  Two fundamental issues about the relation between the deformed
Heisenberg-Weyl algebra in noncommutative space and the undeformed one in
commutative space are elucidated. First the un-equivalency theorem between two
algebras is proved: the deformed algebra related to the undeformed one by a
non-orthogonal similarity transformation is explored; furthermore,
non-existence of a unitary similarity transformation which transforms the
deformed algebra to the undeformed one is demonstrated. Secondly the uniqueness
of realizing the deformed phase space variables via the undeformed ones is
elucidated: both the deformed Heisenberg-Weyl algebra and the deformed bosonic
algebra should be maintained under a linear transformation between two sets of
phase space variables which fixes that such a linear transformation is unique.
Elucidation of this un-equivalency theorem has basic meaning both in theory and
experiment.
