  We investigate the topological nilpotence degree, in the sense of
Henn-Lannes-Schwartz, of a connected Noetherian unstable algebra $R$. When $R$
is the mod $p$ cohomology ring of a compact Lie group, Kuhn showed how this
invariant is controlled by centralizers of elementary abelian $p$-subgroups. By
replacing centralizers of elementary abelian $p$-subgroups with components of
Lannes' $T$-functor, and utilizing the techniques of unstable algebras over the
Steenrod algebra, we are able to generalize Kuhn's result to a large class of
connected Noetherian unstable algebras. We show how this generalizes Kuhn's
result to more general classes of groups, such as groups of finite virtual
cohomological dimension, profinite groups, and Kac-Moody groups. In fact, our
results apply much more generally, for example, we establish results for
$p$-local compact groups in the sense of Broto-Levi-Oliver, for connected
$H$-spaces with Noetherian mod $p$ cohomology, and for the Borel equivariant
cohomology of a compact Lie group acting on a manifold. Along the way we
establish several results of independent interest. For example, we formulate
and prove a version of Carlson's depth conjecture in the case of a Noetherian
unstable algebra of minimal depth.
