The Group of Quasisymmetric Homeomorphisms of the Circle and Quantization of the Universal Teichmüller Space

In the first part of the paper we describe the complex geometry of the universal Teichmüller space $\mathcal T$, which may be realized as an open subset in the complex Banach space of holomorphic quadratic differentials in the unit disc. The quotient $\mathcal S$ of the diffeomorphism group of the circle modulo Möbius transformations may be treated as a smooth part of $\mathcal T$. In the second part we consider the quantization of universal Teichmüller space $\mathcal T$. We explain first how to quantize the smooth part $\mathcal S$ by embedding it into a Hilbert–Schmidt Siegel disc. This quantization method, however, does not apply to the whole universal Teichmüller space $\mathcal T$, for its quantization we use an approach, due to Connes.