  We study the complex powers $A^{z}$ of an elliptic, strictly positive
pseudodifferential operator $A$ using an axiomatic method that combines the
approaches of Guillemin and Seeley. In particular, we introduce a class of
algebras, ``extended Weyl algebras,'' whose definition was inspired by
Guillemin's paper on the subject. An extended Weyl algebra can be thought of as
an algebra of ``abstract pseudodifferential operators.'' Many algebras of
pseudodifferential operators are extended Weyl algebras. Several results
typical for algebras of pseudodifferential operators (asymptotic completeness,
construction of Sobolev spaces, boundedness between apropriate Sobolev spaces,
>...) generalize to extended Weyl algebras. Most important, our results may be
used to obtain precise estimates at infinity for $A^{z}$, when $A > 0 $ is
elliptic and defined on a non-compact manifold, provided that a suitable ideal
of regularizing operators is specified (a submultiplicative $\Psi^*$--algebra).
We shall use these results in a forthcoming paper to study pseudodifferential
operators and Sobolev spaces on manifolds with a Lie structure at infinity (a
certain class of non-compact manifolds).
