$\ast$-preserving homomorphism between $C^{\ast}$-algebras

Question

An elementary question: A $\ast$-preserving homomorphism between $C^{\ast}$-algebras is positive. Is there any condition that makes a positive homomorphism, $\ast$-preserving?

Answer 1

A positive homomorphism is automatically $\ast$-preserving. First note that any self-adjoint element can be written as the difference of two positive elements, so a positive homomorphism preserves self-adjoint elements. Next note that a homomorphism of complex algebras preserves $i$. Finally, note that an arbitrary element $a$ can be written

$$a = \frac{a + a^{\ast}}{2} + i \frac{a - a^{\ast}}{2i}$$

and a positive homomorphism preserves this decomposition. The adjoint is

$$a^{\ast} = \frac{a + a^{\ast}}{2} - i \frac{a - a^{\ast}}{2i}$$

so positive homomorphisms preserve adjoints.