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Projective modules over higher-dimensional noncommutative tori. (English) Zbl 0663.46073

Following G. A. Elliott, by a non-commutative n-dimensional torus one means the universal \(C^*\)-algebra \(A_{\theta}\) generated by the products \(u_ x\), \(x\in {\mathbb{Z}}^ n\), of n unitary generators, satisfying the relation \[ u_ yu_ x=\exp (\pi i\theta (x,y))u_{x+y}, \] where \(\theta\) is a real skew bilinear form on \({\mathbb{Z}}^ n\). Any non-commutative torus \(A_{\theta}\) has a canonical torus \(\tau\).
The main results of the paper under review are as follows: If \(\theta\) is not rational (i.e., the values of \(\theta\) on the integral lattice \({\mathbb{Z}}^ n\subset {\mathbb{R}}^ n\) are not all rational), then:
- the positive cone of the algebraic \(K_ 0\)-group \(K_ 0(A_{\theta})\) consists exactly of the elements on which \(\tau\) is strictly positive together with zero,
- any two projective modules which represent the same element of \(K_ 0(A_{\theta})\) are isomorphic (cancellation),
- up to isomorphism every projective \(A_{\theta}\)-module is quite explicitly constructed,
- the projection in \(A_{\theta}\) itself generate all of \(K_ 0(A_{\theta}),\)
- the natural map from the quotient group \(UA_{\theta}/U^ 0A_{\theta}\) of the group of unitary elements factorized by its connected component of identity to the algebraic \(K_ 1\)-group \(K_ 1(A_{\theta})\) is an isomorphism,
- and finally, any two projections in \(A_{\theta}\) which represent the same element of \(K_ 0(A_{\theta})\) are in the same path component of the set of projections in \(A_{\theta}.\)
The paper provides a good presentation of the so-called non-commutative differential topology, see also [A. Connes, C. R. Acad. Sci. Paris 290, 599-604 (1980; Zbl 0433.46057); A. Connes and A. Rieffel, Proc. Conf. Oper. Algebr. Math. Phys., Univ. Iowa (1985), Contemp. Math. 62, 237-266 (1987; Zbl 0633.46069)].
Reviewer: Do Ngoc Diep

MSC:

46L80 \(K\)-theory and operator algebras (including cyclic theory)
46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
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