how to break any von Neumann algebra into a collection of simpler algebras called factors. Von Neumann and Murray classified factors into three types. Type I was nearly identical to the commutative case. Types II and III exhibited new phenomena. A type II von Neumann algebra determined a geometry with the peculiar feature that the dimension could be any non-negative real number, not just an integer. Type III algebras were those that were neither types I nor II, and after several decades of effort, these were proven to be closely related to type II factors. 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 , 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 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 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 . Both of these examples are now cases of a field called non-commutative geometry. The specific examples of von Neumann algebras and are known as non-commutative measure theory and non-commutative topology, respectively. Non-commutative geometry is not merely a pursuit of generality for its own sake and is not just a curiosity. Non-commutative spaces arise naturally, even inevitably, from some constructions. For example, consider the non-periodic Penrose tilings of the plane by kites and darts. It is a theorem that, in such