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

They form a category with usual structures.

Is this category equivalent to the categiry of $C^*$ algebras.

The equivalency means in terms of natural transformation between functores.

If the answer would be affirmative then the study of (category of) $C^*$ algebras would be equivalent to study of (category of) $Z^*$ algebras.

A $Z^*$ algebra is a $C^*$ algebra whose all positive elements are zero divisor.
They form a category with usual structures.

Question. Is this category equivalent to the category of $C^*$ algebras?

The equivalency means in terms of natural transformation between functors.
If the answer would be affirmative then the study of (category of) $C^*$ algebras would be equivalent to study of (category of) $Z^*$ algebras.