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Thermal Wightman functions of a massless scalar field are studied within theframework of a ``near horizon'' static background model of an extremal R-Nblack hole. This model is built up by using global Carter-like coordinates overan infinite set of Bertotti-Robinson submanifolds glued together. Theanalytical extendibility beyond the horizon is imposed as constraints on(thermal) Wightman's functions defined on a Bertotti-Robinson sub manifold. Itturns out that only the Bertotti-Robinson vacuum state, i.e. $T=0$, satisfiesthe above requirement. Furthermore the extension of this state onto the wholemanifold is proved to coincide exactly with the vacuum state in the globalCarter-like coordinates. Hence a theorem similar to Bisognano-Wichmann theoremfor the Minkowski space-time in terms of Wightman functions holds withvanishing ``Unruh-Rindler temperature''. Furtermore, the Carter-like vacuumrestricted to a Bertotti-Robinson region, resulting a pure state there, hasvanishing entropy despite of the presence of event horizons. Some comments onthe real extreme R-N black hole are given

A Bisognano - Wichmann-like theorem in a certain case of a non-bifurcate event horizon related to an extreme Reissner - Nordström black hole