  On a doubling metric measure space $(M,d,\mu)$ endowed with a "carr\'e du
champ", let $\mathcal{L}$ be the associated Markov generator and $\dot
L^{p}_\alpha(M,\mathcal{L},\mu)$ the corresponding homogeneous Sobolev space of
order $0<\alpha<1$ in $L^p$, $1<p<+\infty$, with norm
$\left\|\mathcal{L}^{\alpha/2}f\right\|_p$. We give sufficient conditions on
the heat semigroup $(e^{-t\mathcal{L}})_{t>0}$ for the spaces $\dot
L^{p}_\alpha(M,\mathcal{L},\mu) \cap L^\infty(M,\mu)$ to be algebras for the
pointwise product. Two approaches are developed, one using paraproducts
(relying on extrapolation to prove their boundedness) and a second one through
geometrical square functionals (relying on sharp estimates involving
oscillations). A chain rule and a paralinearisation result are also given. In
comparison with previous results ([29,11]), the main improvements consist in
the fact that we neither require any Poincar\'e inequalities nor
$L^p$-boundedness of Riesz transforms, but only $L^p$-boundedness of the
gradient of the semigroup. As a consequence, in the range $p\in(1,2]$, the
Sobolev algebra property is shown under Gaussian upper estimates of the heat
kernel only.
