Is the category of Z^* algebras equivalent to the category of C^* algebras?

Question

A  Z*  algebra is a  C^* algebra whose all positive elements are zero divisor.

The family of all Z^* algebras with C^* morphisms forms a category.

Is the category of  Z^* algebras equivalent to the category of C^* algebras?

If the answer would be positive then the theory of C* algebras can be reduced to study of Z^* algebras

The equivalence of two categories is meant in the following sense:

https://en.wikipedia.org/wiki/Equivalence_of_categories#Definition

Comments

What is the relevance for physics?

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@ArnoldNeumaier  Thank you and my +1 for your comment.

May be a more general question would be "What is the relevance of  C^* algebras for physics?" so with the same motivations one can consider some possible physical interpretations for Z^* algebras. Any way there are a some pure math tags in Physicsoverflow but there are physical interpretations for those subject

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@ArnoldNeumaier  I think that if we have some topology-dynamical interpretation for this concept, then actualy we have a physical interpretation too.