Automorphisms and nonselfadjoint crossed products

Abstract

We are interested in the invariant subspace structure of the nonselfadjoint crossed product determined by a finite von Neumann algebra M and a trace preserving automorphism α. In this paper we investigate the form of two-sided invariant subspaces for the case that a is ergodic on the center of M. 1. Introduction* In this paper, we consider the typical finite maximal subdiagonal algebras which are called nonselfadjoint crossed products. These algebras are constructed as certain subalgebras of crossed products of finite von Neumann algebras by trace preserving automorphisms. Recently, McAsey, Muhly and the author studied the invariant subspace structure and the maximality of these algebras (cf. [4], [5], [6], [7]). Let M be a von Neumann algebra with a faithful normal tracial state τ and let a be a *-automorphism of M such that τ°α = r. We regard M as acting on the noncommutative Lebesgue space L2(M, τ) (cf. [10]) and consider the Hubert space Π - {/: Z >L\M, τ)\Σ\\f(n)\\l } which may be identified with l\Z) (x) L\M, τ). Let 2 (resp. 31) be the left (resp. right) crossed product of M and a, and let £+ (resp. 9t+) be the left (resp. right) nonselfadjoint crossed product of S (resp. 3t) (cf. §2). In [6], we showed that the following three conditions are equivalent; (i) M is a factor; (ii) a conditioned form of the Beurling-Lax-Halmos theorem is valid; and (iii) 2+ is a maximal (M). We now suppose that a is ergodic on S(Λf). Then every two-sided invariant subspace of U which is not left-reducing is left-pure, left-full, right-pure and right-full (Theorems 3.2 and 4.5). Further, if S is a factor, then every proper two-sided invariant subspace of U is of