An elementary question: A $\ast$-preserving homomorphism between $C^{\ast}$-algebras is positive. Is there any condition that makes a positive homomorphism, $\ast$-preserving?
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There is a long way to go from positivity-preserving to being a $$-homomorphism. In decreasing order of generality: positive maps, $n$-positive maps (positivity preserving on $n \times n$ matrices), completely-positive maps ($n$-positive for all $n$), order-zero maps(CP and orthogonality preserving), -homomorphisms. – Michael Jun 14 '13 at 06:59
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A positive homomorphism is automatically $\ast$-preserving. First note that any self-adjoint element can be written as the difference of two positive elements, so a positive homomorphism preserves self-adjoint elements. Next note that a homomorphism of complex algebras preserves $i$. Finally, note that an arbitrary element $a$ can be written
$$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.
Qiaochu Yuan
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