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MOTIVATION In mathematics, there is a close relationship between ''spaces'', which are geometric in nature, and the numerical Functions on them. In general, such functions will form a Commutative Ring . For instance, one may take the ring ''C''(''X'') of Continuous Complex -valued functions on a Topological Space ''X''. In many important cases (''e.g.'', if ''X'' is a Compact Hausdorff Space ), we can recover ''X'' from ''C''(''X''), and therefore it makes some sense to say that ''X'' has ''commutative geometry''. For other cases and applications, including mathematical Physics 1 and Functional Analysis , Non-commutative Ring s arise as the natural candidates for a ring of functions on some non-commutative "space". "Non-commutative spaces", however defined, cannot be too similar to ordinary Topological Space s, as these are known to correspond to commutative rings in many important cases. For this reason, the field is also called non-commutative topology — some of the motivating questions of the theory are concerned with extending known Topological Invariant s to these new spaces. That is, the "space" itself is used as some sort of Middle Term .
For the Duality between Locally Compact Measure Space s and commutative Von Neumann Algebra s, Noncommutative Von Neumann Algebra s are called ''non-commutative Measure Space s''. NON-COMMUTATIVE DIFFERENTIABLE MANIFOLDS Another area of study is that of Non-commutative Differentiable Manifold s. An ordinary Differentiable Manifold can be characterized by the commutative algebra of Smooth Function s defined on the manifold, and the space of smooth Section s of its Tangent Bundle , Cotangent Bundle and other Fiber Bundle s. All these spaces are Modules over the commutative algebra of smooth functions. The concepts of Exterior Derivative , Lie Derivative and Covariant Derivative are also important elements in understanding Derivation s over this algebra. In the non-commutative case, the algebras in question are non-commutative. To handle Differential Form s, one must work with the Graded Exterior Algebra bundle of all P-form s under the Wedge Product and look at its algebra of smooth sections. A "differential" is taken to be an Antiderivation (or something more general) on this algebra which increases the grading by 1 and is Quadratically Nilpotent . NON-COMMUTATIVE AFFINE SCHEMES In analogy to the Duality between Affine Scheme s and Commutative Ring s, we can also have noncommutative affine schemes. EXAMPLES OF NON-COMMUTATIVE SPACES
HISTORY Some of the theory developed by Alain Connes to handle noncommutative geometry at a technical level has roots in older attempts, in particular in Ergodic Theory . The proposal of George Mackey to create a ''virtual subgroup'' theory, with respect to which ergodic Group Action s would become Homogeneous Space s of an extended kind, has by now been subsumed. SEE ALSO {Link without Title} The applications in particle physics are described on the entry for Noncommutative Quantum Field Theory |
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