Homogeneous domains connected with factors of type \(\text{II}_ \infty\) and representations of infinite-dimensional groups (Q1316824)

It is well-known that the group \(\text{Diff}_ 0 (S^ 1)\) of diffeomorphisms of the circle \(S^ 1\) can be embedded into the infinite dimensional symplectic group. This embedding leads to a construction of projective representations of the group \(\text{Diff}_ 0 (S^ 1)\). The aim of this paper is to modify the corresponding theory replacing the symplectic group by the analogue \(G\) of this group related to the factor of type \(\text{II}_ \infty\) and the group of diffeomorphisms \(\text{Diff}_ 0 (S^ 1)\) by the group \(D_ \theta\) of diffeomorphisms of a two-dimensional torus \(T^ 2\) preserving an irrational winding on it. The group \(G\) acts in a certain domain \(D\) of operators belonging to the factor of type \(\text{II}_ \infty\). This domain is an analogue of the corresponding classical domain. The author constructs projective representations of the group \(G\). The group \(D_ \theta\) can be embedded into \(G\). An explicit mapping which fulfils this embedding is given. By making use of the representations of the group \(G\) the author constructs representations of \(D_ \theta\).