Quantization on a one-sheeted hyperboloid (Q1109346)

Let M be a symplectic manifold. A quantization on M is a parametrized family of associative algebras \(A_ h\) whose elements belong to a vector space \(A\subset C^{\infty}(M)\) together with the representations of these algebras in a Hilbert space. Moreover the following conditions have to be satisfied: for \(a,b\in A\) \[ a*_ hb\to ab,\quad h^{-1}(a*_ hb-b*_ ha)\to i\{a,b\} \] when \(h\to 0\), where \(*_ h\) is the multiplication in \(A_ h\) and \(\{\) a,b\(\}\) is a Poisson bracket on M. In the present paper the above construction is extended to a Gårding space A of a quasi-regular representation of a group of motions of M when M is a one-sheeted hyperboloid. Moreover the author shows that the above convergence holds not only in the topology of uniform convergence on M but also in the topology of A.