  From a string theory point of view the most natural gauge action on the fuzzy
sphere {\bf S}^2_L is the Alekseev-Recknagel-Schomerus action which is a
particular combination of the Yang-Mills action and the Chern-Simons term .
Since the differential calculus on the fuzzy sphere is 3-dimensional the field
content of this model consists naturally of a 2-dimensional gauge field
together with a scalar fluctuation normal to the sphere . For U(1) gauge theory
we compute the quadratic effective action and shows explicitly that the tadpole
diagrams and the vacuum polarization tensor contain a gauge-invariant UV-IR
mixing in the continuum limit L{\longrightarrow}{\infty} where L is the matrix
size of the fuzzy sphere. In other words the quantum U(1) effective action does
not vanish in the commutative limit and a noncommutative anomaly survives . We
compute the scalar effective potential and prove the gauge-fixing-independence
of the limiting model L={\infty} and then show explicitly that the one-loop
result predicts a first order phase transition which was observed recently in
simulation . The one-loop result for the U(1) theory is exact in this limit .
It is also argued that if we add a large mass term for the scalar mode the
UV-IR mixing will be completely removed from the gauge sector . It is found in
this case to be confined to the scalar sector only. This is in accordance with
the large L analysis of the model . Finally we show that the phase transition
becomes harder to reach starting from small couplings when we increase M .
