Normed combinatorial homology and noncommutative tori

Marco Grandis

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

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

e-mail: grandis@dima.unige.it
http://www.dima.unige.it/~grandis/

Notes: Dip. Mat. Univ. Genova, Preprint 484 (2003), 14 p.

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.