xviii REPRESENTATION THEORY AND NUMERICAL AF-INVARIANTS Proposition 13.3. The same is true if the condition Prim (A) = Prim(m2/A) is replaced by Prim (A + ^- ) C Prim (A) by Proposition 13.4. In Chapter 16 we give a complete classification of the class A = 2, T V = 2, 3,4. This class contains 28 specimens, and it turns out that all of them are non- isomorphic except for a subset consisting of the three specimens in Figure 19. The most striking classification result for a restricted, but infinite, class of examples in this memoir is that if A = ra^r, then (TV, Prim A,/ (J)) is a complete invariant. This is proved in Theorem 17.18. In addition to these formal invariants there are very efficient methods to decide non-isomorphism when A is rational based on a quantity r (v) — (a\v) defined in (11.3)-(11.4) see Theorem 11.10, Remark 11.11, Corollaries 11.12-11.13, Scholium 11.24. In fact / (J) = A^ - 1 (a | v) = XN~X times the inner product of the right and left Perron-Frobenius eigenvectors a, v of J, normalized so that ci\ = v\ = 1. In two new papers [14, 15] written jointly with K.H. Kim and F. Roush, we give a completely general algorithm to decide C*-equivalence of two square matrices A and B. In its most general form, however, this general algorithm is not merely based on the computation of a complete set of invariants, and the methods of the present monograph are still most useful for the particular examples arising from the noncommutative dynamical systems we are considering.

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