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 - t ℒ ) 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.