Continuous geometry

In mathematics, continuous geometry is an analogue of complex projective geometry introduced by von Neumann (1936, 1998), where instead of the dimension of a subspace being in a discrete set 0, 1, ..., n, it can be an element of the unit interval [0,1]. Neumann was motivated by his discovery of von Neumann algebras with a dimension function taking a continuous range of dimensions, and the first example of a continuous geometry other than projective space was the projections of the hyperfinite type II factor.

Menger and Birkhoff gave axioms for projective geometry in terms of the lattice of linear subspaces of projective space. Von Neumann's axioms for continuous geometry are a weakened form of these axioms.

A continuous geometry is a lattice L with the following properties

This section summarizes some of the results of von Neumann (1998, Part I). These results are similar to, and were motivated by, von Neumann's work on projections in von Neumann algebras.

Two elements a and b of L are called perspective, written a ∼ b, if they have a common complement. This is an equivalence relation on L; the proof that it is transitive is quite hard.

The equivalence classes A, B, ... of L have a total order on them defined by A ≤ B if there is some a in A and b in B with a ≤ b. (This need not hold for all a in A and b in B.)

The dimension function D from L to the unit interval is defined as follows.

The image of D can be the whole unit interval, or the set of numbers 0, 1/n, 2/n, ..., 1 for some positive integer n. Two elements of L have the same image under D if and only if they are perspective, so it gives an injection from the equivalence classes to a subset of the unit interval. The dimension function D has the properties:

In projective geometry, the Veblen–Young theorem states that a projective geometry of dimension at least 3 is isomorphic to the projective geometry of a vector space over a division ring. This can be restated as saying that the subspaces in the projective geometry correspond to the principal right ideals of a matrix algebra over a division ring.

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