The Sobolev space of half-differentiable functions and quasisymmetric homeomorphisms

In this paper, we give an interpretation of some classical objects of function theory in terms of Banach algebras of linear operators in a Hilbert space. We are especially interested in quasisymmetric homeomorphisms of the circle. They are boundary values of quasiconformal homeomorphisms of the disk and form a group QS(S-1) with respect to composition. This group acts on the Sobolev space H-0(1/2) (S-1, R) of half-differentiable functions on the circle by reparameterization. We give an interpretation of the group QS(S-1) and the space H-0(1/2) (S-1, R) in terms of noncommutative geometry.