  We consider a free quantized Klein-Gordon field in a spacetime (M, g)
containing a stationary black hole, more precisely a spacetime with a
stationary bifurcate Killing horizon in the sense of Kay and Wald. We prove the
existence of the Hartle-Hawking-Israel ground state, which is a pure state on
the whole spacetime whose restriction to the exterior of the black hole is a
thermal state at Hawking temperature.We show that the HHI state is a Hadamard
state and is the unique Hadamard extension of the above thermal state to the
whole spacetime. We construct the HHI state by Wick rotation in Killing time
coordinates, using the notion of the Calderon projector for elliptic boundary
value problems.
