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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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