THE CROSSED PRODUCT VON NEUMANN ALGEBRAS ASSOCIATED WITH SL2(R)

Let A be the abelian von Neumann subalgebra {M-f : f is an element of L-infinity(H,mu(r))} of B(L-2(H, mu(r))), where H is the upper half plane and the measure d mu(r) = dxdy/y(2-r). For any integers r > 1, the linear fractional action of SL2(R) on H induces a continuous action a of SL2(R) on A. It is shown that the crossed product R(A, alpha) of A under the action a of SL2(R) is *-isomorphic to B(L-2(P,2dxdy/y(3-2r)))(circle times) over bar )L-K where SL2(R) = PK is the Iwasawa decomposition of SL2(R) Thus R(A, alpha) is of type I