Question: What are the applications of operator algebras to other areas?
More precisely, I would like to know the results in mathematical areas outside of operator algebras which were proved by using operator algebras' techniques, or which are corollaries of operator algebras' theorems.
I ask this question for seeing how operator algebras are connected to the other mathematical areas and for better understanding what concrete role it is currently playing in the mathematical world.
Best Answer
I'm a little puzzled by the tone of the original question. My personal view is that operator algebras are intrinsically interesting, and if there are good applications to other fields, so much the better ... I think this is a pretty common attitude, probably among people in most areas of pure math.
Anyway, I am not the most qualified to describe some of these applications, but here are a few of the main ones.
(1) Connes' index theorem for foliated manifolds. For instance see here and here. Connes 1982 Fields medal was awarded in part for his work on foliations.
(2) Jones' work connecting von Neumann algebras and geometric topology, which gave rise to a new knot invariant. See here for a nice overview. Jones was awarded the Fields medal in 1990 in part for this work.
(3) Mathematical physics. Many connections, some more established and some more conjectural. The KMS theory is surely one of the standouts.
(4) An early motivation was the theory of group representations, e.g., J. Dixmier, Anneaux d'operateurs et representations des groups, Seminaire Bourbaki, Vol. 1 (1995), 331-336.
(5) As mkreisel mentions, there are applications to the Novikov conjecture. See G. G. Kasparov, Equivariant KK-theory and the Novikov conjecture, Invent. Math. 91 (1977), 147-201.
(6) The Kadison-Singer problem originally arose as a problem in operator algebras. It now has connections to many other areas (harmonic analysis, Banach space theory, signal analysis, ...).
I'm sure I am forgetting some important ones.