  We study real linear scalar field theory on two simple non-globally
hyperbolic spacetimes containing closed timelike curves within the framework
proposed by Kay for algebraic quantum field theory on non-globally hyperbolic
spacetimes. In this context, a spacetime (M,g) is said to be `F-quantum
compatible' with a field theory if it admits a *-algebra of local observables
for that theory which satisfies a locality condition known as `F-locality'.
Kay's proposal is that, in formulating algebraic quantum field theory on
$(M,g)$, F-locality should be imposed as a necessary condition on the *-algebra
of observables. The spacetimes studied are the 2- and 4-dimensional spacelike
cylinders (Minkowski space quotiented by a timelike translation). Kay has shown
that the 4-dimensional spacelike cylinder is F-quantum compatible with massless
fields. We prove that it is also F-quantum compatible with massive fields and
prove the F-quantum compatibility of the 2-dimensional spacelike cylinder with
both massive and massless fields. In each case, F-quantum compatibility is
proved by constructing a suitable F-local algebra.
