Let *T *be a compact
Hausdorff space. In 1985, Raeburn and Taylor introduced a family of continuous
trace *C**-algebras with the property that any Dixmier-Douady class in *H*^{3}(*T
*;Z) can be realized by such an algebra. A Raeburn-Taylor *C**-algebra
can be presented as a twisted groupoid *C**-algebra *C**(* G*_{U},s),
where *G*_{U} is a groupoid corresponding to an open cover *U *of
*T*, and s is a 2-cocycle of *G*_{U}. In this work we explore
the asymptotic behavior of the algebras *C**(* G*_{U},s), as
one refines the covers *U*.

Let *W*^{(0)}*,W*^{(1)}*,W*^{(2)}*,*…
be a sequence of open covers of *T *, where *W*^{(0)} = *U *and
each *W*^{(i)} is a refinement (of a particular sort) of *W*^{(i+1)}.
Denote by *G*_{n} the groupoid corresponding to the cover *W*^{(n)}.
The asymptotic behavior of the sequence *C**(* G*_{n},s_{n}),
for compatible 2-cocycles s_{n}, is encoded in the *C**-algebra of
a groupoid *G *which we introduce, and a 2-cocycle s of G.

In our main result we construct
for every *n* an isometric ***-homomorphism from *C**(* G*_{n},s_{n})
to *C**(* G*,s). As a very special case, when *T *is a point,
our construction produces all UHF *C**-algebras as algebras of the form *C**(*G*).
In this case *C**(*G*) is the direct limit of the algebras *C**(*
G*_{n}). In light of this, in a more general setting we regard our
construction as a certain generalized direct limit of the sequence {*C**(*
G*_{n},s_{n})}. We also
study the properties of the groupoid *G
*and several other groupoids related
to covers and to sequences of refinements, which we introduce.

This
research was in part joint work with Daniel Markiewicz.