ≥ , of compact operators from to , the space of all continuous -valued functions defined in the interval of ordinals [1, ] and equipped with the supremum norm. In particular, under the Continuum Hypothesis, we extend a recent result of C. Samuel by classifying, up to isomorphism, the spaces ( , 0 (Γ) ), where ≤ < 1 , ≥ , Γ is a countable set, contains no complemented copy of 1 , * has the Mazur property and the density character of * * is less than or equal to ℵ 1 .