I am interested in quantum channels which are completely positive trace preserving maps. For simplicity, let us assume $\Phi: \mathcal{B(H)} \rightarrow \mathcal{B(H)}$ be a quantum channel. Further we assume that the channel be unital.
My confusion is in the case when $\mathcal{H}$ is of infinite dimension, because identity operator is not a trace class operator in this case. So what would be the analogous problem for this case? A simple googling did not give me satisfactory results. Can we just write the similar definition in terms of Krauss operator in this case too? Advanced thanks for any answer/reference/suggestion.
The notion of "trace-preserving" still makes sense if you restrict it to positive operators: the trace of a positive operator is well-defined, if you allow infinity as a value (or the different cardinalities of infinity if $\mathcal H$ is not separable).
Usually, though, one considers trace-preserving maps on the trace-class operators, and their duals (ucp maps) on all of $\mathcal B(\mathcal H)$.