The simultaneous study of a space $X$ and its observables $F(X)$ (real, complex, or operator-valued functions on $X$) is an old topic, but with quantum groups and non-commutative geometry it has been the source of much modern mathematics. The introductory paper by Connes does a really nice job at explaining this.
In the paper Connes underlines the pioneering work of I.M. Gelfand in this area. However he misses one little thing. Gelfand's work on integral geometry was also motivated by this philosophy. The idea is to consider the incidence relation as a special type of multivalued map between the two spaces and to consider how functions, forms, densities, and other functional objects correspond under the map.
