A slightly different approach to the geometry of function spaces developed at the same time as von Neumann and Murray's work on the classification of factors. This approach is the theory of Here, the motivating example is the formula_5, where "X" is a locally compact Hausdorff topological space. By definition, this is the algebra of continuous complex-valued functions on "X" that vanish at infinity (which loosely means that the farther you go from a chosen point, the closer the function gets to zero) with the operations of pointwise addition and multiplication. The Gelfand–Naimark theorem implied that there is a correspondence between commutative and geometric objects: Every commutative is of the form formula_5 for some locally compact Hausdorff space "X". Consequently it is possible to study locally compact Hausdorff spaces purely in terms of commutative Non-commutative geometry takes this as inspiration for the study of non-commutative If there were such a thing as a "non-commutative space "X"," then its formula_5 would be a non-commutative ; if in addition the Gelfand–Naimark theorem applied to these non-existent objects, then spaces (commutative or not) would be the same as so, for lack of a direct approach to the definition of a non-commutative space, a non-commutative space is "defined" to be a non-commutative Many standard geometric tools can be restated in terms of and this gives geometrically-inspired techniques for studying non-commutative .