Jordan and Local Multipliers on Certain Banach Algebras are Multipliers

We prove that every continuous Jordan multiplier $T$ from a $C^*$-algebra $A$ into a Banach $A$-bimodule $X$ is a multiplier. We also characterize continuous linear maps on $C^*$-algebras and standard operator algebras determined by preserving some identities involving zero products. Additionally, we show that each continuous local multiplier from a Banach algebra $A$ with property $(\mathbb{B})$ into a Banach $A$-bimodule $X$ is a multiplier.