  We study a quantum field in elliptic de Sitter space dS_4/Z_2 - the spacetime
obtained from identifying antipodal points in dS_4. We find that the operator
algebra and Hilbert space cannot be defined for the entire space, but only for
observable causal patches. This makes the system into an explicit realization
of the horizon complementarity principle. In the absence of a global quantum
theory, we propose a recipe for translating operators and states between
observers. This translation involves information loss, in accordance with the
fact that two observers see different patches of the spacetime. As a check, we
recover the thermal state at the de Sitter temperature as a state that appears
the same to all observers. This thermal state arises from the same functional
that, in ordinary dS_4, describes the Bunch-Davies vacuum.
