  We prove a number of results linking properties of actions by compact groups
(both quantum and classical) on Banach spaces, such as uniform continuity,
spectrum finiteness and extensibility of the actions across several
constructions. Examples include: (a) a unitary representation of a compact
quantum group induces a continuous action on the $C^*$-algebra of bounded
operators if and only if it has finitely many isotypic components, and hence is
uniformly continuous; (b) a compact quantum group is finite if and only if its
continuous actions on $C^*$-algebras lift to continuous actions on either the
multiplier algebras or von Neumann envelopes thereof; (c) a (classical) locally
compact group $\mathbb{G}$ is discrete if and only if the forgetful functor
from $\mathbb{G}$-acted-upon compact $T_2$ spaces back to compact $T_2$ spaces
creates coproducts; (d) a representation of a linearly reductive quantum group
has finitely many isotypic components if and only if its restrictions to two
topologically-generating quantum subgroups, one of which is normal, do; (e)
equivalent characterizations of uniform continuity for actions of compact
groups on Banach spaces, e.g. that such an action is uniformly continuous if
and only if its restrictions to a pro-torus and to pro-$p$ subgroups are.
