continuous geometry

continuous geometry

Let

V

be a finite dimensional

vector space

(over some field) with

dimension

n

. Let

P

⁢

G

⁢

(

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

∈

P

⁢

G

⁢

(

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

P

⁢

G

⁢

(

V

)

to

ℤ

. One

property

of

dim

is that for every

i

between

0

and

n

, there is an

a

∈

P

⁢

G

⁢

(

V

)

such that

dim

⁡

(

a

)

=

i

. If we

normalize

dim

by dividing its values by

n

, then we get a function

d

:

P

⁢

G

⁢

(

V

)

→

[

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

∈

[

0

,

1

]

there is an

a

∈

L

such that

d

⁢

(

a

)

=

r

. Furthermore,

d

takes

infinite

independent

joins to infinite sums:

d

⁢

(

⋁

i

=

1

∞

a

i

)

=

∑

i

=

1

∞

d

⁢

(

a

i

)

⁢

whenever

⁢

a

j

+

1

∧

(

⋁

i

=

1

j

a

i

)

=

0

⁢

for

⁢

j

≥

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

∼

on elements of

L

is a

transitive relation

(Von Neumann). Since

∼

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

P

⁢

G

⁢

(

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

P

⁢

G

⁢

(

n

-

1

)

can be embedded into

P

⁢

G

⁢

(

2

⁢

n

-

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

ϕ

:

P

⁢

G

⁢

(

n

-

1

)

→

P

⁢

G

⁢

(

2

⁢

n

-

1

)

is the

embedding

, and

d

n

is the dimension function on

P

⁢

G

⁢

(

n

-

1

)

and

d

2

⁢

n

is the dimension function on

P

⁢

G

⁢

(

2

⁢

n

-

1

)

, then

d

n

⁢

(

a

)

=

d

2

⁢

n

⁢

(

ϕ

⁢

(

a

)

)

. As a result, we get a chain of embeddings

P

⁢

G

⁢

(

1

)

↪

P

⁢

G

⁢

(

3

)

↪

⋯

↪

P

⁢

G

⁢

(

2

n

-

1

)

↪

⋯

.

Taking the union of these lattices, we get a lattice

P

⁢

G

⁢

(

∞

)

, 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

P

⁢

G

⁢

(

∞

)

, turning it into a metric lattice, which in turn can be completed to a lattice

C

⁢

G

⁢

(

D

)

. This

C

⁢

G

⁢

(

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

C

⁢

G

⁢

(

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

Canonical

name

ContinuousGeometry

Date of creation

2013-03-22 16:42:21

Last modified on

2013-03-22 16:42:21

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 06C20

Classification

msc 51D30

Synonym

von Neumann lattice

Related topic

LatticeOfProjections

Defines

irreducible continuous geometry

Generated on Fri Feb 9 10:00:15 2018 by

LaTeXML