Continuous geometry
In mathematics, continuous geometry is an analogue of complex projective geometry introduced by von Neumann (1936, 1998), where instead of the dimension of a subspace being in a discrete set 0, 1, ..., 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,
- , where A is a directed set and if α < β then aα < aβ, 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 a ∧ b = 0, a ∨ b = 1, 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/n, 2/n, ..., 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].
- Kaplansky (1955) 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⊗F2 that multiplies dimensions by 2. So we can take a direct limit of
- 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 von Neumann (1936b) , and is called the continuous geometry over "F"
Dimension[]
This section summarizes some of the results of von Neumann (1998, Part I) . 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 a ∼ b, 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 A ≤ B if there is some a in A and b in B with a ≤ b. (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 a ∧ b = 0 then their sum A + B is defined to be the equivalence class of a ∨ b. Otherwise the sum A + B 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 [B : A] is defined to be the unique integer n ≥ 0 such that B = nA + C with C < B.
- For equivalence classes A and B with A not {0} the real number (B : A) is defined to be the limit of [B : C] / [A : C] 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 ({a} : {1}), 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/n, 2/n, ..., 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 a < b then D(a) < D(b)
- D(a ∨ b) + D(a ∧ b) = D(a) + D(b)
- D(a) = 0 if and only if a = 0, and D(a) = 1 if and only if a = 1
- 0 ≤ D(a) ≤ 1
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.
Neumann generalized this to continuous geometries, and more generally to complemented modular lattices, as follows (Neumann 1998, Part II). His theorem states that if a complemented modular lattice L has order[when defined as?] 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 Mn(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 a1, ..., an such that ai ∧ aj = 0 if i ≠ j, and a1 ∨ ... ∨ 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 (Neumann 1998, Part II theorem 2.4).
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[]
- Birkhoff, Garrett (1979) [1940], Lattice theory, American Mathematical Society Colloquium Publications, 25 (3rd ed.), Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1025-5, MR 0598630
- Fofanova, T.S. (2001) [1994], "Orthomodular lattice", Encyclopedia of Mathematics, EMS Press
- Halperin, Israel (1960), "Introduction to von Neumann algebras and continuous geometry", Canadian Mathematical Bulletin, 3 (3): 273–288, doi:10.4153/CMB-1960-034-5, ISSN 0008-4395, MR 0123923
- Halperin, Israel (1985), "Books in Review: A survey of John von Neumann's books on continuous geometry", Order, 1 (3): 301–305, doi:10.1007/BF00383607, ISSN 0167-8094, MR 1554221, S2CID 122594481
- Kaplansky, Irving (1955), "Any orthocomplemented complete modular lattice is a continuous geometry", Annals of Mathematics, Second Series, 61 (3): 524–541, doi:10.2307/1969811, ISSN 0003-486X, JSTOR 1969811, MR 0088476
- Neumann, John von (1936), "Continuous geometry", Proceedings of the National Academy of Sciences of the United States of America, 22 (2): 92–100, Bibcode:1936PNAS...22...92N, doi:10.1073/pnas.22.2.92, ISSN 0027-8424, JSTOR 86390, PMC 1076712, PMID 16588062, Zbl 0014.22307
- Neumann, John von (1936b), "Examples of continuous geometries", Proc. Natl. Acad. Sci. USA, 22 (2): 101–108, Bibcode:1936PNAS...22..101N, doi:10.1073/pnas.22.2.101, JFM 62.0648.03, JSTOR 86391, PMC 1076713, PMID 16588050
- Neumann, John von (1998) [1960], Continuous geometry, Princeton Landmarks in Mathematics, Princeton University Press, ISBN 978-0-691-05893-1, MR 0120174
- Neumann, John von (1962), Taub, A. H. (ed.), Collected works. Vol. IV: Continuous geometry and other topics, Oxford: Pergamon Press, MR 0157874
- Neumann, John von (1981) [1937], Halperin, Israel (ed.), "Continuous geometries with a transition probability", Memoirs of the American Mathematical Society, 34 (252), ISBN 978-0-8218-2252-4, ISSN 0065-9266, MR 0634656
- Skornyakov, L. A. (1964), Complemented modular lattices and regular rings, London: Oliver & Boyd, MR 0166126
- Projective geometry
- Von Neumann algebras
- Lattice theory