This book develops tools to handle C*-algebras arising as completions of convolution algebras
of sections of line bundles over possibly non-Hausdorff groupoids. A fundamental result of
Gelfand describes commutative C*-algebras as continuous functions on locally compact Hausdorff
spaces. Kumjian and later Renault showed that Gelfand's result can be extended to include
non-commutative C*-algebras containing a commutative C*-algebra. In their setting the
C*-algebras in question may be described as the completion of convolution algebras of functions
on twisted Hausdorff groupoids with respect to a certain norm. However there are many natural
settings in which the Kumjian-Renault theory does not apply in part because the groupoids
which arise are not Hausdorff. In fact non-Hausdorff groupoids have been a source of
surprising counterexamples and technical difficulties for decades. Including numerous
illustrative examples this book extends the Kumjian-Renault theory to a much broader class of
C*-algebras. This work will be of interest to researchers and graduate students in the area of
groupoid C*-algebras the interface between dynamical systems and C*-algebras and related
fields.
