I know that lots of effort is being put into quantization of geometry(NCG).This effort of course comes with the idea of operator algebra being a powerful machinery. Has any effort been given in the other direction.I mean to make some analogue of geometrical structures in Operator Algebras. This question may be a bit absurd and arises from one of my "RANDOM" thoughts. Precisely I mean "what are the differntial geometry ideas that are applicable in operator algebras"
The theory of Banach manifolds has many important applications to the theory of operator algebras, and more generally to the theory of Banach spaces. For example, one can use it to very effectively apply the theory of Jordan structures to operator algebraic questions. The best introduction to this topic that I know of is the book of Cho-Ho Chu: Jordan Structures in Geometry and Analysis