Noncommutative geometry

Remarks

Example: the space $X$ is compact if and only if $C_0(X)$ has a unit. So a unital C*-algebra is read as a "compact noncommutative space". Another example: vector bundles over a compact space $X$ correspond to finitely generated projective $C(X)$-modules. So these modules play the role of vector bundles for any unital C*-algebra, and this leads to operator K-theory.

Non-example: the notion "a point of the space" does not carry over. A point $x$ of $X$ gives the character $f \mapsto f(x)$ of $C_0(X)$, that is, a nonzero multiplicative linear functional. The matrix algebra $M_2(\C)$ has no such functional. Indeed, 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 $\varphi(I) = \varphi(E_{11}) + \varphi(E_{22}) = 0$. Then $\varphi(a) = \varphi(I)\varphi(a) = 0$ for every $a$, a contradiction.

The approach of Alain Connes adds a spectral triple to $A$ to describe metric and differential structure.