TITLE: Applications of functional analysis beyond analysis(towards algebra, geometry, number theory...)
QUESTION [10 upvotes]: So far, We have seen the applications of functional analysis in PDE, probability and many areas in applied mathematics. On the other hand, methods of algebraic topology are introduced to functional analysis via operator K-theory.
From the mathematical research point of view, (as a crazy fan of pure math) I wonder if functional analysis(theory of Banach and Hilbert spaces and operator algebras), as a well-built theory in mathematics, has any applications to the areas in pure math like algebraic topology, algebraic geometry and number theory, solving problems for which purely algebraic or geometric methods seem to be powerless.

REPLY [5 votes]: Recently a new application of functional analysis in geometry appeared, the study of envelopes of topological algebras. It allows to look at "big" geometric disciplines -- complex geometry, differential geometry, topology -- from the point of view of category theory, so that these disciplines become "purely categorical constructions". This can be considered as a developement of Klein's Erlangen program.
According to this view, different geometric disciplines are just pictures that appear in the imagination of an outlooker after applying different "observation tools" for studying a given category of topological algebras. Formally, this "projection of functional analysis to geometry" is established by a categorical  construction, called envelope (and there are many different envelopes that give different geometries as disciplines).      
This activity allows to build different generalizations of Pontryagin duality to classes of non-commutative groups (including some quantum groups).