The crossed product von Neumann algebras associated with \(\mathrm{SL}_2(\mathbb R)\) (Q532963)

The group \(\text{SL}_2(\mathbb R)\) of real \(2\times 2\) matrices of unit determinant acts on the upper half plane \(\mathbb H\) as fractional linear transformations. Consider the measure \(d\mu_r= dx\,dy\,y^{2-r}\) on \(\mathbb H\) and let \({\mathcal H}^r_a\) be the Hilbert space of analytic functions in \(L^2(\mathbb H,\mu_r)\). When \(r>1\) is an integer, an irreducible unitary representation \(\pi_r\) of \(\text{SL}_2(\mathbb R)\) on \({\mathcal H}^r_a\) is defined, which, in turn, gives a natural action \(\alpha\) of \(\text{SL}_2(\mathbb R)\) on the maximal abelian subalgebra \({\mathcal A}\) of multiplication operators \(M_f\), \(f\in L^\infty(\mathbb H,\mu_r),\) on \(L^2 (\mathbb H,\mu_r)\). This paper studies the structure of the crossed product von Neumann algebra \({\mathcal R}({\mathcal A}, \alpha).\) As a consequence of the main result, it is shown that this algebra is of type I.