ar X iv : 0 70 4 . 39 88 v 1 [ m at h . O A ] 3 0 A pr 2 00 7 THREE APPLICATIONS OF THE CUNTZ SEMIGROUP
نویسنده
چکیده
Building on work of Elliott and coworkers, we present three applications of the Cuntz semigroup: (i) for many simple C∗-algebras, the Thomsen semigroup is recovered functorially from the Elliott invariant, and this yields a new proof of Elliott’s classification theorem for simple, unital AI algebras; (ii) for the algebras in (i), classification of their Hilbert modules is similar to the von Neumann algebra context; (iii) for the algebras in (i), approximate unitary equivalence of self-adjoint operators is characterised in terms of the Elliott invariant.
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ar X iv : 0 70 4 . 39 02 v 1 [ m at h . PR ] 3 0 A pr 2 00 7 On the number of collisions in Λ - coalescents ∗
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The 3x + 1 semigroup is the multiplicative semigroup S of positive rational numbers generated by { 2k+1 3k+2 : k ≥ 0} together with {2}. This semigroup encodes backwards iteration under the 3x + 1 map, and the 3x + 1 conjecture implies that it contains every positive integer. This semigroup is proved to be the set of positive rationals a b in lowest terms with b 6≡ 0( mod 3), and so contains al...
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