Seminars

2pm Tuesday 24 January 2012, Room 345, Building 28, Clayton campus
Solution to the problem of Hölder classification of infinite-dimensional spheres
Dr Sergey Ajiev, University of New South Wales

The uniform classification of infinite-dimensional spheres, developed in relation with the solution of the distortion problem is more balanced than the continuous, isometric, Lipschitz or uniform classifications of infinite-dimensional Banach spaces. It allows to transfer a group structure, group actions and other metric-related constructions from one space onto another. We show that the uniformly continuous homeomorphisms can be "upgraded" to the Hölder ones in the classical setting and establish the explicit and, occasionally, sharp exponents of the Hölder regularity for pairs of concrete spaces, including various Besov, Lizorkin-Triebel, Sobolev, sequence, Schatten-von Neumann and other Banach spaces (including lattices and more general non-commutative spaces). Our function spaces are allowed to be anisotropic and can be defined in terms of differences, local approximation by polynomials, the coefficients of wavelet expansions, Littlewood-Paley decompositions or a functional calculus. Not every equivalent norm is geometrically friendly.

These results appear to have close ties with the presence of a remarkable phenomenon from the infinite-dimensional approximation theory discovered by Tsar'kov for the uniform mappings between pairs of Lebesgue spaces and the problems of extension and interpolation of the mappings between the pairs of the spaces under consideration. Among the applications of the main results are multiple examples of spaces that do not allow any C*-algebra structure but can be endowed with a homogeneous Hölder group structure.