Quantization of the Sobolev space of half-differentiable functions

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A quantization of the Sobolev space of half- differentiable functions on the circle, which is closely connected with string theory, is constructed. The group of quasisymmetric circle homeomorphisms acts on by reparametrizations, but this action is not smooth. Nevertheless, a quantum infinitesimal action of on can be defined, which enables one to construct a quantum algebra of observables which is associated with the system . Bibliography: 7 titles.