Positive maps and $*$-homomorphisms

Question

If $\varphi:A \to B$ is a linear map between $C^*$-algebras, it is said to be positive if it sends positive elements in $A$ to positive elements in $B$.

We know that every $*$-homomorphism is positive. But how about the converse? Does every positive map between $C^*$-algebras necessarily a $*$-homomorphism?

Thanks in advance!

Answer 1

No, and by a long shot.

Let $A=B=\mathbb C $, and $\varphi (z)=2z $.

Or, let $A=C [0,1] $, $B=\mathbb C $, and $\varphi (f)=\int_0^1 f $.

Or, $A=M_n (\mathbb C) $, $B=\mathbb C $, and $\varphi (x)=\text {Tr}\, (x) $.

Or, $A=M_n (\mathbb C) $, $B=M_n (\mathbb C )$, and $\varphi (x)=cxc^*$ with $c $ not a unitary.

There are countless examples. And a particular flavour of positive maps, the completely positive maps, play a very important role.