Homological algebra in operator spaces with applications to harmonic analysis

Abstract

A homological algebra theory is developed in the category of operator spaces which closely matches the theory developed in general algebra and its extension to the Banach space setting. Using this category, we establish several results regarding the question of classifying which ideals in the Fourier algebra of a locally compact group are complemented. Furthermore we classify the groups for which the Fourier algebra is operator biprojective.

Additionally, the notion of operator weak amenability for completely contractive Banach algebras is introduced. We then study the potential operator weak amenability for the Fourier algebra and various sub-algebras of its second dual.