  In earlier work a crossed product of a Banach algebra was constructed from a
Banach algebra dynamical system $(A,G,\alpha)$ and a class $\mathcal{R}$ of
continuous covariant representations, and its representations were determined.
In this paper the theory is developed further. We consider the dependence of
the crossed product on the class $\mathcal{R}$ and its essential uniqueness.
Next we study generalized Beurling algebras: weighted Bochner spaces of
$A$-valued functions on $G$ with a continuous multiplication. Though not Banach
algebras in general, they are isomorphic to a crossed product of a Banach
algebra, and the earlier work therefore predicts the structure of their
representations. Classical results for the usual Beurling (Banach) algebras of
scalar valued functions are then retrieved as special cases. We also show how,
e.g., an anti-covariant pair of anti-representations of $A$ and $G$ can be
viewed as a covariant pair for a related Banach algebra dynamical system, so
that the earlier work becomes applicable to classes of such other pairs. After
including material on the representations of the projective tensor product of
Banach algebras, we combine this idea with the results already obtained and
describe the two-sided modules over the generalized Beurling algebras, where
again specializing to the scalars gives a classical result.
