Zero product preserving linear maps of CCR -algebras with Hausdorff spectrum

In this paper, we try to attack a conjecture of Araujo and Jarosz that every bijective linear map θ between C ∗ -algebras, with both θ and its inverse θ −1 preserving zero products, arises from an algebra isomorphism followed by a central multiplier. We show it is true for CCR C ∗ -algebras with Hausdorff spectrum, and in general, some special C ∗ -algebras associated to continuous fields of C ∗ -algebras.