In mathematics,continuous geometry is an analogue of complexprojective geometry introduced byvon Neumann (1936,1998), where instead of the dimension of a subspace being in a discrete set, it can be an element of the unit interval. Von Neumann was motivated by his discovery ofvon Neumann algebras with a dimension function taking a continuous range of dimensions, and the first example of a continuous geometry other than projective space was the projections of thehyperfinite type II factor.
Menger and Birkhoff gave axioms for projective geometry in terms of the lattice of linear subspaces of projective space. Von Neumann's axioms for continuous geometry are a weakened form of these axioms.
A continuous geometry is alatticeL with the following properties
The lattice operations ∧, ∨ satisfy a certain continuity property,
, whereA is adirected set and ifα <β thenaα <aβ, and the same condition with ∧ and ∨ reversed.
Every element inL has a complement (not necessarily unique). A complement of an elementa is an elementb witha ∧b = 0,a ∨b = 1, where 0 and 1 are the minimal and maximal elements ofL.
L is irreducible: this means that the only elements with unique complements are 0 and 1.
Finite-dimensional complex projective space, or rather its set of linear subspaces, is a continuous geometry, with dimensions taking values in the discrete set
The projections of a finite type II von Neumann algebra form a continuous geometry with dimensions taking values in the unit interval.
IfV is a vector space over afield (ordivision ring)F, then there is a natural map from the lattice PG(V) of subspaces ofV to the lattice of subspaces of that multiplies dimensions by 2. So we can take adirect limit of
This has a dimension function taking values alldyadic rationals between 0 and 1. Its completion is a continuous geometry containing elements of every dimension in. This geometry was constructed byvon Neumann (1936b), and is called the continuous geometry overF
This section summarizes some of the results ofvon Neumann (1998, Part I). These results are similar to, and were motivated by, von Neumann's work on projections in von Neumann algebras.
Two elementsa andb ofL are calledperspective, writtena ∼b, if they have a common complement. This is anequivalence relation onL; the proof that it is transitive is quite hard.
The equivalence classesA,B, ... ofL have a total order on them defined byA ≤B if there is somea inA andb inB witha ≤b. (This need not hold for alla inA andb inB.)
The dimension functionD fromL to the unit interval is defined as follows.
If equivalence classesA andB contain elementsa andb witha ∧b = 0 then their sumA +B is defined to be the equivalence class ofa ∨b. Otherwise the sumA +B is not defined. For a positive integern, the productnA is defined to be the sum ofn copies ofA, if this sum is defined.
For equivalence classesA andB withA not {0} the integer[B :A] is defined to be the unique integern ≥ 0 such thatB =nA +C withC <B.
For equivalence classesA andB withA not {0} the real number(B :A) is defined to be the limit of[B :C] / [A :C] asC runs through a minimal sequence: this means that eitherC contains a minimal nonzero element, or an infinite sequence of nonzero elements each of which is at most half the preceding one.
D(a) is defined to be({a} : {1}), where {a} and {1} are the equivalence classes containinga and 1.
The image ofD can be the whole unit interval, or the set of numbers for some positive integern. Two elements ofL have the same image underD if and only if they are perspective, so it gives an injection from the equivalence classes to a subset of the unit interval. The dimension functionD has the properties:
Ifa <b thenD(a) <D(b)
D(a ∨b) +D(a ∧b) =D(a) +D(b)
D(a) = 0 if and only ifa = 0, andD(a) = 1 if and only ifa = 1
In projective geometry, theVeblen–Young theorem states that a projective geometry of dimension at least 3 isisomorphic to the projective geometry of a vector space over a division ring. This can be restated as saying that the subspaces in the projective geometry correspond to theprincipal right ideals of a matrix algebra over a division ring.
Von Neumann generalized this to continuous geometries, and more generally to complemented modular lattices, as follows (von Neumann 1998, Part II). His theorem states that if a complemented modular latticeL has order[when defined as?] at least 4, then the elements ofL correspond to the principal right ideals of avon Neumann regular ring. More precisely if the lattice has ordern then the von Neumann regular ring can be taken to be ann byn matrix ringMn(R) over another von Neumann regular ringR. Here a complemented modular lattice has ordern if it has a homogeneous basis ofn elements, where a basis isn elementsa1, ...,an such thatai ∧aj = 0 ifi ≠j, anda1 ∨ ... ∨an = 1, and a basis is called homogeneous if any two elements are perspective. The order of a lattice need not be unique; for example, any lattice has order 1. The condition that the lattice has order at least 4 corresponds to the condition that the dimension is at least 3 in the Veblen–Young theorem, as a projective space has dimension at least 3 if and only if it has a set of at least 4 independent points.
Conversely, the principal right ideals of a von Neumann regular ring form a complemented modular lattice (von Neumann 1998, Part II theorem 2.4).
Suppose thatR is a von Neumann regular ring andL its lattice of principal right ideals, so thatL is a complemented modular lattice. Neumann showed thatL is a continuous geometry if and only ifR is an irreducible completerank ring.
