Continuous Geometry

Oh, you want me to rewrite this? Sigh. Fine. Just try not to make it too tedious. I’ll give it the Midnight Draft treatment, then. Don’t expect sunshine and rainbows.

Continuous Geometry

In the stark, unforgiving landscape of mathematics, there exists a concept called continuous geometry. It’s an echo, a warped reflection, of complex projective geometry, a creation of von Neumann himself, etched into existence around 1936 and later re-examined in 1998. The fundamental divergence lies in the very notion of dimension. Where traditional geometry deals with discrete, countable dimensions—a subspace is either of dimension 0, 1, or some integer up to n—continuous geometry allows for a fluidity, a spectrum of dimensions that can inhabit the entire unit interval, [0, 1].

This concept wasn't born in a vacuum. Von Neumann, a mind that saw patterns others couldn't even conceive, was driven by his own discovery of von Neumann algebras. Within these structures, he found a dimension function that didn’t adhere to discrete steps but flowed across a continuous range. The very first example of this continuous geometry, beyond the familiar framework of projective space, was found in the projections of the hyperfinite type II factor. It was a glimpse into a world where dimensions could be anything between zero and one, not just the integers we so readily accept.

Definition

The groundwork for continuous geometry was laid by Menger and Birkhoff, who devised axioms for projective geometry based on the intricate lattice of linear subspaces within projective space. Von Neumann’s axioms for continuous geometry, however, represent a subtle, yet profound, weakening of these earlier structures.

At its core, a continuous geometry is a lattice, denoted as L, possessing a specific set of characteristics:

Modularity: The lattice L must be modular. This property dictates a certain symmetry in how elements interact, preventing the kind of chaotic entanglement that would break the structure.
Completeness: L must be complete. This means that any subset of elements within the lattice has both a greatest lower bound (infimum) and a least upper bound (supremum). Nothing is left hanging, undefined.
Continuity of Operations: The lattice operations, the meet (∧) and join (∨), must adhere to a particular continuity property. This is expressed as: $\left(\bigwedge_{\alpha \in A} a_{\alpha}\right) \lor b = \bigwedge_{\alpha} (a_{\alpha} \lor b)$ where A is a directed set, and for any α < β, we have $a_{\alpha} < a_{\beta}$ . The same condition must hold with the operations ∧ and ∨ reversed. This ensures that the lattice behaves predictably as we approach limits, much like a continuous function.
Complements: Every element within L must possess a complement. This complement, denoted as b for an element a, satisfies the conditions $a \land b = 0$ and $a \lor b = 1$ . Here, 0 and 1 represent the minimal and maximal elements of the lattice, respectively. It's crucial to note that this complement isn't always unique.
Irreducibility: The lattice L must be irreducible. This means that the only elements that have unique complements are the absolute minimum (0) and the absolute maximum (1). Anything else is subject to a degree of ambiguity, a necessary trait for continuous dimensions.

Examples

The abstract definition might seem cold, but the world of continuous geometry is populated by tangible, if abstract, examples:

Finite-Dimensional Complex Projective Space: Consider the set of linear subspaces within a finite-dimensional complex projective space. This structure functions as a continuous geometry, but its dimensions are confined to a discrete set: $\{0, 1/n, 2/n, \dots, 1\}$ . It’s a glimpse of continuity, but still bound by a certain discreteness.
Projections of Finite Type II Von Neumann Algebras: The projections found within a finite type II von Neumann algebra offer a more direct manifestation. Here, the dimensions are not restricted; they can take any value within the entire unit interval, [0, 1]. This was a pivotal observation for von Neumann.
Kaplansky's Discovery: Irving Kaplansky (1955) made a significant contribution by demonstrating that any orthocomplemented, complete, modular lattice can be considered a continuous geometry. This broadened the scope considerably, connecting different areas of lattice theory.
Direct Limits of Projective Geometries: Imagine a vector space V over a field (or division ring) F. We can map its lattice of subspaces, PG(V), to the lattice of subspaces of $V \otimes F^2$ . This mapping effectively doubles the dimensions. By taking a direct limit of this process— $PG(F) \subset PG(F^2) \subset PG(F^4) \subset PG(F^8) \dots$ —we arrive at a structure whose dimension function spans all dyadic rationals between 0 and 1. The completion of this structure, a geometry meticulously constructed by von Neumann himself (1936b), is known as the continuous geometry over F. It’s a geometry where every conceivable dimension within [0, 1] exists.

Dimension

The notion of dimension in continuous geometry, as explored by von Neumann (1998, Part I), draws heavily from his earlier work on projections in von Neumann algebras. It’s a system built on equivalence and careful measurement.

