  A conjecture in quantum mechanics states that any quantum canonical
transformation can decompose into a sequence of three basic canonical
transformations; gauge, point and interchange of coordinates and momenta. It is
shown that if one attempts to construct the three basic transformations in
star-product form, while gauge and point transformations are immediate in
star-exponential form, interchange has no correspondent, but it is possible in
an ordinary exponential form. As an alternative approach, it is shown that all
three basic transformations can be constructed in the ordinary exponential form
and that in some cases this approach provides more useful tools than the
star-exponential form in finding the generating function for given canonical
transformation or vice versa. It is also shown that transforms of c-number
phase space functions under linear-nonlinear canonical transformations and
intertwining method can be treated within this argument.
