A strong boundedness result for separable Rosenthal compacta

It is proved that the class of separable Rosenthal compacta on the Cantor set having a uniformly bounded dense sequence of continuous functions, is strongly bounded.Comment: 13 pages, no figure

On classes of Banach spaces admitting "small" universal spaces

We characterize those classes \ccc of separable Banach spaces admitting a separable universal space $Y$ (that is, a space $Y$ containing, up to isomorphism, all members of \ccc) which is not universal for all separable Banach spaces. The characterization is a byproduct of the fact, proved in the paper, that the class $\mathrm{NU}$ of non-universal separable Banach spaces is strongly bounded. This settles in the affirmative the main conjecture form \cite{AD}. Our approach is based, among others, on a construction of \llll_\infty-spaces, due to J. Bourgain and G. Pisier. As a consequence we show that there exists a family $\{Y_\xi:\xi<\omega_1\}$ of separable, non-universal, \llll_\infty-spaces which uniformly exhausts all separable Banach spaces. A number of other natural classes of separable Banach spaces are shown to be strongly bounded as well.Comment: 26 pages, no figures. Transactions of AMS (to appear

Quotients of Banach spaces and surjectively universal spaces

We characterize those classes $\mathcal{C}$ of separable Banach spaces for which there exists a separable Banach space $Y$ not containing $\ell_1$ and such that every space in the class $\mathcal{C}$ is a quotient of $Y$.Comment: 23 pages, no figure

The Steinhaus property and Haar-null sets

It is shown that if $G$ is an uncountable Polish group and $A\subseteq G$ is a universally measurable set such that $A^{-1}A$ is meager, then the set $T_l(A)=\{\mu\in P(G): \mu(gA)=0 \text{for all} g\in G\}$ is co-meager. In particular, if $A$ is analytic and not left Haar-null, then $1\in\mathrm{Int}(A^{-1}AA^{-1}A)$.Comment: 9 pages, no figure

Some strongly bounded classes of Banach spaces

We show that the classes of separable reflexive Banach spaces and of spaces with separable dual are strongly bounded. This gives a new proof of a recent result of E. Odell and Th. Schlumprecht, asserting that there exists a separable reflexive Banach space containing isomorphic copies of every separable uniformly convex Banach spaces.Comment: 10 page

On pairs of definable orthogonal families

We introduce the notion of an M-family of infinite subsets of \nn which is implicitly contained in the work of A. R. D. Mathias. We study the structure of a pair of orthogonal hereditary families \aaa and \bbb, where \aaa is analytic and \bbb is $C$-measurable and an M-family.Comment: 21 pages, no figures. Illinois Journal of Mathematics (to appear

Uniformity norms, their weaker versions, and applications

We show that, under some mild hypotheses, the Gowers uniformity norms (both in the additive and in the hypergraph setting) are essentially equivalent to certain weaker norms which are easier to understand. We present two applications of this equivalence: a variant of the Koopman--von Neumann decomposition, and a proof of the relative inverse theorem for the Gowers $U^s[N]$-norm using a norm-type pseudorandomness condition