On a doubling metric measure space (M,d,μ) endowed with a “carré du champ”, let ℒ be the associated Markov generator and L ˙ α p (M,ℒ,μ) the corresponding homogeneous Sobolev space of order 0<α<1 in L p , 1<p<+∞, with norm ∥ℒ α/2 f∥ p . We give sufficient conditions on the heat semigroup (e -tℒ ) t>0 for the spaces L ˙ α p (M,ℒ,μ)∩L ∞ (M,μ) 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 Poincaré 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∈(1,2], the Sobolev algebra property is shown under Gaussian upper estimates of the heat kernel only.