Let be a finite dimensional vector space (over some field) with dimension . Let be its lattice of subspaces, also known as the projective geometry of . It is well-known that we can associate each element a unique integer , namely, the dimension of the as a subspace of . can be seen as a function from to . One property of is that for every between and , there is an such that . If we normalize by dividing its values by , then we get a function . As (the dimension of ) increases, the range of begins to “fill up” . Of course, we know this is impossible as long as is finite dimensional.
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 is a generalization of a projective geometry so that a “continuous” dimension function can be defined on such that for every real number there is an such that . Furthermore, takes infinite independent joins to infinite sums:
From a continuous geometry , it can be shown that the perspective (http://planetmath.org/ComplementedLattice) relation on elements of is a transitive relation (Von Neumann). Since is also reflexive and symmetric, it is an equivalence relation. 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 into .
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 the projective geometry of dimension over (lattice of left (right) subspaces of left (right) -dimensional vector space over ). Von Neumann found that can be embedded into in such a way that not only the lattice operations are preserved, but the values of the “normalized dimension function” described above are also preserved. In other words, if is the embedding, and is the dimension function on and is the dimension function on , then . As a result, we get a chain of embeddings
Taking the union of these lattices, we get a lattice , which is complemented and modular, which has a “normalized dimension function” into whose values take the form ( positive integers). This is also a valuation on , turning it into a metric lattice, which in turn can be completed to a lattice . This is the first example of a continuous geometry having a “continuous” dimension function.
An irreducible continuous geometry is a continuous geometry whose center is trivial (consisting of just and ). It turns out that an irreducible continuous geometry is just for some division ring .
- 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).
|Date of creation||2013-03-22 16:42:21|
|Last modified on||2013-03-22 16:42:21|
|Last modified by||CWoo (3771)|
|Synonym||von Neumann lattice|
|Defines||irreducible continuous geometry|