RGTs. Although in both cases a rigorous mathematical formalism is used, a physical approximation usually enters the problem by choosing an appropriate auxiliary space. Only for a certain class of models we will be able to find auxiliary spaces with a structure that allows for describing the non commutativity effects without any approximation.\
We stress again that in the exact as well as in the perfect RGT ${\cal O}_{\cal H}$ and ${\cal O}_{{\cal H}'}$ are known so that we can determine $G$ and $G^+$ in both cases according to the explanations in section \[constr\].\
If the auxiliary space is active it may happen that it vanishes by truncation during the RG procedure. In such a case no auxiliary space is available after the local transformation has been worked out and the previously provided information concerning the correlations between adjacent system blocks is lost. Therefore the RGT is at most perfect.\
In the case of an auxiliary space which (only) allows for an approximate description of the correlations between adjacent system blocks we would like to have some insight into the accuracy of the approximation. Here we remember the numerical DMRG procedure in which convergence of numerical values of ground state 