In Nielsen and Chuang's Quantum Computation and Quantum information there is an axiomatic definition of the quantum operation (as one of the 3 approaches to quantum operations).
A quantum operation is defined as a map between two sets of density operators satisfying 3 axioms. The axiom 1 states the trace of the output of the operation is less or equal to 1, i.e., $tr(\mathcal{E}(\rho))\leq 1$ which according to the book's explanation includes non-trace-preserving case when $tr(\mathcal{E}(\rho))<1$.
I understand in practical terms when it's useful to consider non-trace preserving maps (non-complete description of a process), but I don't understand how does it fit the definition of the quantum operation. It's defined as a map between spaces of density operators, so the output of the operation must be the density operator and as such it must have a trace equal to 1. Is Axiom 1 an attempt to introduce some more general notion of the quantum operation, that is not a map between spaces of density operators? If so, what is it?