Continuous geometry

In mathematics, continuous geometry is an analogue of complex projective geometry introduced by , where instead of the dimension of a subspace being in a discrete set 0, 1, \dots, \textit{n} , it can be an element of the unit interval [0,1] . Von Neumann was motivated by his discovery of von 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 the hyperfinite type II factor.

In mathematics, continuous geometry is an analogue of complex projective geometry introduced by , where instead of the dimension of a subspace being in a discrete set 0, 1, \dots, \textit{n} , it can be an element of the unit interval [0,1] . Von Neumann was motivated by his discovery of von 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 the hyperfinite type II factor. ==Definition== 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 a lattice L with the following properties L is modular. L is complete. The lattice operations ∧, ∨ satisfy a certain continuity property, \Big(\bigwedge_{\alpha\in A} a_\alpha\Big)\lor b = \bigwedge_\alpha (a_\alpha\lor b) , where A is a directed set and if then , and the same condition with ∧ and ∨ reversed. Every element in L has a complement (not necessarily unique). A complement of an element a is an element b with , , where 0 and 1 are the minimal and maximal elements of L. L is irreducible: this means that the only elements with unique complements are 0 and 1. ==Examples== Finite-dimensional complex projective space, or rather its set of linear subspaces, is a continuous geometry, with dimensions taking values in the discrete set \{0, 1/\textit{n}\,, 2/\textit{n}\,, \dots, 1\} The projections of a finite type II von Neumann algebra form a continuous geometry with dimensions taking values in the unit interval [0,1] . showed that any orthocomplemented complete modular lattice is a continuous geometry. If V is a vector space over a field (or division ring) F, then there is a natural map from the lattice PG(V) of subspaces of V to the lattice of subspaces of V \otimes F^2 that multiplies dimensions by 2. So we can take a direct limit of PG(F)\subset PG(F^2)\subset PG(F^4)\subset PG(F^8)\cdots This has a dimension function taking values all dyadic rationals between 0 and 1. Its completion is a continuous geometry containing elements of every dimension in [0,1] . This geometry was constructed by , and is called the continuous geometry over F ==Dimension== This section summarizes some of the results of . These results are similar to, and were motivated by, von Neumann's work on projections in von Neumann algebras. Two elements a and b of L are called perspective, written , if they have a common complement. This is an equivalence relation on L; the proof that it is transitive is quite hard. The equivalence classes A, B, ... of L have a total order on them defined by if there is some a in A and b in B with . (This need not hold for all a in A and b in B.) The dimension function D from L to the unit interval is defined as follows. If equivalence classes A and B contain elements a and b with then their sum is defined to be the equivalence class of . Otherwise the sum is not defined. For a positive integer n, the product nA is defined to be the sum of n copies of A, if this sum is defined. For equivalence classes A and B with A not {0} the integer is defined to be the unique integer such that with . For equivalence classes A and B with A not {0} the real number is defined to be the limit of as C runs through a minimal sequence: this means that either C 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 , where {a} and {1} are the equivalence classes containing a and 1. The image of D can be the whole unit interval, or the set of numbers 0, 1/\textit{n}\,, 2/\textit{n}\,, \dots, 1 for some positive integer n. Two elements of L have the same image under D 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 function D has the properties: If then D(a ∨ b) + D(a ∧ b) = D(a) + D(b) if and only if , and if and only if ==Coordinatization theorem== In projective geometry, the Veblen–Young theorem states that a projective geometry of dimension at least 3 is isomorphic 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 the principal 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 . His theorem states that if a complemented modular lattice L has order at least 4, then the elements of L correspond to the principal right ideals of a von Neumann regular ring. More precisely if the lattice has order n then the von Neumann regular ring can be taken to be an n by n matrix ring M n (R) over another von Neumann regular ring R. Here a complemented modular lattice has order n if it has a homogeneous basis of n elements, where a basis is n elements a 1 , ..., a n such that if , and , 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 . Suppose that R is a von Neumann regular ring and L its lattice of principal right ideals, so that L is a complemented modular lattice. Neumann showed that L is a continuous geometry if and only if R is an irreducible complete rank ring. ==References==

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