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  "body": "\n## Definition\n\nA *noncommutative space* is a C*-algebra $A$, possibly not commutative, often with extra structure such as a dense $*$-subalgebra or a spectral triple.\n*Noncommutative geometry* is the study of such algebras with the methods of topology and geometry, through the following dictionary.\n\n1. A locally compact Hausdorff space $X$ corresponds to the commutative C*-algebra $C_0(X)$ of continuous complex functions on $X$ that vanish at infinity.\n2. By the [[commutative Gelfand–Naimark theorem|NT564Y315M]], every commutative C*-algebra is isomorphic to $C_0(X)$ for a locally compact Hausdorff space $X$, unique up to homeomorphism.\n3. A notion of the geometry of $X$ that can be expressed through $C_0(X)$ alone is then defined for every C*-algebra by the same formula.\n\n## Remarks\n\nExample: the space $X$ is compact if and only if $C_0(X)$ has a unit.\nSo a unital C*-algebra is read as a \"compact noncommutative space\".\nAnother example: vector bundles over a compact space $X$ correspond to finitely generated projective $C(X)$-modules.\nSo these modules play the role of vector bundles for any unital C*-algebra, and this leads to operator K-theory.\n\nNon-example: the notion \"a point of the space\" does not carry over.\nA point $x$ of $X$ gives the character $f \\mapsto f(x)$ of $C_0(X)$, that is, a nonzero multiplicative linear functional.\nThe matrix algebra $M_2(\\C)$ has no such functional.\nIndeed, if $\\varphi$ is one and $E_{ij}$ are the matrix units, then $\\varphi(E_{11}) = \\varphi(E_{12})\\varphi(E_{21}) = \\varphi(E_{22})$ and $\\varphi(E_{11})\\varphi(E_{22}) = \\varphi(E_{11}E_{22}) = 0$, so $\\varphi(I) = \\varphi(E_{11}) + \\varphi(E_{22}) = 0$.\nThen $\\varphi(a) = \\varphi(I)\\varphi(a) = 0$ for every $a$, a contradiction.\n\nThe approach of Alain Connes adds a [[spectral triple|FTQMQR3YS7]] to $A$ to describe metric and differential structure.\n",
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