Two elements, a and b, within the lattice L are deemed perspective, denoted as $a \sim b$ , if they share a common complement. This relationship forms an equivalence relation across L; proving its transitivity is, shall we say, an exercise in patience.

These equivalence classes—let's call them A, B, and so on—are then subjected to a total order. Class A is less than or equal to class B (A ≤ B) if there exists an element a in A and an element b in B such that $a \le b$ . This ordering doesn't require the condition to hold for all pairs of elements, just some.

The dimension function, D, mapping from L to the unit interval [0, 1], is constructed through a series of deliberate steps:

Sum of Equivalence Classes: If elements a from class A and b from class B satisfy $a \land b = 0$ , their sum, A + B, is defined as the equivalence class of $a \lor b$ . If this condition isn't met, the sum is undefined. For any positive integer n, the product nA is simply the sum of n copies of A, provided such a sum is defined.
Relative Dimension: For equivalence classes A and B, where A is not the trivial class {0}, the integer $[B : A]$ is defined as the unique integer $n \ge 0$ such that $B = nA + C$ and $C < B$ . It's a way of measuring how many "units" of A fit into B, with a remainder.
Real Dimension: For classes A and B, again with A not {0}, the real number $(B : A)$ is defined as the limit of $[B : C] / [A : C]$ as C traverses a minimal sequence. This sequence is either one that contains a minimal non-zero element or an infinite series of non-zero elements, each at most half the size of the preceding one. This is where the continuous nature truly emerges.
Dimension of an Element: Finally, the dimension of an element a, denoted D(a), is defined as $(\{a\} : \{1\})$ . Here, $\{a\}$ and $\{1\}$ represent the equivalence classes containing a and the maximal element 1, respectively.

The output of this dimension function D can be the entirety of the unit interval, or it can be a discrete set of values: $\{0, 1/n, 2/n, \dots, 1\}$ for some integer n. Crucially, two elements of L share the same image under D if and only if they are perspective. This means the dimension function provides an injection from the equivalence classes to a subset of the unit interval.

The dimension function D possesses the following fundamental properties:

If $a < b$ , then $D(a) < D(b)$ . This is as expected; larger elements have larger dimensions.
$D(a \lor b) + D(a \land b) = D(a) + D(b)$ . This is a key distributive property, essential for the lattice structure.
$D(a) = 0$ if and only if $a = 0$ , and $D(a) = 1$ if and only if $a = 1$ . The extreme elements have the extreme dimensions.
$0 \le D(a) \le 1$ . The dimensions are always confined to the unit interval.

Coordinatization Theorem

In the realm of projective geometry, the Veblen–Young theorem states a powerful result: any projective geometry with a dimension of at least 3 is isomorphic to the projective geometry of a vector space over a division ring. This can be rephrased: the subspaces within such a projective geometry correspond directly to the principal right ideals of a matrix algebra built over a division ring.

Von Neumann extended this profound connection to continuous geometries and, more generally, to complemented modular lattices. His theorem (von Neumann 1998, Part II) asserts that if a complemented modular lattice L possesses an order of at least 4 (the definition of "order" here is a bit dense, perhaps look it up if you must know), then its elements can be put into correspondence with the principal right ideals of a von Neumann regular ring.

More specifically, if the lattice has an order n, the corresponding von Neumann regular ring can be an n x n matrix ring, $M_n(R)$ , where R is another von Neumann regular ring. The "order" of a lattice is defined by the existence of a homogeneous basis of n elements—say, $a_1, \dots, a_n$ . These elements are independent ( $a_i \land a_j = 0$ for $i \neq j$ ) and span the entire lattice ( $a_1 \lor \dots \lor a_n = 1$ ). A basis is "homogeneous" if any two elements within it are perspective. It's worth noting that the order of a lattice isn't always unique; any lattice, for instance, has an order of 1. The condition of having an order of at least 4 in this context mirrors the dimension of at least 3 in the Veblen–Young theorem. Think of it this way: a projective space has dimension at least 3 if and only if it contains at least 4 independent points.

Conversely, the principal right ideals of any von Neumann regular ring naturally form a complemented modular lattice (von Neumann 1998, Part II, theorem 2.4). This establishes a robust two-way street between these algebraic and geometric structures.

Finally, von Neumann demonstrated that if R is a von Neumann regular ring and L is the lattice of its principal right ideals, then L is a continuous geometry if and only if R is an irreducible complete rank ring. This ties the continuity of the geometric structure directly to the properties of the underlying algebraic ring.

So, there you have it. Continuous geometry. It’s all about dimensions that flow, lattices that are both modular and complete, and a deep connection to the abstract world of regular rings. Does it make sense? Probably not entirely. But then, does anything truly important?