Let
L
\mathcal {L}
be the von Neumann algebra crossed product determined by a finite von Neumann algebra M and a trace preserving automorphism. In this paper we investigate the invariant subspace structure of the subalgebra
L
+
{\mathcal {L}_ + }
of
L
\mathcal {L}
consisting of those operators whose spectrum with respect to the dual automorphism group on
L
\mathcal {L}
is nonnegative, and we determine conditions under which
L
+
{\mathcal {L}_ + }
is maximal among the
σ
\sigma
-weakly closed subalgebras of
L
\mathcal {L}
. Our main result asserts that the following statements are equivalent: (1) M is a factor; (2)
L
+
{\mathcal {L}_ + }
is a maximal
σ
\sigma
-weakly closed subalgebra of
L
\mathcal {L}
; and (3) a version of the Beurling, Lax, Halmos theorem is valid for
L
+
{\mathcal {L}_ + }
. In addition, we prove that if
A
\mathfrak {A}
is a subdiagonal algebra in a von Neumann algebra
B
\mathcal {B}
and if a form of the Beurling, Lax, Halmos theorem holds for
A
\mathfrak {A}
, then
B
\mathcal {B}
is isomorphic to a crossed product of the form
L
\mathcal {L}
and
A
\mathfrak {A}
is isomorphic to
L
+
{\mathcal {L}_ + }
.