Noncommutative geometry (NCG) is a branch of mathematics that studies geometric ideas through noncommutative algebras. In ordinary geometry, a space can often be studied by means of a commutative algebra of functions on it; noncommutative geometry extends this viewpoint to algebras in which the product of two elements need not commute. Such algebras are treated as analogues of algebras of functions on generalized, or "noncommutative", spaces. The subject is not a single formalism. It includes operator-algebraic methods based on C*-algebras, von Neumann algebras, and spectral triples; algebraic approaches to noncommutative rings and graded algebras; and constructions related to deformation quantization, groupoid C*-algebras, cyclic homology, and K-theory. A standard example is the noncommutative torus, whose algebra is generated by two unitary elements satisfying a twisted commutation relation and which has served as a test case for noncommutative versions of vector bundles, connections, curvature and index theory.
History and scope Ideas now grouped under noncommutative geometry developed from several areas, including operator algebra theory, index theory, algebraic geometry, quantum mechanics and ergodic theory. The term is particularly associated with work of Alain Connes, who introduced a framework in which operator algebras, cyclic cohomology and generalized differential forms could be used to study spaces that are poorly described by ordinary point-set methods. The scope of the field is broad. In the operator-algebraic tradition, noncommutative algebras are interpreted as algebras of functions on noncommutative topological, measure-theoretic, smooth or metric spaces. In noncommutative algebraic geometry, one studies associative rings and categories of modules as analogues of coordinate rings and categories of sheaves. Deformation quantization and quantum groups are related areas when their noncommutative algebras are interpreted geometrically, although they are also studied as independent subjects.
Motivation The main motivation is to extend the duality between spaces and algebras of functions. If X {\displaystyle X} is a compact Hausdorff space, the commutative C*-algebra C ( X ) {\displaystyle C(X)} of complex-valued continuous functions determines X {\displaystyle X} up to homeomorphism, by the commutative Gelfand representation. Similarly, the category of affine schemes in algebraic geometry is dual to the category of commutative rings, and many geometric properties of a scheme can be studied through categories of sheaves or modules. In these classical examples, geometry is encoded algebraically. Addition and multiplication of functions are defined pointwise, and the commutativity of multiplication reflects the fact that functions take scalar values on an ordinary set of points. Noncommutative geometry starts from the observation that many algebras arising naturally in analysis, geometry and physics are not commutative but still retain geometric features. Rather than first defining a space and then its functions, one studies a noncommutative algebra directly and interprets it as the algebra of functions on a generalized space. A noncommutative algebra generally has too few characters to reconstruct a point-set space in the usual way. Consequently, noncommutative geometry often replaces points by other structures, such as representations, modules, traces, states, K-theory classes, cyclic cocycles or categories. This shift is one reason that different versions of noncommutative geometry emphasize different invariants and notions of equivalence, such as Morita equivalence.
Motivation from ergodic theory Some of the operator-algebraic constructions used in noncommutative geometry have roots in ergodic theory. In particular, crossed-product algebras associated with group actions can be viewed as algebras of functions on quotient spaces that may be singular or may not have satisfactory ordinary quotient spaces. Earlier ideas such as George Mackey's virtual subgroup theory anticipated the use of operator algebras to treat ergodic actions as generalized homogeneous spaces.
Operator-algebraic noncommutative spaces
C*-algebras and von Neumann algebras The formal duals of noncommutative C*-algebras are often called noncommutative topological spaces. This terminology is motivated by the Gelfand–Naimark theorem, which identifies commutative C*-algebras with algebras of continuous functions on locally compact Hausdorff spaces. A noncommutative C*-algebra may therefore be studied as though it were the algebra of continuous functions on a space whose ordinary set of points has been replaced by operator-algebraic data. In a similar way, commutative von Neumann algebras correspond to measure-theoretic objects, while noncommutative von Neumann algebras are often regarded as noncommutative measure spaces. This viewpoint is useful in areas such as noncommutative integration, Tomita–Takesaki theory and the classification theory of operator algebras.
Groupoids and crossed products Groupoid C*-algebras and crossed product algebras provide many basic examples. If a group acts on a space, the crossed product combines functions on the space with the action of the group. When the quotient by the action is singular, non-Hausdorff or otherwise ill behaved, the crossed product may still retain useful geometric and dynamical information. This approach is important for foliations, tilings, dynamical systems and examples from mathematical physics.
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