and noncommutative tori

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
NC_\theta
associated with `irrational' rotations have thus the same
classification up to isomorphism as the well-known irrational
rotation C* -algebras A_\theta.

Keywords: Cubical sets, noncommutative C*-algebras, combinatorial homology, normed abelian groups

2000 MSC: 55U10, 81R60, 55Nxx

*Theory and Applications of Categories,*
Vol. 13, 2004,
No. 7, pp 114-128.

http://www.tac.mta.ca/tac/volumes/13/7/13-07.dvi

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http://www.tac.mta.ca/tac/volumes/13/7/13-07.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/13/7/13-07.dvi

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