An analogue of Kakutani’s representation theorem for Banach lattice algebras is provided. We characterize Banach lattice algebras that embed as a closed sublattice-algebra of C ( K ) C(K) precisely as those with a positive approximate identity ( e γ ) (e_\gamma ) such that x ∗ ( e γ ) → ‖ x ∗ ‖ x^{*}(e_\gamma )\to \|x^{*}\| for every positive functional x ∗ x^{*} . We also show that every Banach lattice algebra with identity other than C ( K ) C(K) admits different product operations which are compatible with the order and the algebraic identity. This complements the classical result, due to Martignon, that on C ( K ) C(K) spaces pointwise multiplication is the unique compatible product.