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continuous geometry (Definition)

Let $ V$ be a finite dimensional vector space (over some field) with dimension $ n$. Let $ PG(V)$ be its lattice of subspaces, also known as the projective geometry of $ V$. It is well-known that we can associate each element $ a\in PG(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 $ \mathbb{Z}$. One property of $ \dim$ is that for every $ i$ between 0 and $ n$, there is an $ a\in PG(V)$ such that $ \dim(a)=i$. If we normalize $ \dim$ by dividing its values by $ n$, then we get a function $ d: PG(V)\to [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 “geometry” 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 generalization of a projective geometry so that a “continuous” dimension function $ d$ can be defined on $ L$ such that for every real number $ r\in [0,1]$ there is an $ a\in L$ such that $ d(a)=r$. Furthermore, $ d$ takes infinite independent joins to infinite sums:

$\displaystyle d(\bigvee_{i=1}^{\infty} a_i)=\sum_{i=1}^{\infty} d(a_i)$ whenever $\displaystyle a_{j+1}\wedge (\bigvee_{i=1}^{j} a_i)=0$ for $\displaystyle j\ge 1.$

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 relation $ \thicksim$ on elements of $ L$ is a transitive relation (Von Neumann). Since $ \thicksim$ 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 $ 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 operations are preserved, but the values of the “normalized dimension function” $ d$ described above are also preserved. In other words, if $ \phi:PG(n-1)\to PG(2n-1)$ is the embedding, and $ d_{n}$ is the dimension function on $ PG(n-1)$ and $ d_{2n}$ is the dimension function on $ PG(2n-1)$, then $ d_{n}(a)=d_{2n}(\phi(a))$. As a result, we get a chain of embeddings

$\displaystyle PG(1)\hookrightarrow PG(3)\hookrightarrow \cdots \hookrightarrow PG(2^n-1) \hookrightarrow \cdots.$
Taking the union of these lattices, we get a lattice $ PG(\infty)$, which is complemented and modular, which has a “normalized dimension function” $ d$ into $ [0,1]$ whose values take the form $ p/2^m$ ($ p,m$ positive integers). This $ d$ is also a valuation on $ PG(\infty)$, 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.

Remarks.

Bibliography

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).



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See Also: lattice of projections

Other names:  von Neumann lattice
Also defines:  irreducible continuous geometry
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Cross-references: modular lattice, complete, orthocomplemented, division ring, center, topology, topological lattice, complete lattice, metric, valuation, positive, union, chain, embedding, operations, right, Hilbert spaces, algebras, operator, theory, equivalence relation, symmetric, Reflexive, transitive relation, relation, join continuous, meet continuous, modular, complemented, sums, joins, independent, infinite, real number, geometry, interval, unit, closed, onto, range, normalize, property, function, integer, associate, projective geometry, subspaces, lattice, dimension, field, vector space, finite dimensional

This is version 4 of continuous geometry, born on 2007-02-17, modified 2008-08-20.
Object id is 8921, canonical name is ContinuousGeometry.
Accessed 1764 times total.

Classification:
AMS MSC: 06C20 (Order, lattices, ordered algebraic structures :: Modular lattices, complemented lattices :: Complemented modular lattices, continuous geometries)
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