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A classification of some noncommutative tori. (English) Zbl 0727.46050

Summary: A detailed description of the isomorphism classes of rational noncommutative tori is given using a classification of rational antisymmetric bicharacters on \({\mathbb{Z}}^ d\). A canonical form for such a torus is presented. An illustration of the extent to which \(K_ 0\) can fail to distinguish the isomorphism classes of these tori is also given.
Isomorphism classes of the canonical smooth subalgebras of the \(C^*\)- algebras associated with an arbitrary antisymmetric bicharacter \(\rho\) on \({\mathbb{Z}}^ 3\) are in a one-to-one correspondence with the isomorphism classes of \(\rho\). The same is true for the \(C^*\)-algebras themselves except in some cases where the possibility exists that (at most) two different bicharacteris on \({\mathbb{Z}}^ 3\) may yield isomorphic \(C^*\)- algebras.

MSC:

46L87 Noncommutative differential geometry
46L55 Noncommutative dynamical systems
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[1] B. Brenken, Classification of skew symmetric matrices, Proc. Amer. Math. Soc., to appear. JSTOR: · Zbl 0726.46040 · doi:10.2307/2047709
[2] ——–, J. Cuntz, G.A. Elliott and R. Nest, On the classification of noncommutative tori, III, Operator algebras and mathematical physics, P.E.T. Jorgensen and P.S. Muhly ed., Comtemp. Math. Amer. Math. Soc. 60 (1986), 503-526. · Zbl 0631.46055
[3] J. Cuntz, G.A. Elliot, F.M. Goodman and P.E.T. Jorgensen, On the classification of noncommutative tori , II, C.R. Math. Rep. Acad. Sci. Canada 7 (1985), 189-194. · Zbl 0601.46059
[4] S. Disney, G.A. Elliott, A. Kumjian and I. Raeburn, On the classification of noncommutative tori, C.R. Math. Rep. Acad. Sci. Canada 7 (1985), 137-141. · Zbl 0573.46036
[5] G.A. Elliot, On the \(K\)-theory of the \(C^*\)-algebra generated by a projective representation of a torsion-free discrete abelian group, Operator algebras and group representations, Pitman, London, 1984, 157-184. · Zbl 0542.46030
[6] M. Newman, Integral Matrices, Academic Press, New York, 1972. · Zbl 0254.15009
[7] M. Pimsner and D. Voiculescu, Exact sequences for \(K\)-groups and Ext-groups of certain crossed product \(C^*\)-algebras, J. Operator Theory 4 (1980), 93-118. · Zbl 0474.46059
[8] M.A. Rieffel, \(C^*\)-algebras associated with irrational rotations, Pacific J. Math, 93 (1981), 415-29. · Zbl 0499.46039 · doi:10.2140/pjm.1981.93.415
[9] H. Yin, Classification of \(C^*\)-crossed products associated with characters on free groups, C.R. Math. Rep. Acad. Sci. Canada, Vol. IX, No 1. (1987). · Zbl 0619.46058
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