continuous geometry

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:

 $d(\bigvee_{i=1}^{\infty}a_{i})=\sum_{i=1}^{\infty}d(a_{i})\mbox{ whenever }a_{% j+1}\wedge(\bigvee_{i=1}^{j}a_{i})=0\mbox{ for }j\geq 1.$

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 (http://planetmath.org/ComplementedLattice) 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

 $PG(1)\hookrightarrow PG(3)\hookrightarrow\cdots\hookrightarrow PG(2^{n}-1)% \hookrightarrow\cdots.$

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.

• Any continuous geometry is a complete lattice and a topological lattice if order convergence is used to define a topology on it.

• An irreducible continuous geometry is a continuous geometry whose center is trivial (consisting of just $0$ and $1$). It turns out that an irreducible continuous geometry is just $CG(D)$ for some division ring $D$.

• (Kaplansky) Any orthocomplemented complete modular lattice is a continuous geometry.

References

• 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 ContinuousGeometry 2013-03-22 16:42:21 2013-03-22 16:42:21 CWoo (3771) CWoo (3771) 8 CWoo (3771) Definition msc 06C20 msc 51D30 von Neumann lattice LatticeOfProjections irreducible continuous geometry