We study the relationship between metric and algebraic structures on section rings of polarized projective manifolds. More precisely, we prove that once the kernel is factored out, the multiplication operator of the section ring becomes an approximate isometry (up to normalization) with respect to the L^2-norm. We then show that this algebraic property characterizes L^2-norms and describe some applications of this characterisation. The semiclassical version of Ohsawa-Takegoshi theorem, describing holomorphic extensions from submanifolds to global manifolds of holomorphic sections of sufficiently large tensor powers, lies at the heart of our approach.