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The Sobolev space of half-differentiable functions and quasisymmetric homeomorphisms. (English) Zbl 1350.81018

Summary: 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 \({\operatorname{QS}(S^{1})}\) with respect to composition. This group acts on the Sobolev space \({H^{1/2}_{0}(S^{1},\mathbb{R})}\) of half-differentiable functions on the circle by reparameterization. We give an interpretation of the group \({\operatorname{QS}(S^{1})}\) and the space \({H^{1/2}_{0}(S^{1},\mathbb{R})}\) in terms of noncommutative geometry.

MSC:

81S10 Geometry and quantization, symplectic methods
81R60 Noncommutative geometry in quantum theory
46E39 Sobolev (and similar kinds of) spaces of functions of discrete variables
58B34 Noncommutative geometry (à la Connes)
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