  We consider the Kolmogorov operator associated with a reaction-diffusion
equation having polynomially growing reaction coefficient and perturbed by a
noise of multiplicative type, in the Banach space $E$ of continuous functions.
By analyzing the smoothing properties of the associated transition semigroup,
we prove a modification of the classical identit\'e du carr\'e di champs that
applies to the present non-Hilbertian setting. As an application of this
identity, we construct the Sobolev space $W^{1,2}(E;\mu)$, where $\mu$ is an
invariant measure for the system, and we prove the validity of the Poincar\'e
inequality and of the spectral gap.
