We bring together ideas in analysis on Hopf *-algebra actions on II
1 subfactors of finite Jones index [9, 24] and algebraic characterizations of Frobenius, Galois and cleft Hopf extensions [3, 13, 14] to prove a non-commutative algebraic analogue of the classical theorem: a finite degree field extension is Galois iff it is separable and normal. Suppose
N
M is a separable Frobenius extension of
k-algebras with trivial centralizer
CM(
N) and split as
N-bimodules. Let
M1![](/images/glyphs/BQT.GIF)
End(
MN) and
M2![](/images/glyphs/BQT.GIF)
End(
M1)
M be the endomorphism algebras in the Jones tower
N
M
M1
M2. We place depth 2 conditions on its second centralizers
A
CM1(
N) and
B
CM2(
M). We prove that
A and
B are semisimple Hopf algebras dual to one another, that
M1 is a smash product of
M and
A, and that
M is a
B-Galois extension of
N.