continuous geometry

Let V be a finite dimensional vector spaceMathworldPlanetmath (over some field) with dimensionMathworldPlanetmathPlanetmathPlanetmath n. Let PG(V) be its latticeMathworldPlanetmath of subspacesMathworldPlanetmathPlanetmathPlanetmath, also known as the projective geometry of V. It is well-known that we can associate each element aPG(V) a unique integer dim(a), namely, the dimension of the a as a subspace of V. dim can be seen as a function from PG(V) to . One property of dim is that for every i between 0 and n, there is an aPG(V) such that dim(a)=i. If we normalize dim by dividing its values by n, then we get a function d:PG(V)[0,1]. As n (the dimension of V) increases, the range of d begins to “fill up” [0,1]. Of course, we know this is impossible as long as V is finite dimensional.

Question: is there a “geometryMathworldPlanetmathPlanetmath” on which a “dimension function” is defined so that it is onto the closed unit interval [0,1]?

The answer is yes, and the geometry is the so-called “continuous geometry”. However, like projective geometries, it is really just a lattice (with some special conditions). A continuous geometry L is a generalizationPlanetmathPlanetmath of a projective geometry so that a “continuousPlanetmathPlanetmath” dimension function d can be defined on L such that for every real number r[0,1] there is an aL such that d(a)=r. Furthermore, d takes infiniteMathworldPlanetmath independent joins to infinite sums:

d(i=1ai)=i=1d(ai) whenever aj+1(i=1jai)=0 for j1.

Definition. A continuous geometry is a lattice L that is complemented, modular, meet continuous, and join continuous.

From a continuous geometry L, it can be shown that the perspective ( relationMathworldPlanetmath on elements of L is a transitive relation (Von Neumann). Since is also reflexiveMathworldPlanetmathPlanetmathPlanetmathPlanetmath and symmetricMathworldPlanetmathPlanetmathPlanetmath, it is an equivalence relationMathworldPlanetmath. In a projective geometry, perspective elements are exactly subspaces having the same dimension. From this equivalence relation, one can proceed to define a “dimension” function from L into [0,1].

Continuous geometry was introduced by Von Neumann in the 1930’s when he was working on the theory of operator algebras in Hilbert spaces. Write PG(n-1) the projective geometry of dimension n-1 over D (lattice of left (right) subspaces of left (right) n-dimensional vector space over D). Von Neumann found that PG(n-1) can be embedded into PG(2n-1) in such a way that not only the lattice operationsMathworldPlanetmath are preserved, but the values of the “normalized dimension function” d described above are also preserved. In other words, if ϕ:PG(n-1)PG(2n-1) is the embeddingMathworldPlanetmathPlanetmath, and dn is the dimension function on PG(n-1) and d2n is the dimension function on PG(2n-1), then dn(a)=d2n(ϕ(a)). As a result, we get a chain of embeddings


Taking the union of these lattices, we get a lattice PG(), which is complemented and modular, which has a “normalized dimension function” d into [0,1] whose values take the form p/2m (p,m positive integers). This d is also a valuation on PG(), turning it into a metric lattice, which in turn can be completed to a lattice CG(D). This CG(D) is the first example of a continuous geometry having a “continuous” dimension function.



  • 1 J. von Neumann, Continuous Geometry, Princeton, (1960).
  • 2 G. Birkhoff Lattice Theory, 3rd Edition, AMS Volume XXV, (1967).
  • 3 G. Grätzer, General Lattice Theory, 2nd Edition, Birkhäuser (1998).
Title continuous geometry
Canonical name ContinuousGeometry
Date of creation 2013-03-22 16:42:21
Last modified on 2013-03-22 16:42:21
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 8
Author CWoo (3771)
Entry type Definition
Classification msc 06C20
Classification msc 51D30
Synonym von Neumann lattice
Related topic LatticeOfProjections
Defines irreducible continuous geometry