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The two-dimensional noncommutative tori (irrational rotation algebras) have been studied by many authors, and have motivated some important work in ^-theory such as the PimsnerVoiculescu exact sequence for crossed products. The FCQ was settled for irrational rotation algebras by Rieffel [Rf 2], who has also obtained the definitive results in the higher-dimensional case [Rf 3] following work of others such as Elliott. Definition 5 J . I . e. UjUk = XjkUkUj for some Xjk e C. The Xjk can be defined by a real bicharacter on Zrt, a homomorphism 0 : Z " A Z n - > R with Xjk = e2iuQ(eJAe*\ w here {ei, • • • ,en} is the standard basis of Z n .

Let A be a C*-algebra. Then A has cancellation if and only if, for every n, equivalent projections in Mn(A) are unitarily equivalent in Mn(A). A C*-algebra with strict cancellation must be stably finite. 5. Let Tn be the rt-torus. 4]. There are, however, some substantial classes of simple C*-algebras for which cancellation can be proved. We will discuss these examples later. 33. Perforation. 1. A C*-algebra A has n-power cancellation if whenever p, q are projections in a matrix algebra over A with n>p ~ n*q, then/?

3], and thus pH(K) = S(H) by the first part of the proof. So if/is any state on G, then there is a state faeK with/1 # =/# IH- As H runs over all finitely generated subgroups of G, fH -> /. 8. If A is JT-simple and stably finite, then K0(A) has the strict ordering from the states coming from quasitraces if and only if it is weakly unperforated and % is surjective. If x is surjective, we say that A has enough quasitraces. 9. Let A be a stably finite simple C*-algebra. e. is weakly unperforated and has strict cancellation) and has enough quasitraces.

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