  We investigate the relations between the (completely bounded) local
Coulhon-Varopoulos dimension and the spectral dimension of spectral triples
associated to sub-Markovian semigroups (or Dirichlet forms) acting on classical
(or noncommutative) $\mathrm{L}^p$-spaces associated to finite measure spaces.
More precisely, we prove that the completely bounded local Coulhon-Varopoulos
dimension $d$ exceeds the spectral dimension and even implies that the
associated Hodge-Dirac operator is $d^+$-summable. We explore different
settings to compare these two values: compact Riemannian manifolds, compact Lie
groups, sublaplacians, metric measure spaces, noncommutative tori and quantum
groups. Specifically, we prove that, while very often equal in smooth compact
settings, these dimensions can diverge. Finally, we show that the existence of
a symmetric sub-Markovian semigroup on a von Neumann algebra with finite
completely bounded local Coulhon-Varopoulos dimension implies that the von
Neumann algebra is necessarily injective.
