The primary objective of this paper is to establish the property of zero product determinacy for the algebra Alg L , where L is either a completely distributive commutative subspace lattice or a subspace lattice with two atoms. This objective is achieved by employing a technical approach that involves demonstrating the isomorphism between the multiplier algebra of the algebra consisting of compact operators belonging to Alg L and Alg L . Furthermore, we investigate the properties of derivations and homomorphisms on these algebras as an application of our main result. Additionally, we prove that in the finite-dimensional case, a unital algebra generated by a single operator is zero product determined if and only if every local derivation from the algebra to any of its Banach bimodules is a derivation. [ABSTRACT FROM AUTHOR]