Infinite index subalgebras of depth two

An algebra extension $A\, \vert\, B$ is right depth two in this paper if its tensor-square is $A$-$B$-isomorphic to a direct summand of any (not necessarily finite) direct sum of $A$ with itself. For example, normal subgroups of infinite groups, infinitely generated Hopf-Galois extensions and infinite-dimensional algebras are depth two in this extended sense. The added generality loses some duality results obtained in the finite theory (Kadison and Szlachányi, 2003) but extends the main theorem of depth two theory, as for example in (Kadison and Nikshych, 2001). That is, a right depth two extension has right bialgebroid $T = (A \otimes_B A)^B$ over its centralizer $R = C_A(B)$. The main theorem: An extension $A\, \vert\, B$ is right depth two and right balanced if and only if $A\, \vert\, B$ is $T$-Galois with respect to left projective, right $R$-bialgebroid $T$.