What is the relevance of preservation of trace in completely postive trace preserving (CPTP) maps?

Why is the trace preserving part necessary? Is it not enough if it can take all matrices to matrices of trace 1?

• I think the question was, why can't we consider something like $$\tilde{C}(\rho)=\frac{C(\rho)}{\mathrm{Tr}[C(\rho)]}$$ for a non trace preserving $C$ a quantum channel? (It said "[...] it can take all matrices to matrices of trace 1") – user2723984 May 8 at 13:54
• In this case $\tilde{C}$ is not linear. – Danylo Y May 8 at 17:31
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