Nonselfadjoint crossed products (invariant subspaces and maximality)

Abstract:

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