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Noncommutative manifolds, the instanton algebra and isospectral deformations. (English) Zbl 0997.81045
A noncommutative manifold is described by a spectral triple \(({\mathcal A},{\mathcal H},D)\), where \({\mathcal A}\) is a noncommutative algebra with involution *, acting on the Hilbert space \({\mathcal H}\), while \(D\) is a selfadjoint operator with compact resolvent and such that \([D,a]\) is bounded for any \(a\in{\mathcal A}\). In this paper, the authors give interesting new examples of noncommutative manifolds including instanton algebra and noncommutative 4-sphere \(S_\theta^4\). Some insteresting properties of this noncommutative 4-sphere have been proved. Also, the authors proved that any complete Riemannian spin manifold whose isometry group has rank \(\geq 2\) admits isospectral deformation to nonfommutative geometry. Certainly, these results have positive impact on physics and mathematics.

MSC:
81R60 Noncommutative geometry in quantum theory
58B34 Noncommutative geometry (à la Connes)
57R99 Differential topology
58J42 Noncommutative global analysis, noncommutative residues
81T75 Noncommutative geometry methods in quantum field theory
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