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, say the ring C(X) of continuous functions on a topological space X, with complex values. In many important cases (X a compact Hausdorff space), we can recover X from C(X); it makes some sense to say that X has commutative geometry. Mathematics is the study of quantity, structure, space and change. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ... Several specialized usages of the terms compact and compactness exist. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ...

For other cases and applications, including in relation with mathematical physics¹, and in particular in functional analysis, non-commutative rings arise. In a general sense non-commutative geometry investigates the possible spatial interpretations of such rings; the point is to get round the lack of commutative multiplication, which is a requirement of any simple-minded spatial theory. Non-commutative spaces, when introduced, cannot be too similar to ordinary spaces. The field is also called non-commutative topology — some of the motivating examples are questions of extending known topological invariants to such. That is, the 'space' itself is used as some sort of middle term. The willingness to question previously held truths and search for new answers resulted in a period of major scientific advancements, now known as the Scientific Revolution. ... Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ... In the mathematical field of topology a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms. ... The middle term is the term that occurs in both premises (but not in the conclusion) of a categorical syllogism. ...

Non-commutative C*-algebras are often now called non-commutative spaces. This is by analogy with the Gelfand representation, which shows that commutative C*-algebras are dual to locally compact Hausdorff spaces. In general, one can associate to any C*-algebra A a topological space Â; see spectrum of a C*-algebra. In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... C*-algebras are an important area of research in functional analysis. ... In mathematics, the Gelfand representation in functional analysis allows a complete characterisation of commutative C*-algebras as algebras of continuous complex-valued functions. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... A dual is a pair – a grouping of two. ... In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... The spectrum of a C*-algebra or dual of a C*-algebra A, denoted Â, is the set of unitary equivalence classes of irreducible *-representations of A. We implicitly assume that irreducible representation means non-null irreducible representation, thus excluding trivial (i. ...

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 actions would become homogeneous spaces of an extended kind, has by now been subsumed. Alain Connes (born April 1, 1947) is a French mathematician, currently Professor at the College de France (Paris, France), IHES (Bures-sur-Yvette, France) and Vanderbilt University (Nashville, Tennessee). ... In mathematics, a measure-preserving transformation T on a probability space is said to be ergodic if the only measurable sets invariant under T have measure 0 or 1. ... George Mackey is an American mathematician, working mainly in the fields of representation theory and group actions, and related parts of functional analysis. ... In mathematics, groups are often used to describe symmetries of objects. ... In mathematics, in particular in the theory of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a manifold or topological space X on which G acts by symmetry in a transitive way; it is not assumed that the action of G is faithful. ...

How about noncommutative differentiable manifolds? Well, first, for an ordinary differentiable manifold, we can look at the commutative algebra of smooth functions over it and also the space of smooth sections of its tangent bundle, cotangent bundle and other fiber bundles. All these spaces are modules of the commutative algebra. We'd also need the concept of exterior derivative and/or Lie derivative and/or covariant derivative. We note that they act as derivations over the algebra in question. So, to go over to the noncommutative case, we simply make the algebras in question noncommutative. To handle differential forms, we work with the graded exterior algebra bundle of all p-forms under the wedge product and look at its algebra of smooth sections. A differential is an antiderivation (or something more general) on this algebra which increases the grading by 1 and is quadratically nilpotent. In mathematics, a manifold M is a type of space, characterised in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ... Section can be: A cross section (in the common sense or the physics sense) In mathematics: A conic section A section of a fiber bundle or sheaf A Caesarean section In UK law, Section 28 In the fictional Star Trek universe, Section 31 A military unit A section (land) is... In mathematics, the tangent bundle of a differentiable manifold is a vector bundle which as a set is the disjoint union of all the tangent spaces at every point in the manifold with natural topology and smooth structure. ... In differential geometry, the cotangent bundle of a manifold is the vector bundle of all the cotangent spaces at every point in the manifold. ... In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure. ... In abstract algebra, a module is a generalization of a vector space. ... In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ... In mathematics, a Lie derivative is a derivation on the algebra of smooth functions over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by The Lie derivatives are represented by vector fields, as... In mathematics, the covariant derivative is a way of specifying a derivative along vector fields on a manifold. ... There are several meanings of derivation: A derivation in abstract algebra is a linear map that satisfies Leibniz law. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a grading. ... In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ... There are several meanings of derivation: A derivation in abstract algebra is a linear map that satisfies Leibniz law. ... In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0. ...

For the duality between locally compact measure spaces and commutative von Neumann algebras, we could call noncommutative von Neumann algebras non-commutative measure spaces. In mathematics, duality has numerous meanings. ... In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ... In mathematics, a measure is a function that assigns a number, e. ... A von Neumann algebra is a *-algebra of bounded operators on a Hilbert space which is closed in the weak operator topology, or equivalently, in the strong operator topology (under pointwise convergence) and contains the identity operator. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... A von Neumann algebra is a *-algebra of bounded operators on a Hilbert space which is closed in the weak operator topology, or equivalently, in the strong operator topology (under pointwise convergence) and contains the identity operator. ... In mathematics, a measure is a function that assigns a number, e. ...

Examples

• The symplectic phase space of classical mechanics is deformed into a non-commutative phase space generated by the position and momentum operators.

Also, in analogy to the duality between affine schemes and polynomial algebras, we can also have noncommutative affine schemes. A symplectic space is either a symplectic manifold or a symplectic vector space. ... A phase diagram or phase space is a useful construct used in mathematics and physics to demonstrate and visualise the changes in a given system. ... Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ... A phase diagram or phase space is a useful construct used in mathematics and physics to demonstrate and visualise the changes in a given system. ... In mathematical formulations of quantum mechanics, an operator is a linear transformation from a Hilbert space to itself. ... The word duality has a variety of different meanings in different contexts: In mathematics, see duality (mathematics). ... In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is defined to be the set of all prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. ... In mathematics, the symmetric algebra S(V) on a vector space V over a field K is a certain commutative unital associative K-algebra containing V. In fact, it is the most general such algebra, which can be expressed by a universal property. ...

[1] The applications in particle physics are described on the entry for Noncommutative quantum field theory Noncommutative quantum field theory (or quantum field theory on noncommutative space-time) is a branch of quantum field theory in which the spatial coordinates1 do not commute. ...