  The quantum Stein's lemma is a fundamental result of quantum hypothesis
testing in the context of distinguishing two quantum states. A recent
conjecture, known as the ``generalized quantum Stein's lemma", asserts that
this result is true in a general framework where one of the states is replaced
by convex sets of quantum states. In this work, we show that the assertion of
the generalized Stein's lemma is true for the setting where the second
hypothesis is the state space of any subalgebra $\mathcal{N}$. This is obtained
through a strong asymptotic equipartition property for smooth subalgebra
entropies that applies for any fixed smoothing parameter $\epsilon\in (0,1)$.
As an application in resource theory, we show that the relative entropy of a
subalgebra is the asymptotic dilution cost under suitable operations. This
provides a scope to establish a connection between different quantum resources.
