Elliptic theory and noncommutative geometry

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Noncommutative geometry aims to replace traditional concepts of classical geometry, such as manifolds and metrics, with abstract operator-algebraic analogs, studying these through operator algebra methods. This pursuit of generality risks overshadowing classical foundations, potentially rendering both questions and answers unrecognizable in traditional terms. However, this is not entirely the case; many problems remain rooted in classical statements but can only be addressed within the framework of noncommutative geometry. One such problem is explored in this work. The classical elliptic theory, particularly the index problem established by Atiyah and Singer, connects an analytic invariant of an elliptic pseudodifferential operator on a smooth compact manifold—its index—to the manifold's topological invariants. In contrast, the index problem for nonlocal (and thus nonpseudodifferential) elliptic operators is significantly more complex, necessitating the application of much more advanced methods than those employed in the classical context. This highlights the importance of noncommutative geometry in solving intricate problems that transcend traditional approaches.