An Approximate Version of the Jordan von Neumann Theorem for Finite Dimensional Real Normed Spaces

It is known that any normed vector space which satisfies the parallelogram law is actually an inner product space. For finite dimensional normed vector spaces over R, we formulate an approximate version of this theorem: if a space approximately satisfies the parallelogram law, then it has a near isometry with Euclidean space. In other words, a small von Neumann Jordan constant E + 1 for X yields a small Banach-Mazur distance with R^n, d(X, R^n) < 1 + B_n E + O(E^2). Finally, we examine how this estimate worsens as the dimension, n, of X increases, with the conclusion that B_n grows quadratically with n.Comment: Version 2 adds contact information for the author and actually states the correct Jordan-von Neumann theorem (oops!

Iterated Joins of Compact Groups

(Joint work with Alexandru Chirvasitu.) The Borsuk-Ulam theorem in algebraic topology indicates restrictions for equivariant maps between spheres; in particular, there is no odd map from a sphere to another sphere of lower dimension. This idea may be generalized greatly in both the topological and operator algebraic settings for actions of compact (quantum) groups, leading to the the noncommutative Borsuk-Ulam conjectures of Baum, Dabrowski, and Hajac. I will present our recent progress (both positive and negative) toward resolving these conjectures using properties of iterated compact group joins