  Given a C*-dynamical system (A,G,\alpha), we say that A is a weakly proper
(X\rtimes G)-algebra if there exists a proper G-space X together with a
nondegenerate G-equivariant *-homomorphism \phi:C_0(X)->M(A). Weakly proper
G-algebras form a large subclass of the class of proper G-algebras in the sense
of Rieffel. In this paper we show that weakly proper (X\rtimes G)-algebras
allow the construction of full fixed-point algebras A^G corresponding to the
full crossed product A\rtimes_{\alpha}G, thus solving, in this setting, a
problem stated by Rieffel in his 1988's original article on proper actions. As
an application we obtain a general Landstad duality result for arbitrary
coactions together with a new and functorial construction of maximalizations of
coactions.
  The same methods also allow the construction of exotic generalized
fixed-point algebras associated to crossed-product norms lying between the
reduced and universal ones. Using these, we give complete answers to some
questions on duality theory for exotic crossed products recently raised by
Kaliszewski, Landstad and Quigg.
