Morse theory for C*-algebras: a geometric interpretation of some noncommutative manifolds

Vida Milani

Iran, Islamic Republic of

Shahid Beheshti University

Dept. of Math., Faculty of Math. Sci., Shahid Beheshti University, Tehran, Iran.

School of Mathematics, Georgia Institute of Technology, Atlanta GA, USA.

vmilani3@math.gatech.edu

Ali Asghar Rezaei

Iran, Islamic Republic of

Shahid Beheshti University

Dept. of Math., Faculty of Math. Sci.

Seyed M.H. Mansourbeigi

United States

Polytechnic University NY

Dept. of Electrical Engineering

Submitted: 2013-07-31

|

Accepted: 2013-07-31

|
DOI: https://doi.org/10.4995/agt.2011.1650
Funding Data

Downloads

Keywords:

C*-algebra, Critical points, CW complexes, Homotopy equivalence, Homotopy type, Morse function, Noncommutative CW complex, Poset, Pseudo-homotopy type, *-representation, Simplicial complex

Supporting agencies:

This research was not funded

Abstract:

The approach we present is a modification of the Morse theory for unital C*-algebras. We provide tools for the geometric interpretation of noncommutative CW complexes. Some examples are given to illustrate these geometric information. The main object of this work is a classification of unital C*-algebras by noncommutative CW complexes and the modified Morse functions on them.
Show more Show less

References:

A. Connes, Noncommutative Geometry, Academic Press, 1994.

J. Cuntz, Noncommutative Simplicial Complexes and the Baum-Connes Conjecture, arxiv: math/0103219.

J. Cuntz, Quantum spaces and their noncommutative topology, Notices Am. Math. Soc. 48, no. 8 (2001), 793-799.

D. N. Diep, The structure of C*-Algebras of type I, Vestn. Mosk. Univ., Ser. I, no. 2 (1978), 81-87.

J. Dixmier, Les C*-Algèbres et Leurs Représentations, Gauthier-Villars, (1969).

A. Duval, A combinatorial decomposition of simplicial complexes, Isr. J. Math. 87 (1994), 77-87. https://doi.org/10.1007/BF02772984

S. Eilers, T. A. Loring and G. K. Pedersen, Stability of anticommutation relations: an application to NCCW complexes, J. Reine Angew Math. 499 (1998), 101-143. https://doi.org/10.1515/crll.1998.055

J. M. G. Fell and R. S. Doran, Representations of *-Algebras, Locally Compact Groups and Banach *-Algebraic Bundles, Academic Press, 1988.

R. Forman, Morse theory for cell complexes, Adv. in math. 134 (1998). https://doi.org/10.1006/aima.1997.1650

R. Forman, Witten-Morse theory for cell complexes, Topology 37 (1998), 945-979. https://doi.org/10.1016/S0040-9383(97)00071-2

J. M. Gracia-Bondia, J. C. Varilly and H. Figueroa, Elements of Noncommutative Geometry, Birkhauser, 2001. https://doi.org/10.1007/978-1-4612-0005-5

A. Hatcher, Algebraic Topology, Cambridge Univ. Press, 2002.

G. Landi, An Introduction to Noncommutative Spaces and Their Geometry, arXiv:hepth/9701078.

J. Milnor, Morse Theory, Annals of Math. Studies, Princeton Univ. Press, (1963).

G. K. Pedersen, Pullback and pushout constructions in C*-algebras, J. Funct. Anal. 167 (1999), 243-344. https://doi.org/10.1006/jfan.1999.3456

Show more Show less