Defining subsystems in Hilbert spaces with non-Euclidean metric

It is well established that pseudo-Hermitian quantum mechanics is a trivial extension of regular quantum mechanics. The modified inner-product space defined through the so-called metric operator, turns out to be the most natural way to represent certain phenomena such as those involving balanced gain and loss. However, for composite systems undergoing pseudo-Hermitian evolution, defining the subsystems is generally considered feasible only when the metric operator is chosen to have a tensor product. In this work, we use arguments from algebraic quantum mechanics to show that the subsystems can be well-defined in every metric space -- irrespective of whether or not the metric is of tensor product form. This is done by identifying subsystems with a decomposition of the underlying algebra into sub-algebras. In fact, we show that different decompositions of the underlying C^*-algebra correspond to choosing different equivalence classes of metric operators. We show how each of the subsystems, defined this way, can be tomographically constructed and that these subsystems satisfy the no-signaling principle. Therefore, we put all the choices of the metric on an equal footing.