  We propose that for every event in de Sitter space, there is a CPT-conjugate
event at its antipode. Such an ``elliptic'' $Z_2$-identification of de Sitter
space provides a concrete realization of observer complementarity: every
observer has complete information. It is possible to define the analog of an
S-matrix for quantum gravity in elliptic de Sitter space that is measurable by
all observers. In a holographic description, S-matrix elements may be
represented by correlation functions of a dual (conformal field) theory that
lives on the single boundary sphere. S-matrix elements are de Sitter-invariant,
but have different interpretations for different observers. We argue that
Hilbert states do not necessarily form representations of the full de Sitter
group, but just of the subgroup of rotations. As a result, the Hilbert space
can be finite-dimensional and still have positive norm. We also discuss the
elliptic interpretation of de Sitter space in the context of type IIB* string
theory.
