The ideal structure of certain nonselfadjoint operator algebras

Abstract:

Let $(X, \phi )$ be a locally compact dynamical system, and ${{\mathbf {Z}}^ + }{ \times _\phi } {C_0}(X)$ the norm-closed subalgebra of the crossed product $Z{ \times _\phi }{C_0}(X)$ generated by the nonnegative powers of $\phi$ in case $\phi$ is a homeomorphism. If $\phi$ is just a continuous map, ${{\mathbf {Z}}^ + }{ \times _\phi }{C_0}$ can still be defined by a crossed product type construction. The ideal structure of these algebras is determined in case $\phi$ acts freely. A class of strictly transitive Banach modules is described, indicating that for the nonselfadjoint operator algebras considered here, not all irreducible representations are on Hilbert space. Finally in a special case, the family of all invariant maximal right ideals is given.