preprint: Normed combinatorial homology and noncommutative tori

preprint: Normed combinatorial homology and noncommutative tori

[Marco Grandis]

The following preprint is available in ps, at:

http://www.dima.unige.it/~grandis/Bsy2.ps
___________

Normed combinatorial homology and noncommutative tori
Marco Grandis

Abstract.

        Cubical sets have a directed homology, studied in a previous paper
and consisting of preordered abelian groups, with a positive cone generated
by the structural cubes. By this additional information, cubical sets can
provide a sort of "noncommutative topology", agreeing with some results of
noncommutative geometry but lacking the metric aspects of C*-algebras.

        Here, we make such similarity stricter by introducing *normed*
cubical sets and their *normed* directed homology, formed of normed
preordered abelian groups. The normed cubical sets associated with
"irrational rotations" have thus the same classification up to isomorphism
as the well-known irrational rotation C*-algebras.

MSC: 55U10, 81R60, 55Nxx.
Keywords: Cubical sets, noncommutative C*-algebras, combinatorial homology,
normed abelian groups.


Marco Grandis

Dipartimento di Matematica
Universita` di Genova
via Dodecaneso 35
16146 GENOVA, Italy

e-mail: grandis@dima.unige.it
tel: +39.010.353 6805   fax: +39.010.353 6752
http://www.dima.unige.it/~grandis/

preprint: Normed combinatorial homology and noncommutative tori

[Marco Grandis]

The following preprint is available in ps:

Normed combinatorial homology and noncommutative tori
Marco Grandis
Dip. Mat. Univ. Genova, Preprint 484 (Jul 2003), 14 p.

http://www.dima.unige.it/~grandis/Bsy2.ps

Abstract.
        Cubical sets have a directed homology, studied in a previous paper
and consisting of preordered abelian groups, with a positive cone generated
by the structural cubes. By this additional information, cubical sets can
provide a sort of "noncommutative topology", agreeing with some results of
noncommutative geometry but lacking the metric aspects of C*-algebras.
        Here, we make such similarity stricter by introducing *normed*
cubical sets and their *normed* directed homology, formed of normed
preordered abelian groups. The normed cubical sets associated with
"irrational rotations" have thus the same classification up to isomorphism
as the well-known irrational rotation C*-algebras.

MSC: 55U10, 81R60, 55Nxx.
Keywords: Cubical sets, noncommutative C*-algebras, combinatorial homology,
normed abelian groups.


Marco Grandis

Dipartimento di Matematica
Universita` di Genova
via Dodecaneso 35
16146 GENOVA, Italy

e-mail: grandis@dima.unige.it
tel: +39.010.353 6805   fax: +39.010.353 6752
http://www.dima.unige.it/~grandis/