  In the present paper, we intent to enlarge the axiomatic framework of
non-commutative quantum field theories (QFT). We consider QFT on
non-commutative spacetimes in terms of the tempered ultrahyperfunctions of
Sebasti\~ao e Silva corresponding to a convex cone, within the framework
formulated by Wightman. Tempered ultrahyperfunctions are representable by means
of holomorphic functions. As is well known there are certain advantages to be
gained from the representation of distributions in terms of holomorphic
functions. In particular, for non-commutative theories the Wightman functions
involving the $\star$-product, ${\mathfrak W}^\star_m$, have the same form as
the standard form ${\mathfrak W}_m$. We conjecture that the functions
${\mathfrak W}^\star_m$ satisfy a set of properties which actually will
characterize a non-commutative QFT in terms of tempered ultrahyperfunctions. In
order to support this conjecture, we prove for this setting the validity of
some important theorems, of which the CPT theorem and the theorem on the
Spin-Statistics connection are the best known. We assume the validity of these
theorems for non-commutative QFT in the case of spatial non-commutativity only.
