Analysis and Probability seminar

Abstract: We review a series of results showing that certain concrete subsets of Banach spaces and algebras, when endowed with the metric induced by the norm, form metric spaces that uniquely determine the underlying Banach space. In particular, these metric structures allow one to recover not only the Banach space itself but, in some cases, even its algebraic structure.

Examples of such metric invariants include the unit sphere of a Banach space, the set of invertible elements in certain unital Banach algebras, the unitary group of a von Neumann algebra, and the positive part of the unit sphere in spaces of operators on a complex Hilbert space, in von Neumann algebras, or in JBW∗-algebras, among other natural instances.