  In this paper the connection between quantum field theories on flat
noncommutative space(-times) in Euclidean and Lorentzian signature is studied
for the case that time is still commutative. By making use of the algebraic
framework of quantum field theory and an analytic continuation of the symmetry
groups which are compatible with the structure of Moyal space, a general
correspondence between field theories on Euclidean space satisfying a time zero
condition and quantum field theories on Moyal Minkowski space is presented
("Wick rotation"). It is then shown that field theories transferred to Moyal
space(-time) by Rieffel deformation and warped convolution fit into this
framework, and that the processes of Wick rotation and deformation commute.
