  We address the challenging issue of how CP violation is realized in higher
dimensional gauge theories without higher dimensional elementary scalar fields.
In such theories interactions are basically governed by a gauge principle and
therefore to get CP violating phases is a non-trivial task. It is demonstrated
that CP violation is achieved as the result of compactification of extra
dimensions, which is incompatible with the 4-dimensional CP transformation. As
a simple example we adopt a 6-dimensional U(1) model compactified on a
2-dimensional orbifold $T^{2}/Z_{4}$. We argue that the 4-dimensional CP
transformation is related to the complex structure of the extra space and show
how the $Z_{4}$ orbifolding leads to CP violation. We confirm by explicit
calculation of the interaction vertices that CP violating phases remain even
after the re-phasing of relevant fields. For completeness, we derive a
re-phasing invariant CP violating quantity, following a similar argument in the
Kobayashi-Maskawa model which led to the Jarlskog parameter. As an example of a
CP violating observable we briefly comment on the electric dipole moment of the
electron.
