  We consider an action of the circle group, T on a von Neumann algebra, M.
Similarly to the case when the algebra of essentially bounded functions on T is
acted upon by translations, we define the generalized Hardy subspace of H,where
H is the Hilbert space of a standard representation of M and the subalgebra of
analytic elements of M with respect to the action. We prove that the subalgebra
of analytic elements is a reflexive algebra of operators if the Arveson
spectrum is finite or, if the spectrum is infinite, the spectral subspace
corresponding to the least positive element contains an unitary operator. We
also prove that the analytic algebra is reflexive if M is an abelian
W*-algebra. Examples include the algebra of analytic Toeplitz operators,
crossed products, reduced semicrossed products and some reflexive nest
subalgebras of von Neumann algebras.
